portfolio selection with multiple criteria
TRANSCRIPT
Portfolio Selection with Multiple Criteria
Bernhard [email protected]
Invesco Asset Management GmbHFrankfurt am Main
R in Finance, Chicago IL, 20 and 21 May 2016
Pfaff (Invesco) MCO RFinance 2016 1 / 24
Contents
1 Overview
2 Multi Criteria Optimization
3 Example: Multi-asset class portfolio
4 Summary
5 Bibliography
Pfaff (Invesco) MCO RFinance 2016 2 / 24
Overview
Overview
Definition of multiple (conflicting) criteria objectives with respect toportfolio optimization problems.
Concept of non-dominated solutions (Pareto efficiency).
Solutions/points can determined by:1 GA/EMO: elitist non-dominated sorting algorithms (e.g. NSGA-II)2 MCDM: (classical) optimization (e.g. weighted-sum method)
Applied to multi-asset portfolio with objectives for:1 Mean return,2 Total Covariance Risk,3 Diversification with respect to assets’ risk contributions.
Pfaff (Invesco) MCO RFinance 2016 3 / 24
Multi Criteria Optimization
Multi Criteria OptimizationProblem formulation
minimize fm(ω), m = 1, 2, . . . ,M;
subject to gj(ω) ≥ 0, j = 1, 2, . . . , J;
hk(ω) = 0, k = 1, 2, . . . ,K ;
ω(L)i ≤ ωi ≤ ω
(U)i , i = 1, 2, . . . , n.
Problem: M (conflicting) objective functions and n (constrained)variables.
A solution ω ∈ Ω is efficient (Pareto optimal or non-dominated) ifthere is no ω ∈ Ω such that fk(ω) ≤ fk(ω) for k = 1, . . . , p andfi (ω) < fi (ω) for some i ∈ 1, . . . , k.
Pfaff (Invesco) MCO RFinance 2016 4 / 24
Multi Criteria Optimization
Multi Criteria OptimizationGA/EMO: Pareto efficient solutions
Goal: find solutions which lie on Pareto-efficient front and encompassits entire range.
Solutions can be determined by genetic NSGA-II algorithm.
However, no guarantee in finding optimial points on (close to)frontier.
NSGA-II consists of the following steps (a) create random population,(b) selection (non-dominant/constraint-dominant), (c) applyingvariation (crossover, mutation), (d) elitism (crowding distances).
An implementation is provided in the R package mco (see Mersmann,2014).
Pfaff (Invesco) MCO RFinance 2016 5 / 24
Multi Criteria Optimization
Multi Criteria OptimizationMCDM: Weighted-sum method
minimizeM∑
m=1
λmfm(ω), with λm ≥ 0;
subject to gj(ω) ≥ 0, j = 1, 2, . . . , J;
hk(ω) = 0, k = 1, 2, . . . ,K ;
ω(L)i ≤ ωi ≤ ω
(U)i , i = 1, 2, . . . , n.
Scaling/normalization of objective functions is important.
Goals and/or satisfycing levels can be included.
Possibility of a hybrid approach, where some objectives are specifiedas ε-constraints.
Pfaff (Invesco) MCO RFinance 2016 6 / 24
Multi Criteria Optimization
Multi Criteria OptimizationSynopsis of portfolio applications
Computation of non-dominated surface in tri-criterion problems:Hirschberger et al. (2013).
Bi-criterion EMO problem formulation with discontinuities in searchspace (risk & return with (i) either zero or within bounds allocationsand (ii) cardinality constraint on count of included assets): Deb et al.(2011).
Tri-criterion problems (quad-lin-lin):
risk, return and& transaction costs: Steuer et al. (2005), Steuer et al.(2013).risk, return and& ESG index: Utz et al. (2015).
Pfaff (Invesco) MCO RFinance 2016 7 / 24
Example: Multi-asset class portfolio
Example: Multi-asset class portfolioSpecification
Data set MultiAsset contained in FRAPO: month-end data from11/2004 until 11/2011.
MCO conisting of three objectives:1 mean return,2 volatility, and3 dispersion of risk contributions.
Targeted return of 6% p.a.
Targeted volatility of 4% p.a.
Pfaff (Invesco) MCO RFinance 2016 8 / 24
Example: Multi-asset class portfolio
Example: Multi-asset class portfolioR code: Initializing
> library(FRAPO)
> library(mco)
> ## Loading of data
> data(MultiAsset)
> Prices <- timeSeries(MultiAsset,
+ charvec = rownames(MultiAsset))
> NAssets <- ncol(Prices)
> R <- returns(Prices, method = "discrete", percentage = TRUE)
> ## Defining parameters
> TargetRpa <- 6 ## percentage p.a.
> TargetR <- 100 * ((1 + TargetRpa / 100)^(1 / 12) - 1)
> TargetVol <- 4 ## percentage p.a.
> l <- rep(1, 3) ## goal weighting
> WeightedSum <- FALSE
> mu <- colMeans(R)
> S <- cov(R)
Pfaff (Invesco) MCO RFinance 2016 9 / 24
Example: Multi-asset class portfolio
Example: Multi-asset class portfolioR code: Multiple criteria objective and budget constraint
> f <- function(x)
+ y <- numeric(3)
+ y[1] <- -1.0 * l[1] * drop(crossprod(x, mu)) / TargetR
+ y[2] <- l[2] * drop(sqrt(t(x) %*% S %*% x)) *
+ sqrt(12) / TargetVol
+ y[3] <- l[3] * sum((mrc(x, S) / 100)^2)
+ if(WeightedSum)
+ return(sum(y))
+ else
+ return(y)
+
+
> g <- function(x)
+ c(1.01 - sum(x), sum(x) - 0.99)
+
Pfaff (Invesco) MCO RFinance 2016 10 / 24
Example: Multi-asset class portfolio
Example: Multi-asset class portfolioR code: Determining Pareto efficient solutions
> ans <- nsga2(f, NAssets, 3,
+ lower.bounds = rep(0, NAssets),
+ upper.bounds = rep(1, NAssets),
+ constraints = g, cdim = 2, popsize = 500)
> names(ans)
[1] "par" "value" "pareto.optimal"
> ## Preparing objective values for graphics
> mco <- data.frame(ans$value[ans$pareto.optimal, ])
> mco[, 1] <- ((1 + (-1.0 * mco[, 1] * TargetR) / 100)^12
+ - 1.0) * 100
> mco[, 2] <- mco[, 2] * TargetVol
> colnames(mco) <- c("Return", "Risk", "Diversification")
Pfaff (Invesco) MCO RFinance 2016 11 / 24
Example: Multi-asset class portfolio
Example: Multi-asset class portfolioR code: 3D scatterplot
> library(scatterplot3d)
> scatterplot3d(mco,
+ main = "Pareto Efficient Solutions",
+ sub = "Pareto Frontier (Surface)",
+ xlab = "Return Objective",
+ ylab = "Risk Objective",
+ zlab = "Dispersion of MRC",
+ angle = 15,
+ highlight.3d = FALSE,
+ box = TRUE,
+ color = "steelblue",
+ pch = 19, type = "p",
+ cex.symbols = 0.6)
Pareto Efficient Solutions
Pareto Frontier (Surface)
4 6 8 10 12 14 16 18 20
0.0
0.2
0.4
0.6
0.8
1.0
2 4
6 8
1012
1416
Return Objective
Ris
k O
bjec
tive
Dis
pers
ion
of M
RC
Pfaff (Invesco) MCO RFinance 2016 12 / 24
Example: Multi-asset class portfolio
Example: Multi-asset class portfolioR code: image plot
> library(akima)
> library(fields)
> s <- interp(mco[, 2], mco[, 1], mco[, 3],
+ xo = seq(min(mco[, 2]), max(mco[, 2]), length = 100),
+ yo = seq(min(mco[, 1]), max(mco[, 1]), length = 100),
+ duplicate = "mean"
+ )
> par(mar = c(5, 6, 5, 6))
> image.plot(s, nlevel = 50,
+ main = "Image plot of efficient set",
+ legend.lab = "Dispersion of MRC",
+ xlab = "Risk Objective",
+ ylab = "Return Objective",
+ legend.mar = 4,
+ horizontal = TRUE,
+ legend.shrink = 0.7,
+ col = topo.colors(50))
> contour(s, add = TRUE, nlevels = 20, labcex = 0.8)
> points(mco[, 2], mco[, 1], pch = 18, cex = 0.4, col = "orange")
4 6 8 10 12 14
68
1012
1416
18
Image plot of efficient set
Risk ObjectiveR
etur
n O
bjec
tive
0.2 0.4 0.6 0.8Dispersion of MRC
0.15
0.2 0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.8
Pfaff (Invesco) MCO RFinance 2016 13 / 24
Example: Multi-asset class portfolio
Example: Multi-asset class portfolioR code: weighting of objectives
grid <- expand.grid(x = seq(0.05, 0.95, by = 0.05),
y = seq(0.05, 0.95, by = 0.05))
grid <- grid[which(rowSums(grid) <= 1.0), ]
wobj <- as.matrix(cbind(grid, 1 - rowSums(grid)),
nrow = nrow(grid), ncol = 3)
W <- matrix(NA, nrow = nrow(wobj), ncol = NAssets)
WeightedSum <- TRUE
IneqA <- matrix(1, nrow = 1, ncol = NAssets)
ew <- rep(1 / NAssets, NAssets)
library(fPortfolio) ## for donlp2NLP
for(i in 1:nrow(wobj))
l <- c(wobj[i, ])
W[i, ] <- donlp2NLP(start = ew, objective = f,
par.lower = rep(0, NAssets),
ineqA = IneqA, ineqA.lower = 1.0,
ineqA.upper = 1.0)$solution
Pfaff (Invesco) MCO RFinance 2016 14 / 24
Example: Multi-asset class portfolio
Example: Multi-asset class portfolioR code: weighting of objectives & ternary plot
> library(PerformanceAnalytics)
> library(ggtern) ## Wahrschau! version < 2.0.1
> Es95Mod <- apply(W, 1, function(x)
+ r <- timeSeries(R %*% x / 100, time(R))
+ -100 * ES(r)
+ )
> terndat <- data.frame(cbind(wobj, Es95Mod))
> colnames(terndat) <- c("x", "y", "z", "value")
> ## Theme for ternary plot
> terntheme <- function()
+ list(theme_rgbg(),
+ theme(legend.position = c(0, 1),
+ legend.justification = c(0, 1),
+ plot.margin=unit(c(0, 2,0, 2), "cm"))
+ )
+
Pfaff (Invesco) MCO RFinance 2016 15 / 24
Example: Multi-asset class portfolio
Example: Multi-asset class portfolioR code: ternary plot, cont’d
> ## ternary plot
> ggtern(terndat, aes(x = x, y = y, z = z, value = value)) +
+ geom_interpolate_tern(aes(value = value, color = ..level..),
+ binwidth = 1.0) +
+ terntheme() +
+ theme_hidegrid_minor() +
+ theme_showgrid_major() +
+ Lline(0.2, color = "blue", linetype = 2) + ## x
+ Tline(0.3, colour = "red2", linetype = 2) + ## y
+ Rline(0.5, color = "brown", linetype = 2) + ## z
+ scale_color_gradient(low = "green", high = "red") +
+ labs(x = "Return", y = "Risk", z = "MRC",
+ title = "Ternary Plot with ES Contour Lines",
+ color = "Level")
100
80
60
40
20
100
80 60 40 20
100
80
60
40
20
Risk
Return MRC
Ris
k
Return
MR
C
0
4
8
12
16Level
Ternary Plot with ES Contour Lines
Pfaff (Invesco) MCO RFinance 2016 16 / 24
Example: Multi-asset class portfolio
Example: Multi-asset class portfolioR code: backtest, part I
> library(cccp) ## for ERC portolio
> ## backtest, extending window
> ep <- time(R)[-c(1:59)]
> bs <- length(ep)
> sp <- rep(start(R), bs)
> ## initialising object
> Wmco <- matrix(NA, nrow = bs, ncol = NAssets)
> Wmsr <- Wmdp <- Wgmv <- Werc <- Wmco
> l <- c(0.2, 0.1, 0.7) ## goal weighting
Pfaff (Invesco) MCO RFinance 2016 17 / 24
Example: Multi-asset class portfolio
Example: Multi-asset class portfolioR code: backtest, part II
> for(i in 1:bs)
+ rdat <- window(R, start = sp[i], end = ep[i])
+ mu <- colMeans(rdat)
+ S <- cov(rdat)
+ Wmco[i, ] <- donlp2NLP(start = ew, objective = f,
+ par.lower = rep(0, NAssets),
+ ineqA = IneqA, ineqA.lower = 1.0,
+ ineqA.upper = 1.0)$solution
+ ans <- tangencyPortfolio(rdat)
+ Wmsr[i, ] <- getWeights(ans)
+ ans <- PMD(rdat)
+ Wmdp[i, ] <- FRAPO::Weights(ans) / 100
+ ans <- PGMV(rdat)
+ Wgmv[i, ] <- FRAPO::Weights(ans) / 100
+ ans <- rp(ew, S, ew, optctrl = ctrl(trace = FALSE))
+ Werc[i, ] <- c(getx(ans))
+
Iteration: 0
pobj: 0.126801
dobj: -0.96263
pinf: 2.22045e-016
dinf: 3.83253
dgap: 1.08943
Iteration: 1
pobj: 0.124852
dobj: 0.00264633
pinf: 2.22045e-016
dinf: 0.427881
dgap: 0.122206
Iteration: 2
pobj: 0.118802
dobj: 0.0835599
pinf: 4.44089e-016
dinf: 0.111199
dgap: 0.0352419
Iteration: 3
pobj: 0.113548
dobj: 0.0993458
pinf: 7.77156e-016
dinf: 0.0372481
dgap: 0.014202
Iteration: 4
pobj: 0.107482
dobj: 0.104089
pinf: 1.55431e-015
dinf: 1.66533e-016
dgap: 0.00339286
Iteration: 5
pobj: 0.105591
dobj: 0.104892
pinf: 2.9976e-015
dinf: 2.23773e-016
dgap: 0.000698946
Iteration: 6
pobj: 0.105115
dobj: 0.105007
pinf: 5.77316e-015
dinf: 1.8619e-016
dgap: 0.000108626
Iteration: 7
pobj: 0.105047
dobj: 0.105027
pinf: 1.13243e-014
dinf: 1.00074e-016
dgap: 1.98598e-005
Iteration: 8
pobj: 0.105036
dobj: 0.105032
pinf: 2.23155e-014
dinf: 1.90283e-016
dgap: 4.3372e-006
Iteration: 9
pobj: 0.105034
dobj: 0.105033
pinf: 4.44089e-014
dinf: 1.24127e-016
dgap: 8.77542e-007
Iteration: 10
pobj: 0.105033
dobj: 0.105033
pinf: 8.82627e-014
dinf: 1.64204e-016
dgap: 1.12839e-007
Iteration: 11
pobj: 0.105033
dobj: 0.105033
pinf: 1.72196e-013
dinf: 1.94289e-016
dgap: 8.26087e-009
Optimal solution found.
Iteration: 0
pobj: 0.362971
dobj: -0.826572
pinf: 3.36575
dinf: 5.07688
dgap: 17.2127
Iteration: 1
pobj: 0.419846
dobj: -0.610122
pinf: 0.352087
dinf: 0.531087
dgap: 2.61477
Iteration: 2
pobj: 0.422811
dobj: -0.146063
pinf: 0.0836643
dinf: 0.126199
dgap: 0.888119
Iteration: 3
pobj: 0.381137
dobj: 0.149456
pinf: 0.0151466
dinf: 0.0228471
dgap: 0.271697
Iteration: 4
pobj: 0.332313
dobj: 0.250453
pinf: 0.00208538
dinf: 0.00314558
dgap: 0.0846849
Iteration: 5
pobj: 0.313591
dobj: 0.288708
pinf: 0.000394894
dinf: 0.000595657
dgap: 0.0251527
Iteration: 6
pobj: 0.303817
dobj: 0.2957
pinf: 6.39925e-005
dinf: 9.65262e-005
dgap: 0.00815032
Iteration: 7
pobj: 0.299024
dobj: 0.297668
pinf: 3.49767e-006
dinf: 5.27588e-006
dgap: 0.00135742
Iteration: 8
pobj: 0.297976
dobj: 0.297794
pinf: 3.25552e-007
dinf: 4.91062e-007
dgap: 0.000182128
Iteration: 9
pobj: 0.297814
dobj: 0.297802
pinf: 1.91646e-008
dinf: 2.89079e-008
dgap: 1.23223e-005
Iteration: 10
pobj: 0.297803
dobj: 0.297802
pinf: 9.70921e-010
dinf: 1.46454e-009
dgap: 6.32737e-007
Iteration: 11
pobj: 0.297802
dobj: 0.297802
pinf: 4.85793e-011
dinf: 7.32765e-011
dgap: 3.16812e-008
Optimal solution found.
Iteration: 0
pobj: 0.123721
dobj: -0.963402
pinf: 1.11022e-016
dinf: 3.82468
dgap: 1.08712
Iteration: 1
pobj: 0.121862
dobj: 0.00123752
pinf: 2.22045e-016
dinf: 0.422508
dgap: 0.120624
Iteration: 2
pobj: 0.115814
dobj: 0.0810899
pinf: 2.22045e-016
dinf: 0.109938
dgap: 0.0347246
Iteration: 3
pobj: 0.11028
dobj: 0.0972651
pinf: 4.44089e-016
dinf: 0.0338733
dgap: 0.0130149
Iteration: 4
pobj: 0.104565
dobj: 0.101339
pinf: 8.88178e-016
dinf: 0.000902096
dgap: 0.00322598
Iteration: 5
pobj: 0.102671
dobj: 0.102068
pinf: 1.77636e-015
dinf: 5.70661e-005
dgap: 0.000603327
Iteration: 6
pobj: 0.102259
dobj: 0.102169
pinf: 3.44169e-015
dinf: 2.69901e-006
dgap: 9.01822e-005
Iteration: 7
pobj: 0.102204
dobj: 0.102187
pinf: 6.77236e-015
dinf: 1.28266e-007
dgap: 1.73152e-005
Iteration: 8
pobj: 0.102194
dobj: 0.10219
pinf: 1.33227e-014
dinf: 2.63174e-009
dgap: 3.7038e-006
Iteration: 9
pobj: 0.102191
dobj: 0.102191
pinf: 2.65343e-014
dinf: 4.96533e-011
dgap: 6.94795e-007
Iteration: 10
pobj: 0.102191
dobj: 0.102191
pinf: 5.19584e-014
dinf: 1.92505e-012
dgap: 7.89682e-008
Optimal solution found.
Iteration: 0
pobj: 0.35774
dobj: -0.832046
pinf: 3.3757
dinf: 5.0986
dgap: 17.3336
Iteration: 1
pobj: 0.415478
dobj: -0.61496
pinf: 0.362862
dinf: 0.548061
dgap: 2.67242
Iteration: 2
pobj: 0.420214
dobj: -0.158925
pinf: 0.0885245
dinf: 0.133706
dgap: 0.921424
Iteration: 3
pobj: 0.379027
dobj: 0.142668
pinf: 0.0154915
dinf: 0.0233981
dgap: 0.278041
Iteration: 4
pobj: 0.328734
dobj: 0.245164
pinf: 0.00211642
dinf: 0.00319662
dgap: 0.0865185
Iteration: 5
pobj: 0.308997
dobj: 0.284874
pinf: 0.000357781
dinf: 0.000540387
dgap: 0.0243708
Iteration: 6
pobj: 0.299609
dobj: 0.291457
pinf: 6.58067e-005
dinf: 9.93934e-005
dgap: 0.00818694
Iteration: 7
pobj: 0.294797
dobj: 0.2934
pinf: 4.03528e-006
dinf: 6.09483e-006
dgap: 0.00139899
Iteration: 8
pobj: 0.293728
dobj: 0.29354
pinf: 3.8613e-007
dinf: 5.83205e-007
dgap: 0.00018854
Iteration: 9
pobj: 0.293562
dobj: 0.293549
pinf: 2.28877e-008
dinf: 3.45692e-008
dgap: 1.27549e-005
Iteration: 10
pobj: 0.293551
dobj: 0.29355
pinf: 1.16015e-009
dinf: 1.75227e-009
dgap: 6.54844e-007
Iteration: 11
pobj: 0.29355
dobj: 0.29355
pinf: 5.80487e-011
dinf: 8.76747e-011
dgap: 3.27877e-008
Optimal solution found.
Iteration: 0
pobj: 0.123384
dobj: -0.962439
pinf: 1.11022e-016
dinf: 3.81849
dgap: 1.08582
Iteration: 1
pobj: 0.121607
dobj: 0.00201636
pinf: 1.11022e-016
dinf: 0.418756
dgap: 0.11959
Iteration: 2
pobj: 0.115834
dobj: 0.081482
pinf: 2.22045e-016
dinf: 0.108843
dgap: 0.0343518
Iteration: 3
pobj: 0.110506
dobj: 0.0978079
pinf: 3.33067e-016
dinf: 0.0329326
dgap: 0.0126977
Iteration: 4
pobj: 0.10513
dobj: 0.10191
pinf: 6.66134e-016
dinf: 0.00104159
dgap: 0.0032198
Iteration: 5
pobj: 0.103326
dobj: 0.102697
pinf: 1.33227e-015
dinf: 6.32514e-005
dgap: 0.000629017
Iteration: 6
pobj: 0.10292
dobj: 0.102822
pinf: 2.66454e-015
dinf: 2.75908e-006
dgap: 9.83016e-005
Iteration: 7
pobj: 0.102866
dobj: 0.102848
pinf: 5.21805e-015
dinf: 1.30988e-007
dgap: 1.79689e-005
Iteration: 8
pobj: 0.102859
dobj: 0.102856
pinf: 1.03251e-014
dinf: 2.35916e-009
dgap: 3.40098e-006
Iteration: 9
pobj: 0.102859
dobj: 0.102858
pinf: 2.02061e-014
dinf: 7.24439e-011
dgap: 5.93075e-007
Iteration: 10
pobj: 0.102859
dobj: 0.102859
pinf: 3.9746e-014
dinf: 3.26919e-012
dgap: 5.96908e-008
Optimal solution found.
Iteration: 0
pobj: 0.354538
dobj: -0.835719
pinf: 3.37699
dinf: 5.08478
dgap: 17.2936
Iteration: 1
pobj: 0.410595
dobj: -0.619505
pinf: 0.363421
dinf: 0.547207
dgap: 2.66966
Iteration: 2
pobj: 0.415523
dobj: -0.163452
pinf: 0.0886577
dinf: 0.133493
dgap: 0.920655
Iteration: 3
pobj: 0.37489
dobj: 0.138624
pinf: 0.0155736
dinf: 0.0234493
dgap: 0.278065
Iteration: 4
pobj: 0.324781
dobj: 0.240768
pinf: 0.00213858
dinf: 0.00322009
dgap: 0.0869943
Iteration: 5
pobj: 0.304854
dobj: 0.281612
pinf: 0.000325256
dinf: 0.000489741
dgap: 0.0234594
Iteration: 6
pobj: 0.295994
dobj: 0.28797
pinf: 6.22552e-005
dinf: 9.37384e-005
dgap: 0.00805627
Iteration: 7
pobj: 0.291318
dobj: 0.289908
pinf: 3.5414e-006
dinf: 5.33232e-006
dgap: 0.00141155
Iteration: 8
pobj: 0.290252
dobj: 0.29005
pinf: 3.48668e-007
dinf: 5.24993e-007
dgap: 0.000202496
Iteration: 9
pobj: 0.290074
dobj: 0.29006
pinf: 2.12652e-008
dinf: 3.20192e-008
dgap: 1.45576e-005
Iteration: 10
pobj: 0.290061
dobj: 0.29006
pinf: 1.08233e-009
dinf: 1.62968e-009
dgap: 7.5469e-007
Iteration: 11
pobj: 0.290061
dobj: 0.29006
pinf: 5.41673e-011
dinf: 8.15601e-011
dgap: 3.78077e-008
Optimal solution found.
Iteration: 0
pobj: 0.123232
dobj: -0.961476
pinf: 2.22045e-016
dinf: 3.81415
dgap: 1.08471
Iteration: 1
pobj: 0.121498
dobj: 0.00278445
pinf: 2.22045e-016
dinf: 0.415683
dgap: 0.118714
Iteration: 2
pobj: 0.11584
dobj: 0.0819563
pinf: 6.66134e-016
dinf: 0.107489
dgap: 0.0338839
Iteration: 3
pobj: 0.110571
dobj: 0.0978265
pinf: 9.99201e-016
dinf: 0.0331768
dgap: 0.0127444
Iteration: 4
pobj: 0.105146
dobj: 0.102043
pinf: 1.9984e-015
dinf: 0.000770381
dgap: 0.00310324
Iteration: 5
pobj: 0.103416
dobj: 0.102824
pinf: 3.88578e-015
dinf: 4.96563e-005
dgap: 0.000591971
Iteration: 6
pobj: 0.103052
dobj: 0.102966
pinf: 7.54952e-015
dinf: 2.50231e-006
dgap: 8.59119e-005
Iteration: 7
pobj: 0.103009
dobj: 0.102999
pinf: 1.46549e-014
dinf: 1.30899e-007
dgap: 1.08648e-005
Iteration: 8
pobj: 0.103006
dobj: 0.103005
pinf: 2.85327e-014
dinf: 6.69656e-009
dgap: 8.78254e-007
Iteration: 9
pobj: 0.103006
dobj: 0.103006
pinf: 5.56222e-014
dinf: 3.35782e-010
dgap: 4.72658e-008
Optimal solution found.
Iteration: 0
pobj: 0.351675
dobj: -0.837846
pinf: 3.37288
dinf: 5.07138
dgap: 17.2281
Iteration: 1
pobj: 0.407214
dobj: -0.622043
pinf: 0.360178
dinf: 0.541556
dgap: 2.64917
Iteration: 2
pobj: 0.412212
dobj: -0.163113
pinf: 0.0866316
dinf: 0.130257
dgap: 0.906931
Iteration: 3
pobj: 0.371608
dobj: 0.137345
pinf: 0.015385
dinf: 0.0231325
dgap: 0.275101
Iteration: 4
pobj: 0.322178
dobj: 0.238962
pinf: 0.00212085
dinf: 0.00318886
dgap: 0.0861418
Iteration: 5
pobj: 0.302614
dobj: 0.278608
pinf: 0.000351124
dinf: 0.000527943
dgap: 0.0242462
Iteration: 6
pobj: 0.293291
dobj: 0.285236
pinf: 6.33175e-005
dinf: 9.52028e-005
dgap: 0.00808745
Iteration: 7
pobj: 0.288546
dobj: 0.287156
pinf: 3.73612e-006
dinf: 5.61755e-006
dgap: 0.00139091
Iteration: 8
pobj: 0.287471
dobj: 0.287289
pinf: 3.34871e-007
dinf: 5.03505e-007
dgap: 0.000182441
Iteration: 9
pobj: 0.287309
dobj: 0.287297
pinf: 1.91704e-008
dinf: 2.88242e-008
dgap: 1.19711e-005
Iteration: 10
pobj: 0.287298
dobj: 0.287297
pinf: 9.67931e-010
dinf: 1.45536e-009
dgap: 6.11765e-007
Iteration: 11
pobj: 0.287297
dobj: 0.287297
pinf: 4.84209e-011
dinf: 7.28043e-011
dgap: 3.06229e-008
Optimal solution found.
Iteration: 0
pobj: 0.123672
dobj: -0.958803
pinf: 0
dinf: 3.80571
dgap: 1.08247
Iteration: 1
pobj: 0.122002
dobj: 0.00508167
pinf: 7.69561e-017
dinf: 0.409394
dgap: 0.11692
Iteration: 2
pobj: 0.116563
dobj: 0.0834193
pinf: 1.11022e-016
dinf: 0.105204
dgap: 0.0331438
Iteration: 3
pobj: 0.111506
dobj: 0.0990009
pinf: 2.22045e-016
dinf: 0.0324776
dgap: 0.0125053
Iteration: 4
pobj: 0.106235
dobj: 0.103226
pinf: 3.33067e-016
dinf: 0.00030351
dgap: 0.00300823
Iteration: 5
pobj: 0.104614
dobj: 0.104014
pinf: 6.66134e-016
dinf: 1.97027e-005
dgap: 0.000599909
Iteration: 6
pobj: 0.104264
dobj: 0.104166
pinf: 1.33227e-015
dinf: 9.01238e-007
dgap: 9.73663e-005
Iteration: 7
pobj: 0.104215
dobj: 0.104201
pinf: 2.66454e-015
dinf: 4.55005e-008
dgap: 1.42392e-005
Iteration: 8
pobj: 0.104208
dobj: 0.104207
pinf: 5.10703e-015
dinf: 2.3187e-009
dgap: 1.21067e-006
Iteration: 9
pobj: 0.104208
dobj: 0.104208
pinf: 9.99201e-015
dinf: 1.1635e-010
dgap: 6.57644e-008
Optimal solution found.
Iteration: 0
pobj: 0.356938
dobj: -0.834683
pinf: 3.36383
dinf: 5.0603
dgap: 17.1499
Iteration: 1
pobj: 0.410054
dobj: -0.619
pinf: 0.353629
dinf: 0.531974
dgap: 2.61503
Iteration: 2
pobj: 0.413344
dobj: -0.154362
pinf: 0.0829905
dinf: 0.124845
dgap: 0.882619
Iteration: 3
pobj: 0.371293
dobj: 0.140296
pinf: 0.0151086
dinf: 0.0227283
dgap: 0.270588
Iteration: 4
pobj: 0.322159
dobj: 0.240021
pinf: 0.00207286
dinf: 0.00311827
dgap: 0.0849452
Iteration: 5
pobj: 0.302798
dobj: 0.279666
pinf: 0.000323197
dinf: 0.000486195
dgap: 0.0233401
Iteration: 6
pobj: 0.293872
dobj: 0.286032
pinf: 5.80907e-005
dinf: 8.73876e-005
dgap: 0.00786931
Iteration: 7
pobj: 0.289283
dobj: 0.287889
pinf: 3.54359e-006
dinf: 5.33073e-006
dgap: 0.00139609
Iteration: 8
pobj: 0.288218
dobj: 0.288026
pinf: 3.34809e-007
dinf: 5.03664e-007
dgap: 0.000192126
Iteration: 9
pobj: 0.288048
dobj: 0.288035
pinf: 1.97232e-008
dinf: 2.96702e-008
dgap: 1.30708e-005
Iteration: 10
pobj: 0.288036
dobj: 0.288035
pinf: 9.98953e-010
dinf: 1.50275e-009
dgap: 6.71042e-007
Iteration: 11
pobj: 0.288035
dobj: 0.288035
pinf: 4.9981e-011
dinf: 7.5188e-011
dgap: 3.35985e-008
Optimal solution found.
Iteration: 0
pobj: 0.123828
dobj: -0.958768
pinf: 0
dinf: 3.80567
dgap: 1.0826
Iteration: 1
pobj: 0.122163
dobj: 0.00512226
pinf: 8.29771e-017
dinf: 0.409766
dgap: 0.117041
Iteration: 2
pobj: 0.116729
dobj: 0.0835203
pinf: 4.55014e-017
dinf: 0.105423
dgap: 0.0332091
Iteration: 3
pobj: 0.111667
dobj: 0.099081
pinf: 1.11022e-016
dinf: 0.0327148
dgap: 0.0125857
Iteration: 4
pobj: 0.106334
dobj: 0.103347
pinf: 1.11022e-016
dinf: 0.000148925
dgap: 0.00298678
Iteration: 5
pobj: 0.104712
dobj: 0.104116
pinf: 2.22045e-016
dinf: 9.79927e-006
dgap: 0.000595376
Iteration: 6
pobj: 0.104362
dobj: 0.104265
pinf: 5.55112e-016
dinf: 4.55414e-007
dgap: 9.74638e-005
Iteration: 7
pobj: 0.104313
dobj: 0.104298
pinf: 9.99201e-016
dinf: 2.26524e-008
dgap: 1.47448e-005
Iteration: 8
pobj: 0.104307
dobj: 0.104305
pinf: 1.9984e-015
dinf: 1.14862e-009
dgap: 1.35756e-006
Iteration: 9
pobj: 0.104306
dobj: 0.104306
pinf: 3.88578e-015
dinf: 5.76651e-011
dgap: 7.64563e-008
Optimal solution found.
Iteration: 0
pobj: 0.354456
dobj: -0.837507
pinf: 3.35771
dinf: 5.05822
dgap: 17.1142
Iteration: 1
pobj: 0.407408
dobj: -0.621565
pinf: 0.348215
dinf: 0.524568
dgap: 2.58953
Iteration: 2
pobj: 0.410091
dobj: -0.152485
pinf: 0.0799313
dinf: 0.120412
dgap: 0.864013
Iteration: 3
pobj: 0.366946
dobj: 0.138263
pinf: 0.0147853
dinf: 0.0222733
dgap: 0.266957
Iteration: 4
pobj: 0.317962
dobj: 0.236951
pinf: 0.00202432
dinf: 0.00304953
dgap: 0.0837088
Iteration: 5
pobj: 0.298775
dobj: 0.275807
pinf: 0.000320591
dinf: 0.000482953
dgap: 0.0231729
Iteration: 6
pobj: 0.289855
dobj: 0.282122
pinf: 5.67655e-005
dinf: 8.55142e-005
dgap: 0.00776052
Iteration: 7
pobj: 0.285313
dobj: 0.283941
pinf: 3.54152e-006
dinf: 5.33511e-006
dgap: 0.00137392
Iteration: 8
pobj: 0.28426
dobj: 0.284074
pinf: 3.32216e-007
dinf: 5.00466e-007
dgap: 0.000186246
Iteration: 9
pobj: 0.284095
dobj: 0.284082
pinf: 1.94344e-008
dinf: 2.92769e-008
dgap: 1.24804e-005
Iteration: 10
pobj: 0.284083
dobj: 0.284083
pinf: 9.83431e-010
dinf: 1.48149e-009
dgap: 6.39364e-007
Iteration: 11
pobj: 0.284083
dobj: 0.284083
pinf: 4.9202e-011
dinf: 7.41203e-011
dgap: 3.20087e-008
Optimal solution found.
Iteration: 0
pobj: 0.120572
dobj: -0.963489
pinf: 2.22045e-016
dinf: 3.80865
dgap: 1.08406
Iteration: 1
pobj: 0.11892
dobj: 0.000848444
pinf: 4.44089e-016
dinf: 0.413097
dgap: 0.118072
Iteration: 2
pobj: 0.113618
dobj: 0.0799412
pinf: 6.66134e-016
dinf: 0.106839
dgap: 0.0336768
Iteration: 3
pobj: 0.108609
dobj: 0.0974965
pinf: 1.11022e-015
dinf: 0.0282408
dgap: 0.0111121
Iteration: 4
pobj: 0.104363
dobj: 0.101038
pinf: 2.22045e-015
dinf: 0.0024385
dgap: 0.00332474
Iteration: 5
pobj: 0.102658
dobj: 0.101973
pinf: 4.44089e-015
dinf: 6.13585e-005
dgap: 0.000685151
Iteration: 6
pobj: 0.102245
dobj: 0.102123
pinf: 8.65974e-015
dinf: 2.655e-006
dgap: 0.000121783
Iteration: 7
pobj: 0.102174
dobj: 0.102155
pinf: 1.68754e-014
dinf: 1.15466e-007
dgap: 1.91016e-005
Iteration: 8
pobj: 0.102166
dobj: 0.102163
pinf: 3.28626e-014
dinf: 5.84258e-009
dgap: 2.60925e-006
Iteration: 9
pobj: 0.102165
dobj: 0.102165
pinf: 6.40599e-014
dinf: 2.95638e-010
dgap: 2.20304e-007
Iteration: 10
pobj: 0.102165
dobj: 0.102165
pinf: 1.249e-013
dinf: 1.48168e-011
dgap: 1.19855e-008
Optimal solution found.
Iteration: 0
pobj: 0.355536
dobj: -0.837486
pinf: 3.35801
dinf: 5.03847
dgap: 17.0503
Iteration: 1
pobj: 0.406298
dobj: -0.623431
pinf: 0.347674
dinf: 0.521661
dgap: 2.58085
Iteration: 2
pobj: 0.40821
dobj: -0.153157
pinf: 0.0789282
dinf: 0.118427
dgap: 0.856748
Iteration: 3
pobj: 0.365108
dobj: 0.136459
pinf: 0.0147352
dinf: 0.0221091
dgap: 0.266264
Iteration: 4
pobj: 0.316844
dobj: 0.236044
pinf: 0.00202041
dinf: 0.00303149
dgap: 0.0834158
Iteration: 5
pobj: 0.297495
dobj: 0.277706
pinf: 0.0002203
dinf: 0.000330545
dgap: 0.0199108
Iteration: 6
pobj: 0.290699
dobj: 0.283204
pinf: 5.38778e-005
dinf: 8.08401e-005
dgap: 0.00751782
Iteration: 7
pobj: 0.286699
dobj: 0.285187
pinf: 2.36363e-006
dinf: 3.54646e-006
dgap: 0.00151324
Iteration: 8
pobj: 0.285716
dobj: 0.285416
pinf: 2.40962e-007
dinf: 3.61548e-007
dgap: 0.000299523
Iteration: 9
pobj: 0.285478
dobj: 0.285441
pinf: 1.30473e-008
dinf: 1.95765e-008
dgap: 3.67768e-005
Iteration: 10
pobj: 0.285445
dobj: 0.285443
pinf: 7.12605e-010
dinf: 1.06922e-009
dgap: 2.85218e-006
Iteration: 11
pobj: 0.285443
dobj: 0.285443
pinf: 3.60473e-011
dinf: 5.40867e-011
dgap: 1.5117e-007
Optimal solution found.
Iteration: 0
pobj: 0.119834
dobj: -0.963069
pinf: 0
dinf: 3.80328
dgap: 1.0829
Iteration: 1
pobj: 0.118243
dobj: 0.00110299
pinf: 7.00794e-017
dinf: 0.409733
dgap: 0.11714
Iteration: 2
pobj: 0.113155
dobj: 0.0799458
pinf: 3.53816e-017
dinf: 0.105445
dgap: 0.0332088
Iteration: 3
pobj: 0.108351
dobj: 0.097328
pinf: 2.22045e-016
dinf: 0.0281211
dgap: 0.0110234
Iteration: 4
pobj: 0.104218
dobj: 0.10094
pinf: 2.22045e-016
dinf: 0.00229415
dgap: 0.00327804
Iteration: 5
pobj: 0.102588
dobj: 0.101903
pinf: 4.44089e-016
dinf: 5.84166e-005
dgap: 0.000684432
Iteration: 6
pobj: 0.102192
dobj: 0.10207
pinf: 8.88178e-016
dinf: 2.57387e-006
dgap: 0.000122657
Iteration: 7
pobj: 0.102124
dobj: 0.102106
pinf: 1.77636e-015
dinf: 1.18507e-007
dgap: 1.7633e-005
Iteration: 8
pobj: 0.102115
dobj: 0.102113
pinf: 3.55271e-015
dinf: 6.1043e-009
dgap: 1.59537e-006
Iteration: 9
pobj: 0.102114
dobj: 0.102114
pinf: 6.66134e-015
dinf: 3.06637e-010
dgap: 8.73248e-008
Optimal solution found.
Iteration: 0
pobj: 0.352235
dobj: -0.840882
pinf: 3.35497
dinf: 5.02725
dgap: 16.9985
Iteration: 1
pobj: 0.401996
dobj: -0.627414
pinf: 0.34653
dinf: 0.519257
dgap: 2.57143
Iteration: 2
pobj: 0.403933
dobj: -0.15606
pinf: 0.0781291
dinf: 0.117072
dgap: 0.851064
Iteration: 3
pobj: 0.36086
dobj: 0.13277
pinf: 0.0146947
dinf: 0.0220192
dgap: 0.265347
Iteration: 4
pobj: 0.312815
dobj: 0.232108
pinf: 0.00201562
dinf: 0.00302031
dgap: 0.0832983
Iteration: 5
pobj: 0.293472
dobj: 0.273739
pinf: 0.000215875
dinf: 0.000323477
dgap: 0.019851
Iteration: 6
pobj: 0.286152
dobj: 0.279249
pinf: 4.21906e-005
dinf: 6.32204e-005
dgap: 0.00691984
Iteration: 7
pobj: 0.282389
dobj: 0.281014
pinf: 2.10034e-006
dinf: 3.14725e-006
dgap: 0.00137536
Iteration: 8
pobj: 0.281459
dobj: 0.281204
pinf: 1.99725e-007
dinf: 2.99278e-007
dgap: 0.000255898
Iteration: 9
pobj: 0.281251
dobj: 0.281222
pinf: 1.17218e-008
dinf: 1.75644e-008
dgap: 2.96122e-005
Iteration: 10
pobj: 0.281225
dobj: 0.281222
pinf: 6.33569e-010
dinf: 9.49369e-010
dgap: 2.0657e-006
Iteration: 11
pobj: 0.281223
dobj: 0.281222
pinf: 3.19088e-011
dinf: 4.78138e-011
dgap: 1.06887e-007
Optimal solution found.
Iteration: 0
pobj: 0.117959
dobj: -0.965968
pinf: 2.22045e-016
dinf: 3.80402
dgap: 1.08393
Iteration: 1
pobj: 0.116408
dobj: -0.00120716
pinf: 3.33067e-016
dinf: 0.411025
dgap: 0.117615
Iteration: 2
pobj: 0.111886
dobj: 0.0791923
pinf: 6.66134e-016
dinf: 0.103513
dgap: 0.032694
Iteration: 3
pobj: 0.108142
dobj: 0.0986186
pinf: 1.22125e-015
dinf: 0.0236912
dgap: 0.00952298
Iteration: 4
pobj: 0.105629
dobj: 0.102493
pinf: 2.22045e-015
dinf: 0.00298173
dgap: 0.0031366
Iteration: 5
pobj: 0.104677
dobj: 0.103975
pinf: 4.44089e-015
dinf: 1.46869e-016
dgap: 0.000702338
Iteration: 6
pobj: 0.104493
dobj: 0.104337
pinf: 8.65974e-015
dinf: 1.17757e-016
dgap: 0.00015564
Iteration: 7
pobj: 0.104458
dobj: 0.104425
pinf: 1.67644e-014
dinf: 1.46869e-016
dgap: 3.28108e-005
Iteration: 8
pobj: 0.104451
dobj: 0.104443
pinf: 3.21965e-014
dinf: 1.54537e-016
dgap: 7.4908e-006
Iteration: 9
pobj: 0.104449
dobj: 0.104447
pinf: 5.80647e-014
dinf: 1.44222e-016
dgap: 2.03276e-006
Iteration: 10
pobj: 0.104448
dobj: 0.104447
pinf: 1.05915e-013
dinf: 8.77708e-017
dgap: 5.2115e-007
Iteration: 11
pobj: 0.104448
dobj: 0.104448
pinf: 2.06501e-013
dinf: 1.17757e-016
dgap: 8.7804e-008
Optimal solution found.
Iteration: 0
pobj: 0.363208
dobj: -0.831093
pinf: 3.33886
dinf: 4.97613
dgap: 16.7531
Iteration: 1
pobj: 0.405647
dobj: -0.619192
pinf: 0.336719
dinf: 0.501835
dgap: 2.50551
Iteration: 2
pobj: 0.405875
dobj: -0.142641
pinf: 0.0734616
dinf: 0.109485
dgap: 0.816231
Iteration: 3
pobj: 0.363526
dobj: 0.140595
pinf: 0.0140557
dinf: 0.0209482
dgap: 0.257183
Iteration: 4
pobj: 0.319154
dobj: 0.240886
pinf: 0.00188664
dinf: 0.00281179
dgap: 0.0805058
Iteration: 5
pobj: 0.302186
dobj: 0.28447
pinf: 0.000143818
dinf: 0.000214342
dgap: 0.0177788
Iteration: 6
pobj: 0.297486
dobj: 0.291089
pinf: 3.44784e-005
dinf: 5.13855e-005
dgap: 0.00640617
Iteration: 7
pobj: 0.295016
dobj: 0.293756
pinf: 6.77545e-007
dinf: 1.00979e-006
dgap: 0.00126035
Iteration: 8
pobj: 0.294482
dobj: 0.294252
pinf: 5.49922e-008
dinf: 8.19586e-008
dgap: 0.000230268
Iteration: 9
pobj: 0.294385
dobj: 0.294311
pinf: 1.54543e-008
dinf: 2.30325e-008
dgap: 7.38351e-005
Iteration: 10
pobj: 0.294341
dobj: 0.294325
pinf: 2.95609e-009
dinf: 4.40567e-009
dgap: 1.57884e-005
Iteration: 11
pobj: 0.29433
dobj: 0.294329
pinf: 2.68477e-010
dinf: 4.0013e-010
dgap: 1.51719e-006
Iteration: 12
pobj: 0.294329
dobj: 0.294329
pinf: 1.46059e-011
dinf: 2.1768e-011
dgap: 8.34121e-008
Optimal solution found.
Iteration: 0
pobj: 0.117461
dobj: -0.964915
pinf: 0
dinf: 3.79704
dgap: 1.08238
Iteration: 1
pobj: 0.115984
dobj: -0.000549955
pinf: 2.22045e-016
dinf: 0.407162
dgap: 0.116534
Iteration: 2
pobj: 0.111511
dobj: 0.0788346
pinf: 2.22045e-016
dinf: 0.103723
dgap: 0.0326761
Iteration: 3
pobj: 0.10771
dobj: 0.0974338
pinf: 4.44089e-016
dinf: 0.0258255
dgap: 0.0102759
Iteration: 4
pobj: 0.105485
dobj: 0.101423
pinf: 8.88178e-016
dinf: 0.00687743
dgap: 0.00406256
Iteration: 5
pobj: 0.104204
dobj: 0.103265
pinf: 1.77636e-015
dinf: 9.89593e-005
dgap: 0.000938682
Iteration: 6
pobj: 0.103908
dobj: 0.103706
pinf: 3.33067e-015
dinf: 6.15003e-006
dgap: 0.000202417
Iteration: 7
pobj: 0.103838
dobj: 0.10379
pinf: 6.21725e-015
dinf: 7.30475e-007
dgap: 4.77859e-005
Iteration: 8
pobj: 0.103818
dobj: 0.103806
pinf: 1.15463e-014
dinf: 1.1318e-007
dgap: 1.20983e-005
Iteration: 9
pobj: 0.103811
dobj: 0.103807
pinf: 2.08722e-014
dinf: 2.28524e-008
dgap: 3.37276e-006
Iteration: 10
pobj: 0.103808
dobj: 0.103808
pinf: 3.9968e-014
dinf: 1.90124e-009
dgap: 4.65286e-007
Iteration: 11
pobj: 0.103808
dobj: 0.103808
pinf: 7.77156e-014
dinf: 1.0103e-010
dgap: 3.34638e-008
Optimal solution found.
Iteration: 0
pobj: 0.385274
dobj: -0.811764
pinf: 3.34789
dinf: 4.96184
dgap: 16.7501
Iteration: 1
pobj: 0.425933
dobj: -0.60281
pinf: 0.344618
dinf: 0.510751
dgap: 2.53942
Iteration: 2
pobj: 0.427563
dobj: -0.131613
pinf: 0.0765528
dinf: 0.113457
dgap: 0.838096
Iteration: 3
pobj: 0.386747
dobj: 0.158417
pinf: 0.0145386
dinf: 0.0215473
dgap: 0.263701
Iteration: 4
pobj: 0.341808
dobj: 0.261327
pinf: 0.00199159
dinf: 0.0029517
dgap: 0.0828522
Iteration: 5
pobj: 0.324467
dobj: 0.303467
pinf: 0.000243478
dinf: 0.000360853
dgap: 0.021118
Iteration: 6
pobj: 0.318583
dobj: 0.312041
pinf: 4.74853e-005
dinf: 7.03771e-005
dgap: 0.00655609
Iteration: 7
pobj: 0.315852
dobj: 0.314476
pinf: 3.20168e-006
dinf: 4.74514e-006
dgap: 0.00137698
Iteration: 8
pobj: 0.315215
dobj: 0.314771
pinf: 8.31961e-007
dinf: 1.23303e-006
dgap: 0.000443206
Iteration: 9
pobj: 0.314924
dobj: 0.314825
pinf: 1.39334e-007
dinf: 2.06505e-007
dgap: 9.95035e-005
Iteration: 10
pobj: 0.314848
dobj: 0.314836
pinf: 1.43918e-008
dinf: 2.13298e-008
dgap: 1.19034e-005
Iteration: 11
pobj: 0.314838
dobj: 0.314837
pinf: 8.57183e-010
dinf: 1.27041e-009
dgap: 7.43637e-007
Iteration: 12
pobj: 0.314837
dobj: 0.314837
pinf: 4.33867e-011
dinf: 6.43032e-011
dgap: 3.77926e-008
Optimal solution found.
Iteration: 0
pobj: 0.116791
dobj: -0.967377
pinf: 1.11022e-016
dinf: 3.80334
dgap: 1.08417
Iteration: 1
pobj: 0.115279
dobj: -0.00262861
pinf: 2.22045e-016
dinf: 0.411907
dgap: 0.117908
Iteration: 2
pobj: 0.110732
dobj: 0.0774768
pinf: 4.44089e-016
dinf: 0.105455
dgap: 0.0332556
Iteration: 3
pobj: 0.106893
dobj: 0.096494
pinf: 6.66134e-016
dinf: 0.0261378
dgap: 0.0103991
Iteration: 4
pobj: 0.104649
dobj: 0.100586
pinf: 9.99201e-016
dinf: 0.00685832
dgap: 0.0040624
Iteration: 5
pobj: 0.103386
dobj: 0.102436
pinf: 1.9984e-015
dinf: 0.000136677
dgap: 0.000949866
Iteration: 6
pobj: 0.103096
dobj: 0.102896
pinf: 3.88578e-015
dinf: 7.68168e-006
dgap: 0.000199878
Iteration: 7
pobj: 0.103032
dobj: 0.102989
pinf: 7.43849e-015
dinf: 8.37687e-007
dgap: 4.2728e-005
Iteration: 8
pobj: 0.103016
dobj: 0.103002
pinf: 1.31006e-014
dinf: 1.92874e-007
dgap: 1.38698e-005
Iteration: 9
pobj: 0.103009
dobj: 0.103005
pinf: 2.40918e-014
dinf: 2.90994e-008
dgap: 3.67126e-006
Iteration: 10
pobj: 0.103006
dobj: 0.103006
pinf: 4.69624e-014
dinf: 1.43561e-009
dgap: 6.08212e-007
Iteration: 11
pobj: 0.103006
dobj: 0.103006
pinf: 9.18154e-014
dinf: 6.71023e-011
dgap: 8.06956e-008
Optimal solution found.
Iteration: 0
pobj: 0.38066
dobj: -0.816103
pinf: 3.34714
dinf: 4.95893
dgap: 16.7365
Iteration: 1
pobj: 0.420876
dobj: -0.606924
pinf: 0.343878
dinf: 0.509469
dgap: 2.53425
Iteration: 2
pobj: 0.422235
dobj: -0.134453
pinf: 0.0759615
dinf: 0.11254
dgap: 0.833128
Iteration: 3
pobj: 0.381545
dobj: 0.154297
pinf: 0.0144642
dinf: 0.0214293
dgap: 0.262381
Iteration: 4
pobj: 0.336996
dobj: 0.256943
pinf: 0.0019833
dinf: 0.00293834
dgap: 0.0824029
Iteration: 5
pobj: 0.319809
dobj: 0.298993
pinf: 0.000241351
dinf: 0.000357572
dgap: 0.0209314
Iteration: 6
pobj: 0.313918
dobj: 0.307667
pinf: 4.41234e-005
dinf: 6.53705e-005
dgap: 0.00626419
Iteration: 7
pobj: 0.311327
dobj: 0.309963
pinf: 3.80142e-006
dinf: 5.63195e-006
dgap: 0.00136453
Iteration: 8
pobj: 0.310693
dobj: 0.310233
pinf: 1.03309e-006
dinf: 1.53056e-006
dgap: 0.000459589
Iteration: 9
pobj: 0.31038
dobj: 0.310291
pinf: 1.36903e-007
dinf: 2.02828e-007
dgap: 8.92301e-005
Iteration: 10
pobj: 0.310312
dobj: 0.310301
pinf: 1.34995e-008
dinf: 2.00001e-008
dgap: 1.02076e-005
Iteration: 11
pobj: 0.310303
dobj: 0.310302
pinf: 7.81876e-010
dinf: 1.15838e-009
dgap: 6.19152e-007
Iteration: 12
pobj: 0.310302
dobj: 0.310302
pinf: 3.94785e-011
dinf: 5.84893e-011
dgap: 3.13779e-008
Optimal solution found.
Iteration: 0
pobj: 0.115995
dobj: -0.968196
pinf: 2.22045e-016
dinf: 3.80337
dgap: 1.08419
Iteration: 1
pobj: 0.114487
dobj: -0.00342417
pinf: 2.22045e-016
dinf: 0.411915
dgap: 0.117911
Iteration: 2
pobj: 0.109964
dobj: 0.076731
pinf: 4.44089e-016
dinf: 0.105409
dgap: 0.0332335
Iteration: 3
pobj: 0.106061
dobj: 0.0963413
pinf: 8.88178e-016
dinf: 0.0242878
dgap: 0.00971954
Iteration: 4
pobj: 0.103422
dobj: 0.100283
pinf: 1.55431e-015
dinf: 0.00311373
dgap: 0.00313903
Iteration: 5
pobj: 0.102405
dobj: 0.101732
pinf: 3.10862e-015
dinf: 2.40896e-005
dgap: 0.000673131
Iteration: 6
pobj: 0.10219
dobj: 0.102056
pinf: 5.9952e-015
dinf: 1.35684e-006
dgap: 0.000134059
Iteration: 7
pobj: 0.102141
dobj: 0.102112
pinf: 1.13243e-014
dinf: 1.60625e-007
dgap: 2.96809e-005
Iteration: 8
pobj: 0.102128
dobj: 0.102118
pinf: 1.9762e-014
dinf: 4.1199e-008
dgap: 1.00488e-005
Iteration: 9
pobj: 0.102122
dobj: 0.102119
pinf: 3.66374e-014
dinf: 6.07175e-009
dgap: 2.20192e-006
Iteration: 10
pobj: 0.10212
dobj: 0.10212
pinf: 7.06102e-014
dinf: 4.33201e-010
dgap: 2.35553e-007
Iteration: 11
pobj: 0.10212
dobj: 0.10212
pinf: 1.37446e-013
dinf: 2.25349e-011
dgap: 1.38522e-008
Optimal solution found.
Iteration: 0
pobj: 0.380213
dobj: -0.816458
pinf: 3.35571
dinf: 4.94216
dgap: 16.72
Iteration: 1
pobj: 0.419197
dobj: -0.609183
pinf: 0.349697
dinf: 0.51502
dgap: 2.55486
Iteration: 2
pobj: 0.421981
dobj: -0.140946
pinf: 0.0786301
dinf: 0.115803
dgap: 0.849094
Iteration: 3
pobj: 0.383437
dobj: 0.153208
pinf: 0.0148188
dinf: 0.0218246
dgap: 0.266296
Iteration: 4
pobj: 0.33938
dobj: 0.2578
pinf: 0.00205631
dinf: 0.00302845
dgap: 0.0840279
Iteration: 5
pobj: 0.321604
dobj: 0.303741
pinf: 0.000135538
dinf: 0.000199615
dgap: 0.0179219
Iteration: 6
pobj: 0.316139
dobj: 0.310232
pinf: 2.64637e-005
dinf: 3.89746e-005
dgap: 0.0059151
Iteration: 7
pobj: 0.313487
dobj: 0.312476
pinf: 9.80003e-007
dinf: 1.44331e-006
dgap: 0.00101092
Iteration: 8
pobj: 0.312944
dobj: 0.312593
pinf: 2.66492e-007
dinf: 3.92479e-007
dgap: 0.000351696
Iteration: 9
pobj: 0.312672
dobj: 0.312631
pinf: 1.88438e-008
dinf: 2.77524e-008
dgap: 4.11368e-005
Iteration: 10
pobj: 0.312638
dobj: 0.312634
pinf: 1.4419e-009
dinf: 2.12357e-009
dgap: 3.4519e-006
Iteration: 11
pobj: 0.312635
dobj: 0.312635
pinf: 7.56953e-011
dinf: 1.11481e-010
dgap: 1.83903e-007
Optimal solution found.
Iteration: 0
pobj: 0.115338
dobj: -0.968327
pinf: 0
dinf: 3.80189
dgap: 1.08366
Iteration: 1
pobj: 0.113838
dobj: -0.00366892
pinf: 9.2316e-017
dinf: 0.410564
dgap: 0.117507
Iteration: 2
pobj: 0.109368
dobj: 0.0764341
pinf: 7.24442e-017
dinf: 0.104415
dgap: 0.0329334
Iteration: 3
pobj: 0.105623
dobj: 0.0961807
pinf: 1.11022e-016
dinf: 0.0233323
dgap: 0.00944239
Iteration: 4
pobj: 0.103241
dobj: 0.100131
pinf: 8.45578e-017
dinf: 0.00317813
dgap: 0.00311025
Iteration: 5
pobj: 0.102326
dobj: 0.101662
pinf: 1.27305e-016
dinf: 2.47515e-005
dgap: 0.000664339
Iteration: 6
pobj: 0.102133
dobj: 0.102
pinf: 2.22045e-016
dinf: 1.5577e-006
dgap: 0.000133547
Iteration: 7
pobj: 0.102087
dobj: 0.102055
pinf: 2.22045e-016
dinf: 2.05964e-007
dgap: 3.22562e-005
Iteration: 8
pobj: 0.102072
dobj: 0.102062
pinf: 4.44089e-016
dinf: 4.44937e-008
dgap: 1.04288e-005
Iteration: 9
pobj: 0.102067
dobj: 0.102065
pinf: 8.88178e-016
dinf: 1.71902e-009
dgap: 1.8093e-006
Iteration: 10
pobj: 0.102066
dobj: 0.102066
pinf: 1.77636e-015
dinf: 6.13393e-011
dgap: 4.21972e-007
Iteration: 11
pobj: 0.102066
dobj: 0.102066
pinf: 3.44169e-015
dinf: 1.17757e-016
dgap: 9.26229e-008
Optimal solution found.
Iteration: 0
pobj: 0.376361
dobj: -0.820006
pinf: 3.35392
dinf: 4.93227
dgap: 16.6782
Iteration: 1
pobj: 0.414421
dobj: -0.613273
pinf: 0.350025
dinf: 0.514746
dgap: 2.55217
Iteration: 2
pobj: 0.417204
dobj: -0.146965
pinf: 0.0795125
dinf: 0.116931
dgap: 0.853255
Iteration: 3
pobj: 0.379064
dobj: 0.148053
pinf: 0.0149737
dinf: 0.0220204
dgap: 0.267468
Iteration: 4
pobj: 0.335041
dobj: 0.252963
pinf: 0.00207423
dinf: 0.00305036
dgap: 0.0845503
Iteration: 5
pobj: 0.317407
dobj: 0.299617
pinf: 0.000124761
dinf: 0.000183474
dgap: 0.0178442
Iteration: 6
pobj: 0.312192
dobj: 0.306338
pinf: 2.35626e-005
dinf: 3.46512e-005
dgap: 0.0058604
Iteration: 7
pobj: 0.309767
dobj: 0.308611
pinf: 1.55944e-006
dinf: 2.29331e-006
dgap: 0.00115604
Iteration: 8
pobj: 0.309164
dobj: 0.308771
pinf: 3.81572e-007
dinf: 5.61141e-007
dgap: 0.000393093
Iteration: 9
pobj: 0.308865
dobj: 0.308827
pinf: 1.28258e-008
dinf: 1.88616e-008
dgap: 3.86656e-005
Iteration: 10
pobj: 0.308833
dobj: 0.30883
pinf: 8.54524e-010
dinf: 1.25666e-009
dgap: 3.3425e-006
Iteration: 11
pobj: 0.30883
dobj: 0.30883
pinf: 4.41087e-011
dinf: 6.48665e-011
dgap: 1.83208e-007
Optimal solution found.
Iteration: 0
pobj: 0.113578
dobj: -0.970611
pinf: 1.11022e-016
dinf: 3.80243
dgap: 1.08419
Iteration: 1
pobj: 0.112099
dobj: -0.00572224
pinf: 1.11022e-016
dinf: 0.411499
dgap: 0.117821
Iteration: 2
pobj: 0.107756
dobj: 0.074632
pinf: 1.11022e-016
dinf: 0.105031
dgap: 0.0331242
Iteration: 3
pobj: 0.104226
dobj: 0.0951043
pinf: 2.22045e-016
dinf: 0.0224643
dgap: 0.00912126
Iteration: 4
pobj: 0.102363
dobj: 0.0989338
pinf: 4.44089e-016
dinf: 0.00532085
dgap: 0.00342873
Iteration: 5
pobj: 0.101491
dobj: 0.100737
pinf: 7.77156e-016
dinf: 0.000138659
dgap: 0.000753594
Iteration: 6
pobj: 0.101337
dobj: 0.101145
pinf: 1.55431e-015
dinf: 2.07781e-005
dgap: 0.000192457
Iteration: 7
pobj: 0.10129
dobj: 0.101226
pinf: 2.66454e-015
dinf: 4.62642e-006
dgap: 6.40981e-005
Iteration: 8
pobj: 0.101268
dobj: 0.10126
pinf: 5.21805e-015
dinf: 1.8804e-007
dgap: 7.69986e-006
Iteration: 9
pobj: 0.101264
dobj: 0.101263
pinf: 1.0103e-014
dinf: 1.04771e-008
dgap: 8.33675e-007
Iteration: 10
pobj: 0.101264
dobj: 0.101263
pinf: 1.96509e-014
dinf: 5.41637e-010
dgap: 5.89179e-008
Optimal solution found.
Iteration: 0
pobj: 0.374999
dobj: -0.819766
pinf: 3.36109
dinf: 4.91558
dgap: 16.6536
Iteration: 1
pobj: 0.412775
dobj: -0.614757
pinf: 0.3547
dinf: 0.518747
dgap: 2.56685
Iteration: 2
pobj: 0.416925
dobj: -0.151292
pinf: 0.081188
dinf: 0.118737
dgap: 0.862867
Iteration: 3
pobj: 0.38059
dobj: 0.147736
pinf: 0.0151612
dinf: 0.0221732
dgap: 0.269653
Iteration: 4
pobj: 0.337333
dobj: 0.254622
pinf: 0.00212441
dinf: 0.00310694
dgap: 0.0852274
Iteration: 5
pobj: 0.319892
dobj: 0.304305
pinf: 5.30192e-005
dinf: 7.75404e-005
dgap: 0.0156074
Iteration: 6
pobj: 0.315548
dobj: 0.31102
pinf: 9.60636e-006
dinf: 1.40493e-005
dgap: 0.00452997
Iteration: 7
pobj: 0.313995
dobj: 0.312431
pinf: 2.33809e-006
dinf: 3.41944e-006
dgap: 0.00156415
Iteration: 8
pobj: 0.313136
dobj: 0.312869
pinf: 4.9422e-009
dinf: 7.22795e-009
dgap: 0.000266974
Iteration: 9
pobj: 0.312988
dobj: 0.312943
pinf: 4.17764e-010
dinf: 6.10978e-010
dgap: 4.50174e-005
Iteration: 10
pobj: 0.312963
dobj: 0.312958
pinf: 2.27116e-011
dinf: 3.32153e-011
dgap: 5.13114e-006
Iteration: 11
pobj: 0.312961
dobj: 0.312961
pinf: 1.16222e-012
dinf: 1.69991e-012
dgap: 4.02392e-007
Iteration: 12
pobj: 0.312961
dobj: 0.312961
pinf: 5.09592e-013
dinf: 8.49645e-014
dgap: 2.17216e-008
Optimal solution found.
Iteration: 0
pobj: 0.115856
dobj: -0.964936
pinf: 1.11022e-016
dinf: 3.7895
dgap: 1.08079
Iteration: 1
pobj: 0.114505
dobj: -0.00064014
pinf: 2.22045e-016
dinf: 0.402157
dgap: 0.115145
Iteration: 2
pobj: 0.110545
dobj: 0.0786897
pinf: 3.33067e-016
dinf: 0.101424
dgap: 0.031855
Iteration: 3
pobj: 0.10727
dobj: 0.0987058
pinf: 4.44089e-016
dinf: 0.0214124
dgap: 0.00856395
Iteration: 4
pobj: 0.105381
dobj: 0.102591
pinf: 8.88178e-016
dinf: 0.00369991
dgap: 0.0027898
Iteration: 5
pobj: 0.104625
dobj: 0.104105
pinf: 1.88738e-015
dinf: 0.000140424
dgap: 0.000520473
Iteration: 6
pobj: 0.10446
dobj: 0.104275
pinf: 3.10862e-015
dinf: 3.73256e-005
dgap: 0.000185431
Iteration: 7
pobj: 0.104362
dobj: 0.104322
pinf: 6.10623e-015
dinf: 3.28397e-006
dgap: 4.00393e-005
Iteration: 8
pobj: 0.104338
dobj: 0.104331
pinf: 1.16573e-014
dinf: 2.14965e-007
dgap: 6.97598e-006
Iteration: 9
pobj: 0.104333
dobj: 0.104332
pinf: 2.28706e-014
dinf: 9.66859e-009
dgap: 1.08288e-006
Iteration: 10
pobj: 0.104332
dobj: 0.104332
pinf: 4.4631e-014
dinf: 4.73064e-010
dgap: 1.04238e-007
Optimal solution found.
Iteration: 0
pobj: 0.386136
dobj: -0.80738
pinf: 3.38111
dinf: 4.88128
dgap: 16.6285
Iteration: 1
pobj: 0.420447
dobj: -0.605035
pinf: 0.371231
dinf: 0.535943
dgap: 2.62531
Iteration: 2
pobj: 0.428152
dobj: -0.15625
pinf: 0.0918502
dinf: 0.132603
dgap: 0.921431
Iteration: 3
pobj: 0.398811
dobj: 0.158827
pinf: 0.0163072
dinf: 0.0235426
dgap: 0.280851
Iteration: 4
pobj: 0.356177
dobj: 0.268852
pinf: 0.00236023
dinf: 0.00340744
dgap: 0.0903166
Iteration: 5
pobj: 0.338016
dobj: 0.322296
pinf: 2.22343e-016
dinf: 1.18018e-015
dgap: 0.0157196
Iteration: 6
pobj: 0.33352
dobj: 0.328293
pinf: 2.22045e-016
dinf: 6.47366e-016
dgap: 0.0052267
Iteration: 7
pobj: 0.331078
dobj: 0.329859
pinf: 4.44089e-016
dinf: 9.35491e-016
dgap: 0.0012186
Iteration: 8
pobj: 0.330452
dobj: 0.330197
pinf: 8.88178e-016
dinf: 5.43896e-016
dgap: 0.000254928
Iteration: 9
pobj: 0.330271
dobj: 0.330237
pinf: 1.55431e-015
dinf: 8.7419e-016
dgap: 3.37471e-005
Iteration: 10
pobj: 0.330242
dobj: 0.330239
pinf: 3.10862e-015
dinf: 1.35974e-015
dgap: 2.55041e-006
Iteration: 11
pobj: 0.33024
dobj: 0.330239
pinf: 6.21725e-015
dinf: 1.45604e-015
dgap: 1.33747e-007
Optimal solution found.
Iteration: 0
pobj: 0.116146
dobj: -0.96426
pinf: 0
dinf: 3.78728
dgap: 1.08041
Iteration: 1
pobj: 0.114817
dobj: 9.0969e-006
pinf: 5.63718e-017
dinf: 0.400898
dgap: 0.114808
Iteration: 2
pobj: 0.110957
dobj: 0.0792982
pinf: 7.63278e-017
dinf: 0.100825
dgap: 0.0316592
Iteration: 3
pobj: 0.107808
dobj: 0.0991303
pinf: 1.22958e-016
dinf: 0.0218602
dgap: 0.00867721
Iteration: 4
pobj: 0.105946
dobj: 0.103137
pinf: 1.11022e-016
dinf: 0.0036967
dgap: 0.00280938
Iteration: 5
pobj: 0.105199
dobj: 0.104673
pinf: 3.33067e-016
dinf: 0.000138618
dgap: 0.000525296
Iteration: 6
pobj: 0.105032
dobj: 0.104853
pinf: 4.44089e-016
dinf: 3.42385e-005
dgap: 0.000178496
Iteration: 7
pobj: 0.104941
dobj: 0.104897
pinf: 7.77156e-016
dinf: 4.12481e-006
dgap: 4.42235e-005
Iteration: 8
pobj: 0.104915
dobj: 0.104908
pinf: 1.66533e-015
dinf: 2.58393e-007
dgap: 7.91354e-006
Iteration: 9
pobj: 0.104911
dobj: 0.104909
pinf: 3.10862e-015
dinf: 1.03816e-008
dgap: 1.53242e-006
Iteration: 10
pobj: 0.104909
dobj: 0.104909
pinf: 6.21725e-015
dinf: 4.09348e-010
dgap: 2.49718e-007
Iteration: 11
pobj: 0.104909
dobj: 0.104909
pinf: 1.19904e-014
dinf: 1.98054e-011
dgap: 2.22062e-008
Optimal solution found.
Iteration: 0
pobj: 0.385849
dobj: -0.807609
pinf: 3.36414
dinf: 4.8757
dgap: 16.5322
Iteration: 1
pobj: 0.41963
dobj: -0.604388
pinf: 0.357598
dinf: 0.518272
dgap: 2.56214
Iteration: 2
pobj: 0.425749
dobj: -0.147987
pinf: 0.0856397
dinf: 0.124119
dgap: 0.884085
Iteration: 3
pobj: 0.394405
dobj: 0.159673
pinf: 0.015689
dinf: 0.0227382
dgap: 0.27314
Iteration: 4
pobj: 0.352432
dobj: 0.267564
pinf: 0.00226345
dinf: 0.00328044
dgap: 0.087645
Iteration: 5
pobj: 0.334806
dobj: 0.319322
pinf: 8.19094e-006
dinf: 1.18712e-005
dgap: 0.0154874
Iteration: 6
pobj: 0.33039
dobj: 0.325292
pinf: 1.64836e-006
dinf: 2.38899e-006
dgap: 0.00509793
Iteration: 7
pobj: 0.32805
dobj: 0.326833
pinf: 6.21725e-015
dinf: 1.02958e-015
dgap: 0.00121633
Iteration: 8
pobj: 0.327445
dobj: 0.327186
pinf: 1.17684e-014
dinf: 1.01754e-015
dgap: 0.000258891
Iteration: 9
pobj: 0.327267
dobj: 0.32723
pinf: 2.28706e-014
dinf: 1.27555e-015
dgap: 3.7215e-005
Iteration: 10
pobj: 0.327235
dobj: 0.327232
pinf: 4.41869e-014
dinf: 7.19507e-016
dgap: 3.18587e-006
Iteration: 11
pobj: 0.327232
dobj: 0.327232
pinf: 8.61533e-014
dinf: 8.15844e-016
dgap: 1.72663e-007
Optimal solution found.
Iteration: 0
pobj: 0.117079
dobj: -0.962103
pinf: 0
dinf: 3.78066
dgap: 1.07918
Iteration: 1
pobj: 0.115808
dobj: 0.00200619
pinf: 1.11022e-016
dinf: 0.397167
dgap: 0.113802
Iteration: 2
pobj: 0.112228
dobj: 0.081182
pinf: 6.65555e-017
dinf: 0.0988887
dgap: 0.0310461
Iteration: 3
pobj: 0.109478
dobj: 0.101359
pinf: 1.24755e-016
dinf: 0.0204133
dgap: 0.00811975
Iteration: 4
pobj: 0.107975
dobj: 0.10534
pinf: 2.22045e-016
dinf: 0.00375517
dgap: 0.00263493
Iteration: 5
pobj: 0.107366
dobj: 0.106896
pinf: 3.33067e-016
dinf: 0.000179177
dgap: 0.000470257
Iteration: 6
pobj: 0.107234
dobj: 0.10708
pinf: 6.66134e-016
dinf: 4.35616e-005
dgap: 0.000153871
Iteration: 7
pobj: 0.10717
dobj: 0.107145
pinf: 1.22125e-015
dinf: 1.30558e-006
dgap: 2.4619e-005
Iteration: 8
pobj: 0.107161
dobj: 0.107157
pinf: 2.33147e-015
dinf: 1.30306e-007
dgap: 3.51796e-006
Iteration: 9
pobj: 0.107159
dobj: 0.107159
pinf: 4.55191e-015
dinf: 9.18165e-009
dgap: 2.89914e-007
Iteration: 10
pobj: 0.107159
dobj: 0.107159
pinf: 8.88178e-015
dinf: 4.8166e-010
dgap: 1.56171e-008
Optimal solution found.
Iteration: 0
pobj: 0.393465
dobj: -0.79853
pinf: 3.33552
dinf: 4.86131
dgap: 16.3522
Iteration: 1
pobj: 0.426762
dobj: -0.59399
pinf: 0.333281
dinf: 0.485736
dgap: 2.448
Iteration: 2
pobj: 0.430131
dobj: -0.120357
pinf: 0.0735086
dinf: 0.107134
dgap: 0.809646
Iteration: 3
pobj: 0.395132
dobj: 0.171582
pinf: 0.0142669
dinf: 0.0207932
dgap: 0.256665
Iteration: 4
pobj: 0.355217
dobj: 0.27596
pinf: 0.00202361
dinf: 0.00294928
dgap: 0.0815358
Iteration: 5
pobj: 0.33927
dobj: 0.3246
pinf: 2.43872e-005
dinf: 3.55429e-005
dgap: 0.0146787
Iteration: 6
pobj: 0.335384
dobj: 0.330328
pinf: 5.00583e-006
dinf: 7.29568e-006
dgap: 0.00505709
Iteration: 7
pobj: 0.333437
dobj: 0.332259
pinf: 7.77156e-016
dinf: 9.28879e-016
dgap: 0.00117847
Iteration: 8
pobj: 0.332974
dobj: 0.332683
pinf: 1.33227e-015
dinf: 8.00593e-016
dgap: 0.000291088
Iteration: 9
pobj: 0.332831
dobj: 0.332777
pinf: 2.55351e-015
dinf: 6.28037e-016
dgap: 5.43746e-005
Iteration: 10
pobj: 0.332806
dobj: 0.332795
pinf: 4.996e-015
dinf: 1.09906e-015
dgap: 1.04756e-005
Iteration: 11
pobj: 0.332803
dobj: 0.332801
pinf: 9.99201e-015
dinf: 1.01754e-015
dgap: 2.12542e-006
Iteration: 12
pobj: 0.332802
dobj: 0.332802
pinf: 1.95399e-014
dinf: 1.19575e-015
dgap: 3.44637e-007
Iteration: 13
pobj: 0.332802
dobj: 0.332802
pinf: 3.81917e-014
dinf: 1.21111e-015
dgap: 2.99936e-008
Optimal solution found.
Iteration: 0
pobj: 0.119434
dobj: -0.961719
pinf: 1.11022e-016
dinf: 3.78683
dgap: 1.08115
Iteration: 1
pobj: 0.118122
dobj: 0.00270501
pinf: 3.33067e-016
dinf: 0.402671
dgap: 0.115417
Iteration: 2
pobj: 0.114453
dobj: 0.0826542
pinf: 4.44089e-016
dinf: 0.101095
dgap: 0.0317989
Iteration: 3
pobj: 0.111704
dobj: 0.103444
pinf: 8.88178e-016
dinf: 0.0206162
dgap: 0.00826024
Iteration: 4
pobj: 0.110379
dobj: 0.107455
pinf: 1.33227e-015
dinf: 0.00514287
dgap: 0.00292366
Iteration: 5
pobj: 0.109702
dobj: 0.10915
pinf: 2.66454e-015
dinf: 0.000250018
dgap: 0.000552299
Iteration: 6
pobj: 0.109532
dobj: 0.109366
pinf: 4.77396e-015
dinf: 5.00946e-005
dgap: 0.00016602
Iteration: 7
pobj: 0.109462
dobj: 0.109429
pinf: 9.32587e-015
dinf: 1.88675e-006
dgap: 3.32356e-005
Iteration: 8
pobj: 0.10945
dobj: 0.109444
pinf: 1.79856e-014
dinf: 1.21345e-007
dgap: 6.14172e-006
Iteration: 9
pobj: 0.109448
dobj: 0.109447
pinf: 3.54161e-014
dinf: 4.76901e-009
dgap: 8.19382e-007
Iteration: 10
pobj: 0.109448
dobj: 0.109448
pinf: 6.91669e-014
dinf: 2.27442e-010
dgap: 9.04688e-008
Optimal solution found.
Iteration: 0
pobj: 0.397362
dobj: -0.796393
pinf: 3.31572
dinf: 4.89165
dgap: 16.3646
Iteration: 1
pobj: 0.431916
dobj: -0.587479
pinf: 0.315443
dinf: 0.46537
dgap: 2.37876
Iteration: 2
pobj: 0.433101
dobj: -0.0968538
pinf: 0.0637383
dinf: 0.0940325
dgap: 0.751997
Iteration: 3
pobj: 0.393641
dobj: 0.179585
pinf: 0.0128501
dinf: 0.0189577
dgap: 0.242945
Iteration: 4
pobj: 0.35405
dobj: 0.279949
pinf: 0.00176741
dinf: 0.00260745
dgap: 0.0759839
Iteration: 5
pobj: 0.33866
dobj: 0.324075
pinf: 5.69979e-005
dinf: 8.40883e-005
dgap: 0.0146048
Iteration: 6
pobj: 0.33464
dobj: 0.329599
pinf: 1.13094e-005
dinf: 1.66846e-005
dgap: 0.0050443
Iteration: 7
pobj: 0.332671
dobj: 0.331461
pinf: 1.12025e-016
dinf: 1.03555e-015
dgap: 0.00121005
Iteration: 8
pobj: 0.332188
dobj: 0.33188
pinf: 2.22084e-016
dinf: 1.52226e-015
dgap: 0.000308424
Iteration: 9
pobj: 0.33202
dobj: 0.331962
pinf: 1.55171e-017
dinf: 1.05325e-015
dgap: 5.81311e-005
Iteration: 10
pobj: 0.331981
dobj: 0.331971
pinf: 3.33356e-016
dinf: 1.03555e-015
dgap: 1.02146e-005
Iteration: 11
pobj: 0.331973
dobj: 0.331972
pinf: 1.11954e-016
dinf: 1.02358e-015
dgap: 1.19184e-006
Iteration: 12
pobj: 0.331972
dobj: 0.331972
pinf: 1.119e-016
dinf: 8.59975e-016
dgap: 7.76165e-008
Optimal solution found.
Iteration: 0
pobj: 0.118627
dobj: -0.962724
pinf: 2.22045e-016
dinf: 3.78784
dgap: 1.08135
Iteration: 1
pobj: 0.117306
dobj: 0.00177466
pinf: 4.44089e-016
dinf: 0.403095
dgap: 0.115532
Iteration: 2
pobj: 0.113611
dobj: 0.0817504
pinf: 6.66134e-016
dinf: 0.101286
dgap: 0.0318609
Iteration: 3
pobj: 0.11081
dobj: 0.102659
pinf: 1.33227e-015
dinf: 0.0201934
dgap: 0.00815118
Iteration: 4
pobj: 0.109439
dobj: 0.106523
pinf: 2.22045e-015
dinf: 0.00490821
dgap: 0.00291623
Iteration: 5
pobj: 0.108761
dobj: 0.108207
pinf: 4.21885e-015
dinf: 0.000229887
dgap: 0.000554443
Iteration: 6
pobj: 0.108593
dobj: 0.108425
pinf: 7.54952e-015
dinf: 4.71651e-005
dgap: 0.000167987
Iteration: 7
pobj: 0.10852
dobj: 0.108487
pinf: 1.4877e-014
dinf: 1.63072e-006
dgap: 3.36787e-005
Iteration: 8
pobj: 0.108508
dobj: 0.108502
pinf: 2.88658e-014
dinf: 9.17565e-008
dgap: 6.07344e-006
Iteration: 9
pobj: 0.108506
dobj: 0.108505
pinf: 5.66214e-014
dinf: 3.66359e-009
dgap: 7.99799e-007
Iteration: 10
pobj: 0.108506
dobj: 0.108506
pinf: 1.09912e-013
dinf: 2.17213e-010
dgap: 7.12718e-008
Optimal solution found.
Iteration: 0
pobj: 0.396213
dobj: -0.796955
pinf: 3.31709
dinf: 4.88067
dgap: 16.3339
Iteration: 1
pobj: 0.430546
dobj: -0.589339
pinf: 0.316202
dinf: 0.46525
dgap: 2.37906
Iteration: 2
pobj: 0.431926
dobj: -0.098464
pinf: 0.0634511
dinf: 0.0933601
dgap: 0.750465
Iteration: 3
pobj: 0.392846
dobj: 0.178609
pinf: 0.0128326
dinf: 0.0188815
dgap: 0.242841
Iteration: 4
pobj: 0.353657
dobj: 0.279594
pinf: 0.00177424
dinf: 0.00261057
dgap: 0.0759265
Iteration: 5
pobj: 0.338475
dobj: 0.32437
pinf: 4.61853e-005
dinf: 6.79556e-005
dgap: 0.0141207
Iteration: 6
pobj: 0.33464
dobj: 0.3298
pinf: 9.38175e-006
dinf: 1.3804e-005
dgap: 0.00484233
Iteration: 7
pobj: 0.332742
dobj: 0.331622
pinf: 1.79856e-014
dinf: 1.15378e-015
dgap: 0.00111915
Iteration: 8
pobj: 0.3323
dobj: 0.33202
pinf: 3.35287e-014
dinf: 1.9767e-015
dgap: 0.00028012
Iteration: 9
pobj: 0.332161
dobj: 0.332105
pinf: 6.55032e-014
dinf: 1.06489e-015
dgap: 5.55974e-005
Iteration: 10
pobj: 0.332131
dobj: 0.33212
pinf: 1.29674e-013
dinf: 9.15513e-016
dgap: 1.13708e-005
Iteration: 11
pobj: 0.332126
dobj: 0.332123
pinf: 2.58127e-013
dinf: 1.00535e-015
dgap: 2.44323e-006
Iteration: 12
pobj: 0.332124
dobj: 0.332124
pinf: 5.13145e-013
dinf: 1.19575e-015
dgap: 5.87982e-007
Iteration: 13
pobj: 0.332124
dobj: 0.332124
pinf: 1.02618e-012
dinf: 1.43901e-015
dgap: 1.24635e-007
Optimal solution found.
Iteration: 0
pobj: 0.118736
dobj: -0.962961
pinf: 0
dinf: 3.78935
dgap: 1.0817
Iteration: 1
pobj: 0.117401
dobj: 0.00158224
pinf: 9.15303e-017
dinf: 0.404122
dgap: 0.115819
Iteration: 2
pobj: 0.113658
dobj: 0.081688
pinf: 2.22045e-016
dinf: 0.101598
dgap: 0.0319702
Iteration: 3
pobj: 0.110815
dobj: 0.102575
pinf: 2.22045e-016
dinf: 0.0204246
dgap: 0.00824055
Iteration: 4
pobj: 0.109417
dobj: 0.106464
pinf: 4.44089e-016
dinf: 0.00494811
dgap: 0.00295323
Iteration: 5
pobj: 0.108728
dobj: 0.108162
pinf: 8.88178e-016
dinf: 0.000234885
dgap: 0.000566548
Iteration: 6
pobj: 0.108557
dobj: 0.108382
pinf: 1.55431e-015
dinf: 4.97748e-005
dgap: 0.000174904
Iteration: 7
pobj: 0.10848
dobj: 0.108444
pinf: 3.10862e-015
dinf: 1.6261e-006
dgap: 3.54034e-005
Iteration: 8
pobj: 0.108466
dobj: 0.108459
pinf: 5.9952e-015
dinf: 8.34651e-008
dgap: 6.431e-006
Iteration: 9
pobj: 0.108464
dobj: 0.108463
pinf: 1.16573e-014
dinf: 3.25765e-009
dgap: 9.14054e-007
Iteration: 10
pobj: 0.108463
dobj: 0.108463
pinf: 2.28706e-014
dinf: 1.72456e-010
dgap: 7.94426e-008
Optimal solution found.
Iteration: 0
pobj: 0.392365
dobj: -0.800785
pinf: 3.3168
dinf: 4.87906
dgap: 16.3271
Iteration: 1
pobj: 0.426576
dobj: -0.59326
pinf: 0.316057
dinf: 0.464925
dgap: 2.37787
Iteration: 2
pobj: 0.427997
dobj: -0.101583
pinf: 0.0630009
dinf: 0.0926752
dgap: 0.747776
Iteration: 3
pobj: 0.388761
dobj: 0.174835
pinf: 0.0127871
dinf: 0.01881
dgap: 0.242352
Iteration: 4
pobj: 0.349519
dobj: 0.275554
pinf: 0.00176838
dinf: 0.00260131
dgap: 0.0758171
Iteration: 5
pobj: 0.334306
dobj: 0.320292
pinf: 4.26914e-005
dinf: 6.27997e-005
dgap: 0.0140288
Iteration: 6
pobj: 0.330479
dobj: 0.325645
pinf: 8.71698e-006
dinf: 1.28228e-005
dgap: 0.00483663
Iteration: 7
pobj: 0.328572
dobj: 0.327448
pinf: 1.11022e-015
dinf: 1.48122e-015
dgap: 0.00112388
Iteration: 8
pobj: 0.328126
dobj: 0.327846
pinf: 2.10942e-015
dinf: 5.43896e-016
dgap: 0.000280319
Iteration: 9
pobj: 0.327984
dobj: 0.327929
pinf: 4.10783e-015
dinf: 7.36439e-016
dgap: 5.55484e-005
Iteration: 10
pobj: 0.327954
dobj: 0.327942
pinf: 8.21565e-015
dinf: 6.93334e-016
dgap: 1.12352e-005
Iteration: 11
pobj: 0.327947
dobj: 0.327945
pinf: 1.63203e-014
dinf: 1.07066e-015
dgap: 2.33535e-006
Iteration: 12
pobj: 0.327946
dobj: 0.327945
pinf: 3.23075e-014
dinf: 1.06489e-015
dgap: 4.95477e-007
Iteration: 13
pobj: 0.327946
dobj: 0.327945
pinf: 6.36158e-014
dinf: 1.29948e-015
dgap: 7.05679e-008
Optimal solution found.
Iteration: 0
pobj: 0.119679
dobj: -0.961681
pinf: 2.22045e-016
dinf: 3.78626
dgap: 1.08136
Iteration: 1
pobj: 0.118384
dobj: 0.00272727
pinf: 2.22045e-016
dinf: 0.403377
dgap: 0.115657
Iteration: 2
pobj: 0.114706
dobj: 0.0825212
pinf: 4.44089e-016
dinf: 0.102386
dgap: 0.032185
Iteration: 3
pobj: 0.111867
dobj: 0.103416
pinf: 8.88178e-016
dinf: 0.0213116
dgap: 0.00845108
Iteration: 4
pobj: 0.110293
dobj: 0.107551
pinf: 1.55431e-015
dinf: 0.00386282
dgap: 0.00274137
Iteration: 5
pobj: 0.109629
dobj: 0.109107
pinf: 3.10862e-015
dinf: 0.000197921
dgap: 0.00052235
Iteration: 6
pobj: 0.109439
dobj: 0.109312
pinf: 5.77316e-015
dinf: 2.60714e-005
dgap: 0.000127236
Iteration: 7
pobj: 0.109374
dobj: 0.109344
pinf: 1.08802e-014
dinf: 1.79711e-006
dgap: 2.95826e-005
Iteration: 8
pobj: 0.109356
dobj: 0.109351
pinf: 2.17604e-014
dinf: 4.06545e-008
dgap: 5.02064e-006
Iteration: 9
pobj: 0.109352
dobj: 0.109351
pinf: 4.24105e-014
dinf: 1.89586e-009
dgap: 7.21367e-007
Iteration: 10
pobj: 0.109352
dobj: 0.109351
pinf: 8.26006e-014
dinf: 1.05037e-010
dgap: 6.15178e-008
Optimal solution found.
Iteration: 0
pobj: 0.400811
dobj: -0.794445
pinf: 3.31634
dinf: 4.89251
dgap: 16.3718
Iteration: 1
pobj: 0.435729
dobj: -0.58661
pinf: 0.31574
dinf: 0.465803
dgap: 2.38302
Iteration: 2
pobj: 0.436902
dobj: -0.0924833
pinf: 0.0618415
dinf: 0.0912331
dgap: 0.74363
Iteration: 3
pobj: 0.396171
dobj: 0.182021
pinf: 0.0126606
dinf: 0.0186778
dgap: 0.242216
Iteration: 4
pobj: 0.355871
dobj: 0.282165
pinf: 0.00174574
dinf: 0.00257544
dgap: 0.0755348
Iteration: 5
pobj: 0.339889
dobj: 0.326098
pinf: 3.55689e-005
dinf: 5.24738e-005
dgap: 0.0138033
Iteration: 6
pobj: 0.335674
dobj: 0.331253
pinf: 6.18329e-006
dinf: 9.12204e-006
dgap: 0.00442275
Iteration: 7
pobj: 0.333785
dobj: 0.332737
pinf: 1.9762e-014
dinf: 1.55034e-015
dgap: 0.00104714
Iteration: 8
pobj: 0.333341
dobj: 0.333071
pinf: 3.61933e-014
dinf: 1.15378e-015
dgap: 0.000269813
Iteration: 9
pobj: 0.333183
dobj: 0.333138
pinf: 7.01661e-014
dinf: 1.42178e-015
dgap: 4.4317e-005
Iteration: 10
pobj: 0.333152
dobj: 0.333146
pinf: 1.37002e-013
dinf: 1.04148e-015
dgap: 6.50229e-006
Iteration: 11
pobj: 0.333148
dobj: 0.333147
pinf: 2.68452e-013
dinf: 1.7519e-015
dgap: 1.11416e-006
Iteration: 12
pobj: 0.333147
dobj: 0.333147
pinf: 5.24025e-013
dinf: 1.95475e-015
dgap: 1.64139e-007
Optimal solution found.
Iteration: 0
pobj: 0.114165
dobj: -0.966275
pinf: 2.22045e-016
dinf: 3.78424
dgap: 1.08044
Iteration: 1
pobj: 0.112872
dobj: -0.00204408
pinf: 2.22045e-016
dinf: 0.400969
dgap: 0.114916
Iteration: 2
pobj: 0.108962
dobj: 0.0769199
pinf: 2.22045e-016
dinf: 0.102178
dgap: 0.0320425
Iteration: 3
pobj: 0.105721
dobj: 0.0934422
pinf: 4.44089e-016
dinf: 0.032301
dgap: 0.012279
Iteration: 4
pobj: 0.103022
dobj: 0.0998585
pinf: 8.88178e-016
dinf: 0.00211667
dgap: 0.00316377
Iteration: 5
pobj: 0.102134
dobj: 0.101455
pinf: 1.77636e-015
dinf: 0.000122782
dgap: 0.000678932
Iteration: 6
pobj: 0.10187
dobj: 0.101733
pinf: 3.33067e-015
dinf: 1.15584e-005
dgap: 0.00013755
Iteration: 7
pobj: 0.101798
dobj: 0.101771
pinf: 6.21725e-015
dinf: 1.67093e-006
dgap: 2.72901e-005
Iteration: 8
pobj: 0.101783
dobj: 0.101774
pinf: 1.06581e-014
dinf: 4.65597e-007
dgap: 8.87898e-006
Iteration: 9
pobj: 0.101776
dobj: 0.101775
pinf: 2.02061e-014
dinf: 4.9579e-008
dgap: 1.34903e-006
Iteration: 10
pobj: 0.101775
dobj: 0.101775
pinf: 3.88578e-014
dinf: 3.98419e-009
dgap: 1.51041e-007
Iteration: 11
pobj: 0.101775
dobj: 0.101775
pinf: 7.54952e-014
dinf: 2.43325e-010
dgap: 1.08439e-008
Optimal solution found.
Iteration: 0
pobj: 0.399783
dobj: -0.794994
pinf: 3.31488
dinf: 4.88105
dgap: 16.3266
Iteration: 1
pobj: 0.434178
dobj: -0.588119
pinf: 0.314923
dinf: 0.463713
dgap: 2.37577
Iteration: 2
pobj: 0.435608
dobj: -0.0929461
pinf: 0.0610592
dinf: 0.0899077
dgap: 0.738909
Iteration: 3
pobj: 0.395512
dobj: 0.182253
pinf: 0.0125265
dinf: 0.0184449
dgap: 0.240708
Iteration: 4
pobj: 0.355861
dobj: 0.28295
pinf: 0.00175211
dinf: 0.00257992
dgap: 0.0747253
Iteration: 5
pobj: 0.341244
dobj: 0.320681
pinf: 0.000266868
dinf: 0.000392954
dgap: 0.0206834
Iteration: 6
pobj: 0.335556
dobj: 0.329728
pinf: 3.02395e-005
dinf: 4.45267e-005
dgap: 0.00583581
Iteration: 7
pobj: 0.333705
dobj: 0.331908
pinf: 5.47769e-006
dinf: 8.06572e-006
dgap: 0.00179878
Iteration: 8
pobj: 0.332879
dobj: 0.332499
pinf: 2.90878e-014
dinf: 1.21619e-015
dgap: 0.000379786
Iteration: 9
pobj: 0.332692
dobj: 0.332604
pinf: 5.4845e-014
dinf: 1.09344e-015
dgap: 8.79728e-005
Iteration: 10
pobj: 0.332633
dobj: 0.33262
pinf: 1.06803e-013
dinf: 1.16969e-015
dgap: 1.23297e-005
Iteration: 11
pobj: 0.332622
dobj: 0.332621
pinf: 2.08056e-013
dinf: 5.97873e-016
dgap: 1.14505e-006
Iteration: 12
pobj: 0.332621
dobj: 0.332621
pinf: 4.05453e-013
dinf: 9.86788e-016
dgap: 6.83197e-008
Optimal solution found.
Iteration: 0
pobj: 0.11327
dobj: -0.96457
pinf: 0
dinf: 3.77327
dgap: 1.07784
Iteration: 1
pobj: 0.112092
dobj: -0.000673744
pinf: 6.58281e-017
dinf: 0.393348
dgap: 0.112766
Iteration: 2
pobj: 0.108576
dobj: 0.0775103
pinf: 1.11022e-016
dinf: 0.0993329
dgap: 0.031066
Iteration: 3
pobj: 0.105643
dobj: 0.0938764
pinf: 1.11022e-016
dinf: 0.0311067
dgap: 0.0117666
Iteration: 4
pobj: 0.103234
dobj: 0.100227
pinf: 2.22045e-016
dinf: 0.00206712
dgap: 0.0030073
Iteration: 5
pobj: 0.102495
dobj: 0.101849
pinf: 4.44089e-016
dinf: 0.000120302
dgap: 0.000645755
Iteration: 6
pobj: 0.10231
dobj: 0.102173
pinf: 8.88178e-016
dinf: 1.17049e-005
dgap: 0.000137607
Iteration: 7
pobj: 0.102266
dobj: 0.102244
pinf: 1.66533e-015
dinf: 9.16852e-007
dgap: 2.17722e-005
Iteration: 8
pobj: 0.10226
dobj: 0.102255
pinf: 2.9976e-015
dinf: 2.07813e-007
dgap: 5.65601e-006
Iteration: 9
pobj: 0.102258
dobj: 0.102257
pinf: 5.44009e-015
dinf: 4.18585e-008
dgap: 1.41695e-006
Iteration: 10
pobj: 0.102258
dobj: 0.102258
pinf: 1.05471e-014
dinf: 1.9544e-009
dgap: 1.75332e-007
Iteration: 11
pobj: 0.102258
dobj: 0.102258
pinf: 2.04281e-014
dinf: 8.59725e-011
dgap: 3.60605e-008
Optimal solution found.
Iteration: 0
pobj: 0.395682
dobj: -0.799794
pinf: 3.31125
dinf: 4.8802
dgap: 16.308
Iteration: 1
pobj: 0.430324
dobj: -0.593347
pinf: 0.311919
dinf: 0.459714
dgap: 2.36364
Iteration: 2
pobj: 0.430759
dobj: -0.0996301
pinf: 0.0623997
dinf: 0.091966
dgap: 0.745774
Iteration: 3
pobj: 0.391726
dobj: 0.178522
pinf: 0.0126767
dinf: 0.0186832
dgap: 0.241254
Iteration: 4
pobj: 0.352444
dobj: 0.279045
pinf: 0.00179454
dinf: 0.00264483
dgap: 0.0753015
Iteration: 5
pobj: 0.337886
dobj: 0.317213
pinf: 0.000270728
dinf: 0.000399006
dgap: 0.0207992
Iteration: 6
pobj: 0.33229
dobj: 0.326359
pinf: 3.08463e-005
dinf: 4.5462e-005
dgap: 0.00593959
Iteration: 7
pobj: 0.330519
dobj: 0.328551
pinf: 6.26728e-006
dinf: 9.23685e-006
dgap: 0.00196944
Iteration: 8
pobj: 0.329668
dobj: 0.329214
pinf: 1.77636e-015
dinf: 7.61131e-016
dgap: 0.000454159
Iteration: 9
pobj: 0.329481
dobj: 0.329373
pinf: 3.33067e-015
dinf: 1.18018e-015
dgap: 0.000107128
Iteration: 10
pobj: 0.329427
dobj: 0.329407
pinf: 6.43929e-015
dinf: 1.02958e-015
dgap: 1.98478e-005
Iteration: 11
pobj: 0.329416
dobj: 0.329412
pinf: 1.26565e-014
dinf: 8.9509e-016
dgap: 3.90656e-006
Iteration: 12
pobj: 0.329413
dobj: 0.329412
pinf: 2.4869e-014
dinf: 1.20601e-015
dgap: 6.37612e-007
Iteration: 13
pobj: 0.329412
dobj: 0.329412
pinf: 4.86278e-014
dinf: 1.21619e-015
dgap: 6.99926e-008
Optimal solution found.
Iteration: 0
pobj: 0.114706
dobj: -0.96241
pinf: 1.11022e-016
dinf: 3.77192
dgap: 1.07712
Iteration: 1
pobj: 0.113529
dobj: 0.00144866
pinf: 1.11022e-016
dinf: 0.391088
dgap: 0.11208
Iteration: 2
pobj: 0.110034
dobj: 0.0794452
pinf: 1.11022e-016
dinf: 0.0978975
dgap: 0.0305892
Iteration: 3
pobj: 0.107094
dobj: 0.0958683
pinf: 3.33067e-016
dinf: 0.0297254
dgap: 0.0112254
Iteration: 4
pobj: 0.104634
dobj: 0.101833
pinf: 6.66134e-016
dinf: 0.0017786
dgap: 0.00280165
Iteration: 5
pobj: 0.103814
dobj: 0.103204
pinf: 1.33227e-015
dinf: 8.40376e-005
dgap: 0.000610204
Iteration: 6
pobj: 0.103589
dobj: 0.103439
pinf: 2.55351e-015
dinf: 7.05916e-006
dgap: 0.000149631
Iteration: 7
pobj: 0.103526
dobj: 0.103502
pinf: 5.10703e-015
dinf: 3.9425e-008
dgap: 2.4386e-005
Iteration: 8
pobj: 0.103517
dobj: 0.103511
pinf: 9.4369e-015
dinf: 5.93066e-009
dgap: 6.41972e-006
Iteration: 9
pobj: 0.103514
dobj: 0.103512
pinf: 1.70974e-014
dinf: 1.16637e-009
dgap: 2.24055e-006
Iteration: 10
pobj: 0.103513
dobj: 0.103513
pinf: 3.37508e-014
dinf: 2.35238e-011
dgap: 4.19046e-007
Iteration: 11
pobj: 0.103513
dobj: 0.103513
pinf: 6.55032e-014
dinf: 1.41201e-012
dgap: 6.01232e-008
Optimal solution found.
Iteration: 0
pobj: 0.398638
dobj: -0.796719
pinf: 3.31318
dinf: 4.88079
dgap: 16.3186
Iteration: 1
pobj: 0.432431
dobj: -0.589456
pinf: 0.314857
dinf: 0.46383
dgap: 2.37497
Iteration: 2
pobj: 0.432625
dobj: -0.0999254
pinf: 0.0651942
dinf: 0.0960405
dgap: 0.759542
Iteration: 3
pobj: 0.394498
dobj: 0.179915
pinf: 0.0131196
dinf: 0.0193271
dgap: 0.244279
Iteration: 4
pobj: 0.354809
dobj: 0.279984
pinf: 0.00186453
dinf: 0.00274673
dgap: 0.0768724
Iteration: 5
pobj: 0.339842
dobj: 0.31936
pinf: 0.000257823
dinf: 0.000379811
dgap: 0.0206025
Iteration: 6
pobj: 0.334275
dobj: 0.328237
pinf: 3.03067e-005
dinf: 4.46462e-005
dgap: 0.00604621
Iteration: 7
pobj: 0.332443
dobj: 0.330413
pinf: 6.21768e-006
dinf: 9.15954e-006
dgap: 0.00203112
Iteration: 8
pobj: 0.331495
dobj: 0.331002
pinf: 1.13243e-014
dinf: 1.21619e-015
dgap: 0.000492231
Iteration: 9
pobj: 0.331266
dobj: 0.331155
pinf: 2.16493e-014
dinf: 1.54636e-015
dgap: 0.000110334
Iteration: 10
pobj: 0.331204
dobj: 0.331186
pinf: 4.22995e-014
dinf: 5.76889e-016
dgap: 1.80047e-005
Iteration: 11
pobj: 0.331191
dobj: 0.331189
pinf: 8.24896e-014
dinf: 9.48575e-016
dgap: 2.06416e-006
Iteration: 12
pobj: 0.331189
dobj: 0.331189
pinf: 1.60871e-013
dinf: 1.14304e-015
dgap: 1.38534e-007
Optimal solution found.
Iteration: 0
pobj: 0.11477
dobj: -0.961072
pinf: 0
dinf: 3.76706
dgap: 1.07584
Iteration: 1
pobj: 0.113632
dobj: 0.00263219
pinf: 2.22045e-016
dinf: 0.387308
dgap: 0.111
Iteration: 2
pobj: 0.110322
dobj: 0.0805013
pinf: 2.22045e-016
dinf: 0.095544
dgap: 0.0298208
Iteration: 3
pobj: 0.107587
dobj: 0.0965035
pinf: 4.44089e-016
dinf: 0.0295032
dgap: 0.0110831
Iteration: 4
pobj: 0.105282
dobj: 0.102542
pinf: 8.88178e-016
dinf: 0.00180192
dgap: 0.00273982
Iteration: 5
pobj: 0.10454
dobj: 0.103948
pinf: 1.77636e-015
dinf: 9.80713e-005
dgap: 0.000592444
Iteration: 6
pobj: 0.104348
dobj: 0.10421
pinf: 3.33067e-015
dinf: 8.12006e-006
dgap: 0.000138232
Iteration: 7
pobj: 0.104303
dobj: 0.104281
pinf: 6.43929e-015
dinf: 3.19953e-007
dgap: 2.13694e-005
Iteration: 8
pobj: 0.104297
dobj: 0.104295
pinf: 1.24345e-014
dinf: 2.46719e-008
dgap: 2.67608e-006
Iteration: 9
pobj: 0.104297
dobj: 0.104296
pinf: 2.22045e-014
dinf: 5.00253e-009
dgap: 6.11958e-007
Iteration: 10
pobj: 0.104297
dobj: 0.104297
pinf: 4.13003e-014
dinf: 6.97906e-010
dgap: 1.06644e-007
Iteration: 11
pobj: 0.104297
dobj: 0.104297
pinf: 7.92699e-014
dinf: 5.72651e-011
dgap: 1.18903e-008
Optimal solution found.
Iteration: 0
pobj: 0.396694
dobj: -0.797435
pinf: 3.31531
dinf: 4.86496
dgap: 16.2746
Iteration: 1
pobj: 0.429327
dobj: -0.591061
pinf: 0.316147
dinf: 0.463922
dgap: 2.37423
Iteration: 2
pobj: 0.430326
dobj: -0.102681
pinf: 0.0657035
dinf: 0.0964149
dgap: 0.761109
Iteration: 3
pobj: 0.393718
dobj: 0.179723
pinf: 0.0131117
dinf: 0.0192405
dgap: 0.243563
Iteration: 4
pobj: 0.355241
dobj: 0.2805
pinf: 0.00187089
dinf: 0.00274539
dgap: 0.0767899
Iteration: 5
pobj: 0.340985
dobj: 0.320463
pinf: 0.000260152
dinf: 0.000381753
dgap: 0.0206442
Iteration: 6
pobj: 0.335827
dobj: 0.329761
pinf: 3.0625e-005
dinf: 4.49398e-005
dgap: 0.00607401
Iteration: 7
pobj: 0.334207
dobj: 0.332124
pinf: 6.53239e-006
dinf: 9.58579e-006
dgap: 0.00208414
Iteration: 8
pobj: 0.333389
dobj: 0.332899
pinf: 2.84217e-014
dinf: 5.08768e-016
dgap: 0.000490514
Iteration: 9
pobj: 0.333219
dobj: 0.333109
pinf: 5.37348e-014
dinf: 9.08757e-016
dgap: 0.000110014
Iteration: 10
pobj: 0.333181
dobj: 0.333165
pinf: 1.05027e-013
dinf: 1.12127e-015
dgap: 1.65065e-005
Iteration: 11
pobj: 0.333176
dobj: 0.333173
pinf: 2.05169e-013
dinf: 1.41308e-015
dgap: 2.15942e-006
Iteration: 12
pobj: 0.333174
dobj: 0.333174
pinf: 4.00346e-013
dinf: 1.39553e-015
dgap: 2.62284e-007
Optimal solution found.
Pfaff (Invesco) MCO RFinance 2016 18 / 24
Example: Multi-asset class portfolio
Example: Multi-asset class portfolioR code: backtest, part III
> W <- list("MCO" = Wmco, "MSR" = Wmsr, "MDP" = Wmdp,
+ "GMV" = Wgmv, "ERC" = Werc)
> E <- lapply(W, function(x)
+ wTs <- timeSeries(x, charvec = ep)
+ wTsL1 <- lag(wTs, 1)
+ RetFac <- 1 + rowSums(R[ep, ] * wTsL1) / 100.0
+ RetFac[1] <- 100
+ timeSeries(cumprod(RetFac), charvec = ep)
+ )
> cols <- topo.colors(6)
> plot(E[[1]], lwd = 2,
+ ylab = "Index", xlab = "", col = cols[1],
+ main = "Comparison of Allocation Strategies")
> lines(E[[2]], col = cols[2])
> lines(E[[3]], col = cols[3])
> lines(E[[4]], col = cols[4])
> lines(E[[5]], col = cols[5])
> legend("topleft",
+ legend = c("MCO", "MSR", "MDP", "GMW", "ERC"),
+ col = cols, lty = 1, lwd = 2)
> abline(h = 100, col = "gray")
Comparison of Allocation Strategies
Inde
x
2010−01−01 2010−07−01 2011−01−01 2011−07−01
100
102
104
106
108
110
112
MCOMSRMDPGMWERC
Pfaff (Invesco) MCO RFinance 2016 19 / 24
Example: Multi-asset class portfolio
Example: Multi-asset class portfolioR code: backtest, part IV
> Rstrat <- matrix(unlist(lapply(E, Return.calculate)), ncol = 5)
> RstratTs <- na.omit(xts(Rstrat, order.by = as.Date(ep)))
> Bench <- xts(rep(0, nrow(RstratTs)), order.by = as.Date(ep)[-1])
> S1 <- as.matrix(table.AnnualizedReturns(RstratTs, Rf = Bench,
+ scale = 12))
> S2 <- VaR(RstratTs)
> ans <- rbind(S1, -100 * S2)
> colnames(ans) <- c("MCO", "MSR", "MDP", "GMV", "ERC")
> rownames(ans) <- c("Return (p.a.)", "StdDev. Risk (p.a.)",
+ "Sharpe Ratio", "VaR (p.a.)")
> round(ans, 3)
MCO MSR MDP GMV ERC
Return (p.a.) 0.061 0.060 0.058 0.051 0.060
StdDev. Risk (p.a.) 0.038 0.039 0.037 0.034 0.034
Sharpe Ratio 1.605 1.532 1.585 1.513 1.746
VaR (p.a.) 1.301 1.380 1.308 1.211 1.017
Pfaff (Invesco) MCO RFinance 2016 20 / 24
Summary
Summary
Aiding decision makers by making portfolio choices for conflictingobjectives (a posteriori analysis).
Allows amendment of classical portfolio optimization formulations(e.g. GMV, ERC, MDP and/or MSR) by additional goals.
MCDM: For tri-criterion formulations, depiction of solutions by afourth portfolio characteristic/measure is feasible by means of ternaryplots.
Caveat/strength of EMO: It is at the user’s discretion to chose his’optimal’ allocation out of the Pareto efficient set, which might be achallenge on its own.
Pfaff (Invesco) MCO RFinance 2016 21 / 24
Bibliography
Bibliography I
Akima, H. and A. Gebhardt (2015). akima: Interpolation of Irregularly andRegularly Spaced Data. R package version 0.5-12.
Deb, K. (2001). Multi-Objective Optimization using Evolutionary Algorithms.Chichester, UK: John Wiley & Sons, Ltd.
Deb, K. (2011). Multi-objective Evolutionary Optimisation for Product Designand Manufacturing, Chapter Multi-Objective Optimization Using EvolutionaryAlgorithms: An Introduction, pp. 3–34. New York, NY: Springer.
Deb, K., R. Steuer, R. Tewari, and R. Tewari (2011). On the effectiveness of ansga-ii local search approach customized for portfolio optimization. KanGALReport 2011007, Indian Institute of Technology Kanpur, Kanpur, India.
Ehrgott, M. (2005). Multicriteria Optimization (second ed.). Berlin, Heidelberg,New York: Springer.
Goldberg, D. (1989). Genetic Algorithms in Search, Optimization, and MachineLearning. New York, NY: Addison-Wesley.
Pfaff (Invesco) MCO RFinance 2016 22 / 24
Bibliography
Bibliography II
Hamilton, N. (2015). ggtern: An Extension to ’ggplot2’, for the Creation ofTernary Diagrams. R package version 1.0.6.1.
Hirschberger, M., R. Steuer, S. Utz, and M. Wimmer (2013, January–February).Computing the nondominated surface in tri-criterion portfolio selection.Operations Research 61(1), 169–183.
Jones, D. and M. Tamiz (2010). Pratical Goal Programming. International Seriesin Opertations Research and Management Science. New York: Springer.
K., D., A. Pratap, S. Agarwal, and T. Meyarivan (2002). A fast and elitistmultiobjective genetic algorithm: Nsga-ii. IEEE Transaction on EvolutionaryComputation 6(2), 181–197.
Ligges, U. and M. Machler (2003). Scatterplot3d - an r package for visualizingmultivariate data. Journal of Statistical Software 8(11), 1–20.
Mersmann, O. (2014). mco: Multiple Criteria Optimization Algorithms andRelated Functions. R package version 1.0-15.1.
Mitchell, M. (1996). An Introduction to Genetic Algorithms. Cambridge, MA:MIT Press.
Pfaff (Invesco) MCO RFinance 2016 23 / 24
Bibliography
Bibliography III
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