antigone: algorithms for continuous / integer...

Noname manuscript No.(will be inserted by the editor)

ANTIGONE: Algorithms for coNTinuous / Integer GlobalOptimization of Nonlinear Equations

Ruth Misener · Christodoulos A. Floudas

Received: date / Accepted: date

Abstract This manuscript introduces ANTIGONE, Algorithms for coNTinuous / IntegerGlobal Optimization of Nonlinear Equations, a general mixed-integer nonlinear global op-timization framework. ANTIGONE is the evolution of the Global Mixed-Integer QuadraticOptimizer, GloMIQO, to general nonconvex terms. The purpose of this paper is to showhow the extensible structure of ANTIGONE realizes our previously-proposed mixed-integerquadratically-constrained quadratic program (MIQCQP) and mixed-integer signomial opti-mization (MISO) computational frameworks. To demonstrate the capacity of ANTIGONE,this paper presents computational results on a test suite of 2571 problems from standardlibraries and the open literature; we compare ANTIGONE to other state-of-the-art globaloptimization solvers.

Keywords MINLP; deterministic global optimization; optimization software; branch-and-cut

C.A.F. is thankful for support from the National Science Foundation (CBET – 0827907). This material isbased upon work supported by both the National Science Foundation Graduate Research Fellowship to R.M.under Grant No. DGE-0646086 and by the Royal Academy of Engineering Research Fellowship to R.M.

R. Misener · C. A. FloudasDepartment of Chemical and Biological EngineeringPrinceton UniversityPrinceton, NJ 08544-5263; USATel.: +1-609-258-4595Fax: +1-609-258-0211E-mail: [email protected]

Present address: of R. MisenerDepartment of Chemical EngineeringImperial College LondonSouth Kensington SW7 2AZ; UK

2 Ruth Misener, Christodoulos A. Floudas

1 Introduction

Deterministic global optimization of mixed-integer nonlinear programs (MINLP) is broadlyapplicable in diverse domains ranging from molecular biology to refinery operations to com-putational chemistry to synthesizing sustainable processes [17, 24, 50–52, 54, 63, 64, 101,103, 136]. Our recent work developed solution strategies for two classes of deterministicglobal optimization problems: mixed-integer quadratically constrained quadratic programs(MIQCQP) and mixed-integer signomial optimization problems (MISO) [102, 104, 105,107]. A software implementation illustrated and validated the proposed MIQCQP and MISOcomputational frameworks; we further collaborated with the GAMS Development Corpora-tion to publicly release the MIQCQP framework as the Global Mixed-Integer Quadratic Op-timizer (GloMIQO; first available in GAMS 23.8). GloMIQO has been subsequently usedfor applications including: optimal crystal orientation [62]; multiperiod blend scheduling[75, 76]; network flow [131]; water network design [35, 73]; cutting ellipses from area-minimizing rectangles [69].

This manuscript presents ANTIGONE (Algorithms for coNTinuous / Integer Global Op-timization of Nonlinear Expressions; first available in GAMS 24.1), a computational frame-work and associated software implementation for globally optimizing nonconvex MINLP.ANTIGONE integrates our previous MIQCQP [102, 104, 107] and MISO [105] work intoa cohesive whole; ANTIGONE further generalizes the computational framework to includeexponential and logarithmic functions.

The purpose of this paper is to show how the extensible structure of ANTIGONE re-alizes the MIQCQP and MISO computational frameworks [102, 104, 105, 107]. Section2 defines MINLP and mentions several MINLP global optimization solvers. We developANTIGONE in Section 3 as a global optimization framework exploiting an array of specialmathematical structure components. Section 3 may be read as a type of critical literature re-view; the intellectual contribution is the way ANTIGONE ties together its many constitutivealgorithms. After defining a test suite of 2571 optimization problems from standard librariesand the open literature in Section 4, we present extensive computational results in Section 5comparing ANTIGONE to other state-of-the-art global optimization solvers. We concludein Section 6.

2 Problem Definition

MINLP is defined:

min f0(x, y, z)

s.t. bLOm fm(x, y, z) bUP

m 8 m 2 {1, . . . , M}

x 2 RC; y 2 {0, 1}B ; z 2 ZI

(MINLP)

ANTIGONE 3

where C, B, I, and M represent the number of continuous variables, binary variables, in-teger variables, and constraints, respectively. Parameters vectors bLO

m and bUPm bound the

constraints. We assume that it is possible to infer finite bounds⇥xL

i , xUi⇤

on the variables par-ticipating in nonlinear terms fm and that the image of fm is finite on x. Typical expressionsfor f0(x, y, z) and fm(x, y, z) are:

fm(x, y, z) = cm +aTm (x, y, z)+(x, y, z)T Qm (x, y, z)

+Sm

Âs=1

csm ·C

’c=1

xpsm ,cc +

Em

Âe=1

cem · ex +Lm

Â=1

c`m · logx

where the powers psm,c are constant reals; cm, am, Qm, csm , cem , c`m are constant coefficients;(x, y, z) is a combined vector of variables; Sm, Em, Lm are the number of signomial, expo-nential, and logarithmic terms, respectively.

This manuscript refers exclusively to the functional forms fm presented in Table 1, butnote that the ANTIGONE software implementation addresses a wider set of terms throughpreprocessing transformations. For example, ANTIGONE transforms logarithms of baseb into natural logarithms using the transformation logb(x) = log(x)/ log(b) and changespowers with constant base b into exponents (bx = elog(b)·x).

Global optimization of MINLP is a topic of active research. We therefore limit ourselvesto mentioning several excellent review articles and texts [17, 24, 40, 49–52, 54, 63, 64,101, 103, 113, 136, 145] and hereafter discuss the items most relevant to ANTIGONE.The C++ implementation of ANTIGONE belongs to the class of numerical optimizationsoftware addressing MINLP; refer to the review of Bussieck and Vigerske [26] for completetreatment. The existing numerical solvers most relevant to ANTIGONE are: aBB [4, 5, 10,50, 93]; BARON [136–138]; Couenne [16, 88]; GloMIQO [102, 104, 107]; LINDO [58, 85];SCIP [1, 3, 18, 19, 145]; SGO [89–92]. ANTIGONE performs equivalently to GloMIQO[102, 104, 107] when all nonlinearities in MINLP are quadratic.

3 Global optimization Framework

The overarching structure of the ANTIGONE framework is diagrammed in Figure 1; Figure2 illustrates the inheritance structure of four major ANTIGONE base classes. Sections 3.1– 3.3 describe the component parts of: reformulating input; finding special mathematicalstructure; branch-and-cut global optimization.

3.1 Reformulating User Input

Like aBB [4, 5, 10, 50, 93] and SGO [89–92], ANTIGONE emphasizes term-based under-estimators [105]. ANTIGONE therefore begins by reformulating the user-defined MINLP


Fig. 1: Given user-defined MINLP, the framework reformulates the model, detects specialmathematical structure, solves the optimization problem, and returns the model with respectto the original variables

so that each equation is expressed as a sum of the terms outlined in Table 1. As mentionedin Section 2, ANTIGONE will equivalently rewrite the bases of powers and logarithms toreach the terms in Table 1. For more complex terms, ANTIGONE introduces auxiliary vari-ables [16, 57, 130, 136–138, 145]; this strategy is in contrast to systems exclusively usingunderestimators that do not introduce auxiliary variables [4, 5, 92, 95, 108].

Figure 2(a) diagrams the inheritance structure of base class term; this implementationis designed in recognition of an accelerating trend towards developing good underestima-tors for very specific terms (e.g., [32, 61, 71, 72, 83, 86, 89, 91, 93, 94, 98–100, 116, 129,132–136, 139]). Every term needs to know its: participating variable(s); regions of con-vexity/concavity and edge-convexity/edge-concavity; partial derivatives; etc. The polymor-phic strategy illustrated in Figure 2(a) allows us to easily integrate many disparate termtypes into a single framework; we also avoid duplicating code for similar types. Observethat specialized terms which are not at this writing integrated into the framework (e.g.,[61, 71, 72, 86, 129, 139]) could be easily added to Figure 2(a); we anticipate further ad-vances in the area of term-specific underestimators and have architected ANTIGONE ac-cordingly.

The only three term types in Figure 2(a) not analyzed in the open literature are thecomposite univariate terms (exponential; fractional; logarithmic). These composite termsregularly appear in standard libraries: GLOBALLib (e.g., arki0018, chem, ex6 1 1,ex6 1 2, ex14 1 8, filter); MINLPLib (contvar); AMPL Book Library (nltrans);PrincetonLib (e.g., bigbank, s367, s377). ANTIGONE knows analytic expressions forthe second derivatives of these special composite terms and performs interval arithmetic on

ANTIGONE 5

Table 1: Each derived class of base class Term is associated with a specific mathematicalexpression and several relaxation strategies

Term Type Mathematical Form Relaxation Strategies

Constant c –

Bilinear /Quadratic

x1 · x2 McCormick Hull [8, 95]Outer Approximation [45, 46, 49, 59]

MultivariateSignomial

xa11 · xa2

2 · · ·xann Edge-Concave [100, 102, 132, 133]

Exponential Transformation [89–91, 94]Fractional [116, 135]Outer Approximation [59, 93]

AbsoluteValue |a · x+b| Equivalent MILP [105]Representation

Exponential ea·x+b Secant LineOuter Approximation [45, 46, 49, 59]

Linear a · x+b –

Logarithmic log(a · x+b) Secant LineOuter Approximation [45, 46, 49, 59]

UnivariateSignomial

xa Odd Degree Monomial [83]Secant LineOuter Approximation [45, 46, 49, 59]

CompositeExponent (a1 · x+b1) · ea2·x+b2 Secant Line

Fractional a1·x+b1a2·x+b2

Outer Approximation [45, 46, 49, 59]Logarithm (a1 x+b1) log(a2 x+b2)

d2 f/dx2 to determine regions of convexity/concavity:

ExponentialComposite

8><

>:

fe(x) = (a1 · x+b1) · ea2·x+b2

+d2 fedx2 = (a1 ·a2 ·a2 · x+2 ·a1 ·a2 +a2 ·a2 ·b1) · ea2·x+b2

FractionalComposite

8>><

>>:

f f (x) = a1·x+b1a2·x+b2

+d2 f fdx2 =�2 ·a2 · a1·b2�a2·b1

(a2·x+b2)3

LogarithmicComposite

8>><

>>:

f`(x) = (a1 · x+b1) · log(a2 · x+b2)

+d2 f`dx2 = a1·a2·(a2·x+b2)+a1·a2·b2�a2·a2·b1

(a2·x+b2)2


(a) Term

(b) Expression (c) Upper Bounding Solver

(d) Variable

Fig. 2: Inheritance Structure of Four ANTIGONE Base Classes: (a) Term; (b) Expression isa list of pointers to Term objects with associated coefficients; (c) Upper Bounding Solveris the NLP solver interface; (d) Variable

After transforming the initial user-defined MINLP to an equivalent MINLP that exclusivelyuses the terms in Table 1, ANTIGONE performs all of the GloMIQO [102, 104, 107] andthe MISO framework [105] reformulations. The only major additions to the reformulationstrategies specifically address exponential and logarithmic functions. As in the MISO frame-

ANTIGONE 7

work [105], one goal of reformulation is to flatten the expression tree (i.e., find equivalentfunctional forms with fewer layers of auxiliary variables). ANTIGONE eliminates a levelfrom a typical factorable programming tree with each of the following two reformulations:

Flatten Logarithms⇢

log(xa1 · xb

2) ) a · logx1 +b · logx2

Flatten Exponentials⇢

ex1 · ex2 ) ex1+x2

(1)

3.2 Elucidating Special Mathematical Structure

After reformulating the user-defined MINLP, ANTIGONE detects special mathematicalstructure that it will exploit in the branch-and-cut phase (Section 3.3). The types of specialstructure that ANTIGONE considers are: reformulation-linearization technique (RLT) equa-tions; convexity/concavity; edge-convexity/edge-concavity; aBB relaxations; term-specificunderestimators. Our previous work has shown how to find and use these components[102, 104, 105, 107]. To summarize:

– RLT multiplies every pairwise combination of: variables; nonlinear terms; equations[14, 84, 104, 105, 107, 124–128]. ANTIGONE saves the combinations that do not in-troduce new terms into the model formulation and updates these equations at every nodeof the branch-and-cut tree. Special RLT equations are added directly to the model formu-lation; other RLT equations are used as cutting planes and integrated into the feasibility-based bounds tightening routines [102, 104, 105, 107].

– Convexity/Concavity permits the easy generation of a cutting plane at a point x:

f (x)� f (x)+ f 0(x) · (x� x) (convex)f (x) f (x)+ f 0(x) · (x� x) (concave)

Based on interval arithmetic, terms and multi-term expressions are labeled as always/sometimes/never convex/concave; this information is used in the branch-and-cut phase[105, 107].

– Edge-Convexity/Edge-Concavity implies a vertex polyhedral envelope; ANTIGONElabels terms and multi-term expressions as always/sometimes/never edge-convex/edge-concave by applying interval arithmetic to a simple test of Tardella [100, 132–134].

– aBB underestimators convexify an expression with univariate quadratics [4, 5, 10, 50,93]; ANTIGONE uses aBB to relax aggregates of bilinear terms [13, 107].

– Term-Specific Underestimators are outlined in Table 1.

ANTIGONE organizes the special structure analysis of individual equations using thedata structure expression; Figure 2(b) depicts the inheritance structure of this base class.


Each expression object is a list of terms and associated coefficients; each equation alsocontains a list of multi-term expression objects forming disjoint vertex sets in the graphrepresentation of the equation. The class RLT equation is derived from equation; its coef-ficients may change from node to node of the branch-and-cut tree. Objects of type cuttingplane may be developed at any node of the branch-and-cut tree; a cutting plane knows thenode at which it was created and whether it is globally or locally valid (i.e., whether it maybe applied to any node of the branch-and-cut tree or not). For semidefinite programming-like relaxations (see [107]), ANTIGONE also maintains an object with knowledge of everynonlinear term in the MINLP.

3.3 Branch-and-cut Global optimization

After the reformulation and special structure detection phases, ANTIGONE initiates a branch-and-cut global optimization algorithm that: generates tight convex underestimators; dynam-ically generates separating hyperplanes; bounds the variables [4, 5, 10, 14, 16, 43, 44, 77,118, 119, 127, 128]; branches on the search space [2, 16]; finds feasible solutions. Thisstrategy is entirely equivalent to our MISO framework [105]; the additional term types inANTIGONE do not alter the branch-and-cut implementation because the branch-and-cutphase communicates with the term base class rather than with each individual term type.Refer to the book of Floudas [50] for complete treatment.

If ANTIGONE finds, in preprocessing, that an NLP is convex, it will activate a localNLP solver at the root node of the branch-and-cut tree and consider a certificate of optimal-ity from a local solver to be the global optimum. ANTIGONE detects convexity in individualterms and aggregates of quadratic terms; it does not, at this writing, apply function compo-sition rules or symbolic methods (e.g, [56, 111, 145]). For convex problems where the localNLP solver does not return a certificate of optimality at the root node, ANTIGONE revertsto its normal branch-and-cut apprach.

In the current software implementation, ANTIGONE interfaces with: CPLEX for LPand MIP relaxations; CONOPT or SNOPT for finding feasible solutions (default: CONOPT);LAPACK [9] for solving linear systems of equations; Boost [22, 23] for validated intervalarithmetic. The code is itself built as a library and linked to GAMS via an adaptation ofGAMSLinks [88, 144].

4 Definition of the MINLP Test Suite

To validate ANTIGONE, Table 2 defines a test suite of 2571 problems from standard li-braries and the open literature; Tables 9 – 18 in the online supplementary material (Ap-pendix A) give more detailed analysis as to the size and complexity of the individual prob-lems. The examples excluded from the standard test libraries are those which either include

ANTIGONE 9

Table 2: MINLP Test Suite of 2571 Problems

Problem Class # Cases Discrete Source

minlp.org

Belgian Chocolate Problem 4 X [38, 39]Cascading Tanks 8 X [60]Cyclic Scheduling and Control 1 X [48]Distillation Sep. Sequences 1 X [28–31]Heat Exchanger Networks 3 X [47, 146]Metabolic Networks 2 X [66, 67]Multi-Product Batch Plants 2 X [65, 74, 148]Periodic Scheduling 13 X [33, 34]Pooling 19 X [106]Supply Chain Design 2 X [147, 149]Three-Echelon Supply Chain 8 X [114, 115, 150]Unit Commitment 2 X [112, 151]Water Distribution Network 8 X [21, 41]Water Treatment Network 18 X [6, 7, 70, 117]Crude Oil Scheduling 24 X [78, 109, 110]

MIQCQP

BoxQP 90 [25, 142, 143]Circles & Polygon Nesting 35 [68]Crude Oil Scheduling 7 X [79, 80]Multi-Period Blending 7 X [75, 76]Natural Gas Production 3 X [81, 82, 123]Point Packing 14 [12]QAP 67 [11, 87]Randomly-Generated QCQP 135 [15]Reform. GLOBALLib 32 [53, 97, 121]Reform. MINLPLib 10 X [27, 53]Standard QP 15 [122]Water Treatment System 32 [36, 37]Water-Using Network 35 [35, 140, 141]

TestLibraries

AMPL Book Lib 34 [55]Bonmin Test Set 134 X [20, 120]GLOBALLib 368 [53, 97]LINLib 39MacMOOPLib 13MPLLib 21MINLPLib 249 X [27, 53]Convex PrincetonLib 459Nonconvex PrincetonLib 657

functional forms that the current ANTIGONE implementation cannot handle (e.g., trigono-metric, min/max, errorf) or have known feasible points where the objective value is minusinfinity (plus infinity if MINLP is a maximization problem).


The standard libraries are dynamic entities; we are using the latest versions as of 11March 2013. The other test cases are from the open literature. The minlp.org test set isunusually difficult; all 115 problems are less than 5 years old and industrially relevant. Asof 4 March 2013 there is a group of modeling files labeled Test Set on minlp.org; this isunfortunately unusable as a test set for solver comparisons. Each modeling file may containmultiple MINLP problems and some modeling files require user input arguments. As it isunclear how to use the files labeled Test Set on minlp.org, we have compiled a collectionof problems from the minlp.org MINLP Library; Table 2 documents their origin.

5 Testing the Framework

To test the software implementation of the ANTIGONE computational framework, we com-pared the 2571 test problems outlined in Table 2 to several state-of-the-art global optimiza-tion solvers available in GAMS 24.2.1: BARON 12.7.3 [138], Couenne 0.4 [16], LINDO8.0 [58, 85], and SCIP 3.0 [145]. All of the solvers are subject to active development; theseare the latest versions as of February 2014.

We ran each of the 5 · 2571 = 12855 computational experiments under 4 terminationcriteria: (1) a relative optimality gap e = UB�LB

|LB| 1⇥ 10�6 = 1⇥ 10�4%; (2) an absoluteoptimality gap e = UB�LB = 1⇥ 10�6; (3) a 7200 s time limit; (4) an iteration limit of9 · 109. No other parameters were altered from default. Following the recommendation ofDolan and More [42], each of the 5 solvers addressed a test problem successively so asminimize the effect of fluctuating machine load. We performed our computational studiesrunning 64 bit Linux on an Intel Xeon X5650 2.67 GHz processor with 24 GB RAM.

We considered the 2571 test cases outlined in Table 2; the results are in Tables 24 – 43(online supplement). After solving each of the 2571 test cases using GAMS, a Perl scriptasserted the feasibility of the solution returned by each solver. Instances where solvers re-turned infeasible points were automatically relabeled as failures (fl). Therefore, dashes (–)in columns LB and UB of Tables 24 – 43 indicates a benign failure (the solver did not re-turn a solution) while the demarcation fl indicates that the solver returned a point violating avariable bound or constraint equation by max(10�4, 10�4 · |V |) or more where V is the valueunder consideration.

Comparisons between the solvers are based on performance profiles [42]:

tp,s ⌘ Performance metric (e.g., time in seconds) for problem p by solver s 2 S

rp,s ⌘tp,s

min�

tp,s0 : s0 2 S 8 p 2 P

; s 2 S

rs(t) =1np

size�

p 2 P : rp,s t

ANTIGONE 11

Figures 3 – 16, which diagram rs (log10 (t)) as a function of log10 (t), were generated us-ing the GAMS Performance Tools with options: colselect; useobjest; bnd=1e-4;gaptol=1e-4 (PTOOLS 1.1; www.gamsworld.org/performance/paver/; ac-cessed 11 March 2013). PTOOLS takes trace files returned by the solver and automaticallycreates performance profiles. The only processing done in transit between trace files gener-ated by the solvers and the PTOOLS input was to penalize solvers for returning infeasiblesolutions and to add a column for the colselect option.

Each of the Figures 3 – 16 has a subfigure (a) and (b). Subfigures (a) are the output ofPTOOLS using solver reported time as a resource measurement and the objest option toenforce an objective gap of 10�4. Paraphrasing the PTOOLS output, a solver is consideredoptimal if it has the proper model and solver return codes and the relative objective valueerror is within 10�4 of the best possible solution. If the best found objective value is 0, thenthe absolute objective value error is used. Subfigures (b) use the colselect command(and switch off useobjest); this allows us to compare the solvers with respect to theiroptimality gap at termination (100 · UB�LB

|LB| ); this column in the GAMS trace file was createdwith the same Perl script that asserts feasibility of the solution. Only deterministic lowerbounds LB are reported; lower bounds based on guessed variable bounds are discarded.

5.1 Results of the Computational Study

Tables 24 – 43 in the online supplement give the results of the computational study with2571 total problems; Figure 3 displays the performance profiles for the 2112 problems witha nonconvex component (i.e., a second-order test is not enough to prove global optimality).Observe in Figure 3 that the strong performance of ANTIGONE across its entire test suiteis the compilation of strong performance across multiple test libraries (Figures 4 – 9). Seein Figure 3(b) that ANTIGONE is particularly good at coming to a finite optimality gap fordifficult problems; this is useful for contexts where globally optimizing a problem is beyondthe reach of current technology but there is still a need to understand bounds on the globalsolution. Sections 5.2 – 5.7 describe the performance of ANTIGONE with respect to severaltypes of test sets; Sections 5.8 – 5.10 classify the results with respect to types of problems.

5.2 Results: 115 MINLP.ORG Problems

Figure 4 and Tables 40 & 41 in the online supplement display results of ANTIGONE onthe 115 minlp.org test problems; obeserve that ANTIGONE is unusually strong on theserecently-developed, industrially-relevant problems. As an example, ngw-r1-53050 is alarge example with: 3471 continuous variables; 1680 discrete variables; 6581 equations; 80nonlinear terms [114, 115, 150]. While none of the five solvers can globally optimize this


(a) Time s: Log10 Scale (b) Remaining Gap at 7200 s: Log10 Scale

Fig. 3: 2112 Total Test Cases with a Nonconvex Component: Performance Profile (a)compares the CPU time for 2112 Total Test Cases with a Nonconvex Component; Profile(b) diagrams the percent gap remaining at 7200 s

problem, Table 3 shows that ANTIGONE terminates with a 2.20% gap; ANTIGONE andSCIP find better feasible solutions than the other solvers.

Table 3: minlp.org Test Set: Results of ngw-r1-53050 [114, 115, 150]

Time (s) Gap LB UB

ANTIGONE 1.1 – 2.20e+00 3.646e+06 3.728e+06BARON 12.7.3 – 3.57e+01 3.684e+06 5.733e+06Couenne 0.4 – 4.96e+01 2.742e+06 5.437e+06LINDO 8.0 – 6.20e+01 2.677e+06 7.039e+06SCIP 3.0 – 1.55e+01 3.151e+06 3.729e+06Gap ⌘ 100 ·

⇣UB�LB|LB|

⌘; Termination Criteria: Gap = 1⇥10�4% or Time = 7200 s

5.3 Results: 482 MIQCQP Problems

ANTIGONE performs especially well on the 482 MIQCQP problems from the open lit-erature; see the results in Figure 5 and Tables 42 & 43 in the online supplement. Thegood performance on the MIQCQP test set is a result of ANTIGONE performing equiv-alently to GloMIQO in the case of MIQCQP. Table 4 compares solver performance oncastro etal 2007 wts Ex16 M1 [36, 37]; this example has 88 continuous variables,54 equations, and 192 bilinear terms. ANTIGONE is the only solver able to globally opti-

ANTIGONE 13


Fig. 4: 115 MINLP.ORG Problems: Performance Profile (a) compares the CPU time for115 MINLP.ORG Problems; Profile (b) diagrams the percent gap remaining at 7200 s

mize the problem within the time limit. Couenne returns an infeasible solution and is there-fore excluded from Table 4.

Table 4: MIQCQP Test Set: Results of castro etal 2007 wts Ex16 M1 [36, 37]

Time (s) Gap LB UB

ANTIGONE 1.1 26 1.00e-04 1.359e+03 1.359e+03BARON 12.7.3 – 1.85e+01 1.108e+03 1.359e+03LINDO 8.0 – 2.19e+01 1.063e+03 1.361e+03SCIP 3.0 – 1.76e+00 1.335e+03 1.359e+03Gap ⌘ 100 ·

⇣UB�LB|LB|


5.4 Results: 368 GLOBALLib Problems

ANTIGONE performs competitively on the GLOBALLib test set; Figure 6 and Tables 28 &29 in the online supplement present complete results. For example, Table 5 compares solverperformance on a phase equilibrium problem ex6 2 9 [53, 96, 97]. The objective functionof ex6 2 9 is (in part):

(31.483x0 +6x1) log(4.827x0 +0.92x1)+11.503x0 log✓

x0

4.827x0 +0.92x1

◆. (2)

ANTIGONE flattens the expression tree of Equation 2 by distributing the first logarithmacross the linear expression 31.483x0 +6x1 (see [105]) and reformulating the second loga-



Fig. 5: 482 MIQCQP Problems: Performance Profile (a) compares the CPU time for 482MIQCQP Problems; Profile (b) diagrams the percent gap remaining at 7200 s

rithm via Equation 1:

31.483 · x0 · logx2 +6 · x1 · logx2 +11.503 · x0 · logx0 �11.503 · x0 · logx2 (3)

where x2 = 4.827 ·x0 +0.92 ·x1 is an auxiliary variable. Equation 3 subsequently simplifies:

19.98 · x0 · logx2 +6 · x1 · logx2 +11.503 · x0 · logx0 (4)

ANTIGONE replaces the term logx2 in Equation 4 with another auxiliary variable x3 sothat this portion of the ex6 2 9 objective function has: two bilinear terms (x0 x3; x1 x3); onelogarithmic term (logx2); one univariate composite logarithmic term that is convex acrossits entire domain (x0 · logx0; x0 2

⇥10�7, 0.5

⇤).

Using these automatic reformulations, ANTIGONE globally optimizes ex6 2 9 in 19seconds. If ANTIGONE was implemented differently, it could detect an additional compos-ite term in Equation 2 because 31.483 ·x0+6 ·x1 is a constant multiple of 4.827 ·x0+0.92 ·x1.But see in Table 5 that the existing reformulation strategy is very strong with respect to theother solvers. BARON 12.7.3 mistakes ex6 2 9 for an unbounded problem; Couenne 0.4is excluded from Table 5 because it incorrectly evaluates the objective function; LINDO 8.0is more than 100 times slower than ANTIGONE; SCIP 3.0 has a 8% gap at the time limit.

5.5 Results: 249 MINLPLib Problems

ANTIGONE performs competitively on the MINLPLib test set; the complete results are inFigure 7 and Tables 34 & 35 in the online supplement. Table 6 shows the strong performance

ANTIGONE 15

Table 5: GLOBALLib: Results of ex6 2 9 [53, 96, 97]

Time (s) Gap LB UB

ANTIGONE 1.1 22 1.00e-04 -3.407e-02 -3.407e-02BARON 12.7.3 74 – �• -3.407e-02LINDO 8.0 3080 1.00e-04 -3.408e-02 -3.407e-02SCIP 3.0 – 7.96e+00 -3.678e-02 -3.407e-02Gap ⌘ 100 ·

⇣UB�LB|LB|



Fig. 6: 368 GLOBALLib Problems: Performance Profile (a) compares the CPU time for368 GLOBALLib Problems; Profile (b) diagrams the percent gap remaining at 7200 s

of ANTIGONE on tls4; this example has 17 continuous variables, 89 discrete variables,65 equations, and 32 nonlinear terms.

Table 6: MINLPLib: Results of tls4 [27]

Time (s) Gap LB UB

ANTIGONE 1.1 15 1.00e-04 8.300e+00 8.300e+00BARON 12.7.3 473 1.00e-04 8.300e+00 8.300e+00Couenne 0.4 – 5.87e+01 3.424e+00 8.300e+00LINDO 8.0 – 8.31e+01 1.525e+00 9.000e+00SCIP 3.0 42 1.00e-04 8.300e+00 8.300e+00Gap ⌘ 100 ·

⇣UB�LB|LB|




Fig. 7: 249 MINLPLib Problems: Performance Profile (a) compares the CPU time for 249MINLPLib Problems; Profile (b) diagrams the percent gap remaining at 7200 s

5.6 Results: 1116 PrincetonLib Problems

ANTIGONE performs competitively on the PrincetonLib test set; Figure 8 and Tables 38 &39 in the online supplement provide complete results. Consider weapon from Princeton-Lib; weapon is obviously convex after repeated application of the Equation 1 flatten ex-ponentials reformulation. Table 1 shows that ANTIGONE recognizes convexity and solvesweapon quickly. Observe that convexity recognition is a key aspect of the PrincetonLibtest set; 459 of the 1116 problems are convex. Figures 8(c) – 8(f) divide the PrincetonLibproblems into convex and nonconvex test cases.

Table 7: PrincetonLib: Results of weapon

Time (s) Gap LB UB

ANTIGONE 1.1 0 1.00e-04 -1.736e+03 -1.736e+03BARON 12.7.3 0 1.00e-04 -1.736e+03 -1.736e+03Couenne 0.4 – 1.12e+00 -1.755e+03 -1.736e+03LINDO 8.0 – 1.12e+00 -1.755e+03 -1.736e+03SCIP 3.0 – 1.12e+00 -1.755e+03 -1.736e+03Gap ⌘ 100 ·

⇣UB�LB|LB|


ANTIGONE 17

5.7 Results: 241 Other Library Problems

ANTIGONE also performs competitively on a variety of other test libraries (AMPL BookLibrary; Bonmin; LINLib; MacMOOPLib; MPLLib); See Figure 8 and Tables: 24; 25; 26;27; 30; 31; 32; 33; 36; 37 (online supplement). Table 8 gives the performance of ANTIGONEon SLay10H; this MIQCQP has 831 continuous variables, 180 discrete variables, 1306equations, and 20 nonlinear terms.

Table 8: Bonmin Test Set: Results of SLay10H [20, 120]

Time (s) Gap LB UB

ANTIGONE 1.1 52 1.00e-04 1.296e+05 1.296e+05BARON 12.7.3 – 7.70e+01 1.193e+05 5.193e+05Couenne 0.4 – 1.56e+01 1.200e+05 1.422e+05LINDO 8.0 – 6.39e+01 1.191e+05 3.301e+05SCIP 3.0 3476 1.00e-04 1.296e+05 1.296e+05Gap ⌘ 100 ·

⇣UB�LB|LB|


5.8 Results: Classes of Nonlinearities

Beyond individual libraries, which may contain a wide range of different problem types, it isinteresting to consider similar examples across all problem classes. Figures 10 – 12 comparethree classes of nonlinearities: quadratic, signomial, and other test cases. Figure 10 repre-sents the performance of the solvers on the 1397 of the 2571 total test cases with exclusivelyquadratic nonlinearities; Figure 11 diagrams the 759 of the 2571 total test cases that haveonly signomial nonlinearities but are not in Figure 10; Figure 12 diagrams the remaining in-stances. From these three figures, it is evident that the strongest component of ANTIGONEis in addressing problems with quadratic nonlinearities. ANTIGONE is equivalent to theMIQCQP solver GloMIQO [102, 104, 107] in the case where all nonlinearities in MINLPare quadratic; ANTIGONE derives the core of its strength from quadratic instances.

5.9 Results: Discrete and Continuous Problems

Figure 13 diagrams the 528 incorporating at least one discrete variable; Figure 14 repre-sents solver performance on the 2048 instances without a discrete variable. Observe thatANTIGONE is strong in both categories.


5.10 Results: Convex and Nonconvex Nonlinearities

Figure 15 diagrams the 1781 test cases with nonconvex nonlinearities; Figure 16 representsperformance on the 790 test cases on convex NLP and MINLP with nonlinearities that canbe proven to be exclusively convex.

6 Conclusion

This manuscript introduces ANTIGONE, a general mixed-integer nonlinear global opti-mization framework. Analysis on a test set of 2571 shows that ANTIGONE performs stronglyacross an array of standard libraries and problems from the open literature. The performanceof ANTIGONE is especially strong with respect to the industrially-relevant minlp.orgtest set and on MIQCQP. The ANTIGONE framework is the natural evolution of the MIQCQPsolver GloMIQO and the MISO framework to general nonconvex MINLP; its extensibleframework permits easy integration of new algorithmic advances from the open literature.

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ANTIGONE 27

(a) 1116 Cases; Time s (b) 1116 Cases; Remaining Gap at 7200 s

(c) 459 Convex Cases; Time s (d) 459 Convex Cases; Remaining Gap at 7200 s

(e) 657 Nonconvex Cases; Time s (f) 657 Nonconvex Cases; Remaining Gap at 7200 s

Fig. 8: 1116 PrincetonLib Problems: Performance Profiles (a), (c), (e) compare the CPUtime for 1116 PrincetonLib Problems; Profiles (b), (d), (f) diagram the percent gap remainingat 7200 s. All graphs are on a Log10 Scale.



Fig. 9: 241 Other Library Problems: Performance Profile (a) compares the CPU time for241 Other Library Problems; Profile (b) diagrams the percent gap remaining at 7200 s


Fig. 10: 1397 Problems with Exclusively Quadratic Nonlinearities: Performance Profile(a) compares the CPU time for the 1397 test cases with exclusively quadratic nonlinearities;Profile (b) diagrams the percent gap remaining at 7200 s

ANTIGONE 29


Fig. 11: 759 Problems with a Signomial Component: Performance Profile (a) comparesthe CPU time for the 759 test cases with a signomial component; Profile (b) diagrams thepercent gap remaining at 7200 s


Fig. 12: 415 Problems with Exponential, Logarithmic, or Power Terms: PerformanceProfile (a) compares the CPU time for the 415 test cases with an exponential, logarithmic,or power terms; Profile (b) diagrams the percent gap remaining at 7200 s



Fig. 13: 528 Problems with a Discrete Component: Performance Profile (a) compares theCPU time for the 528 test cases with a discrete component; Profile (b) diagrams the percentgap remaining at 7200 s


Fig. 14: 2048 Problems with No Discrete Component: Performance Profile (a) comparesthe CPU time for the 2048 test cases with no discrete component; Profile (b) diagrams thepercent gap remaining at 7200 s

ANTIGONE 31


Fig. 15: 1781 Problems with Nonconvex Nonlinearities: Performance Profile (a) comparesthe CPU time for the 1781 test cases with a nonconvex component; Profile (b) diagrams thepercent gap remaining at 7200 s


Fig. 16: 790 Convex NLP and MINLP with Exclusively Convex Nonlinearities: Perfor-mance Profile (a) compares the CPU time for the 790 test cases with all convex nonlineari-ties; Profile (b) diagrams the percent gap remaining at 7200 s

antigone: algorithms for continuous / integer...

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