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A globally convergent primal-dualinterior-point filter method
for nonlinear programming (ipfilter):New filter optimality measures and computational results
Luis Nunes Vicentewww.mat.uc.pt/~lnv
Dept. Mathematics, Univ. Coimbra
OPTEC, K.U. Leuven, December 2, 2008
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Outline
1 Filter scheme
2 Interior-point framework
3 Primal-dual interior-point filter method
4 Global convergence
5 New filter measures
6 Restoration phase
7 Numerical testing with ipfilter
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Co-Authors
Joint work with M. Ulbrich (TU München), S. Ulbrich (TU Darmstadt)and R. Silva (Coimbra).
Algorithm development and theoretical support:
M. ULBRICH, S. ULBRICH, AND L. N. VICENTE, A globallyconvergent primal-dual interior-point filter method for nonlinearprogramming, Mathematical Programming, 100 (2004) 379-410.
R. SILVA, M. ULBRICH, S. ULBRICH, AND L. N. VICENTE, Aglobally convergent primal-dual interior-point filter method fornonlinear programming: New filter optimality measures andcomputational results, submitted for publication.
OPTEC, December 02, 2008 3/ 58
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Co-Authors
Joint work with M. Ulbrich (TU München), S. Ulbrich (TU Darmstadt)and R. Silva (Coimbra).
Algorithm development and theoretical support:
M. ULBRICH, S. ULBRICH, AND L. N. VICENTE, A globallyconvergent primal-dual interior-point filter method for nonlinearprogramming, Mathematical Programming, 100 (2004) 379-410.
R. SILVA, M. ULBRICH, S. ULBRICH, AND L. N. VICENTE, Aglobally convergent primal-dual interior-point filter method fornonlinear programming: New filter optimality measures andcomputational results, submitted for publication.
OPTEC, December 02, 2008 3/ 58
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NLP Algorithms
The goal is to determine first-order or second-order stationary points.
Newton or quasi-Newton step calculations yield good local behavior(quadratic/superlinear rates of convergence).
NLP algorithms must also converge globally.
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NLP Algorithms
The goal is to determine first-order or second-order stationary points.
Newton or quasi-Newton step calculations yield good local behavior(quadratic/superlinear rates of convergence).
NLP algorithms must also converge globally.
OPTEC, December 02, 2008 4/ 58
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NLP Algorithms
The goal is to determine first-order or second-order stationary points.
Newton or quasi-Newton step calculations yield good local behavior(quadratic/superlinear rates of convergence).
NLP algorithms must also converge globally.
OPTEC, December 02, 2008 4/ 58
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NLP Algorithms
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NLP Globalization Strategies
There are two globalization techniques:
line searches (search along the step direction).
trust regions (step is computed by minimizing a quadratic modelwithin a region of prescribed size).
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NLP Globalization Strategies
There are two globalization techniques:
line searches (search along the step direction).
trust regions (step is computed by minimizing a quadratic modelwithin a region of prescribed size).
OPTEC, December 02, 2008 6/ 58
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NLP Globalization Strategies
There are two globalization techniques:
line searches (search along the step direction).
trust regions (step is computed by minimizing a quadratic modelwithin a region of prescribed size).
OPTEC, December 02, 2008 6/ 58
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NLP Globalization Strategies – Constraints
In the presence of constraints, we must use one of the previoustogether with:
merit function (combining objective function with a penalization ofconstraints).
filter (explained next).
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NLP Globalization Strategies – Constraints
In the presence of constraints, we must use one of the previoustogether with:
merit function (combining objective function with a penalization ofconstraints).
filter (explained next).
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NLP Globalization Strategies – Constraints
In the presence of constraints, we must use one of the previoustogether with:
merit function (combining objective function with a penalization ofconstraints).
filter (explained next).
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Filter Scheme
Globalize without merit functions and penalty parameters.
Borrow the concept of nondominance from multi-criteriaoptimization.
min f (x) s.t. h(x) = 0
↓
min f (x) and θ(x) = ‖h(x)‖
(bi-criteria optimization problem)
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Filter Scheme
Globalize without merit functions and penalty parameters.
Borrow the concept of nondominance from multi-criteriaoptimization.
min f (x) s.t. h(x) = 0
↓
min f (x) and θ(x) = ‖h(x)‖
(bi-criteria optimization problem)
OPTEC, December 02, 2008 8/ 58
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Filter Scheme
Globalize without merit functions and penalty parameters.
Borrow the concept of nondominance from multi-criteriaoptimization.
min f (x) s.t. h(x) = 0
↓
min f (x) and θ(x) = ‖h(x)‖
(bi-criteria optimization problem)
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Filter Scheme – Definitions
A filter is a discrete set of efficient or nondominated points.
A point x or (f (x), θ(x)) is efficient or nondominated if it is notdominated by any other point in the filter.
A point x dominates x ′ (x 6= x ′) if
θ(x) ≤ θ(x ′) and f (x) ≤ f (x ′).
The filter defines the efficient border of the nondominated points.
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Filter Scheme – Definitions
A filter is a discrete set of efficient or nondominated points.
A point x or (f (x), θ(x)) is efficient or nondominated if it is notdominated by any other point in the filter.
A point x dominates x ′ (x 6= x ′) if
θ(x) ≤ θ(x ′) and f (x) ≤ f (x ′).
The filter defines the efficient border of the nondominated points.
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Filter Scheme – Definitions
A filter is a discrete set of efficient or nondominated points.
A point x or (f (x), θ(x)) is efficient or nondominated if it is notdominated by any other point in the filter.
A point x dominates x ′ (x 6= x ′) if
θ(x) ≤ θ(x ′) and f (x) ≤ f (x ′).
The filter defines the efficient border of the nondominated points.
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Filter Scheme – an Example
6
-
f (x)
θ(x)0
q
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Filter Scheme – Definitions
Acceptability is defined in a more stringent way (envelope):
A point x is acceptable to the filter if
θ(xj) > θ(x) + γFθ(xj) or f (xj) > f (x) + γFθ(xj),
for all filter entries xj .
γF ∈ (0, 12 ).
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Filter Scheme – an Example
6
-
f (x)
θ(x)0
q
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Filter Scheme – an Example
6
-
f (x)
θ(x)0
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Forbidden Regions – Merit Function vs. Filter
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Filter Methods Components
Definition of filter entries.
Definition of filter envelope.
Filter management schemes (including removal of filter entries).
Restoration phase when feasibility is not sufficiently small.
Sufficient decrease when feasibility is sufficiently small.
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Filter Methods Components
Definition of filter entries.
Definition of filter envelope.
Filter management schemes (including removal of filter entries).
Restoration phase when feasibility is not sufficiently small.
Sufficient decrease when feasibility is sufficiently small.
OPTEC, December 02, 2008 15/ 58
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Filter Methods Components
Definition of filter entries.
Definition of filter envelope.
Filter management schemes (including removal of filter entries).
Restoration phase when feasibility is not sufficiently small.
Sufficient decrease when feasibility is sufficiently small.
OPTEC, December 02, 2008 15/ 58
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Filter Methods Components
Definition of filter entries.
Definition of filter envelope.
Filter management schemes (including removal of filter entries).
Restoration phase when feasibility is not sufficiently small.
Sufficient decrease when feasibility is sufficiently small.
OPTEC, December 02, 2008 15/ 58
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Filter Methods Components
Definition of filter entries.
Definition of filter envelope.
Filter management schemes (including removal of filter entries).
Restoration phase when feasibility is not sufficiently small.
Sufficient decrease when feasibility is sufficiently small.
OPTEC, December 02, 2008 15/ 58
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Filter – Literature
SQP: Fletcher, Leyffer (Math. Prog. 2002)Fletcher, Gould, Leyffer, Toint, Wächter (SIOPT 2002)Fletcher, Leyffer, Toint (SIOPT 2002)S. Ulbrich (Math. Prog. 2004)SLP: Fletcher, Leyffer, TointChin, Fletcher (Math. Prog. 2003)IPM: Ulbrich, Ulbrich, Vicente (Math. Prog. 2004)Wächter, Biegler (SIOPT 2005)Benson, Shanno, Vanderbei (COAP 2002)Others:Audet, Dennis (SIOPT 2004) – DFOGonzaga, Karas, Vanti (SIOPT 2003)Gould, Leyffer, Toint (SIOPT 2004) – systems of nonlinear equations
nonsmooth optimization, unconstrained optimization...
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Lagrange Prize
Roger Fletcher, Sven Leyffer, and Philippe Toint were the recipients in 2006 of theLagrange Prize for Continuous Optimization, awarded jointly by the MathematicalProgramming Society (MPS) and the Society for Industrial and Applied Mathematics(SIAM), for “outstanding works in the area of continuous optimization”.
“In the development of nonlinear programming over the last decade, anoutstanding new idea has been the introduction of the filter. This newapproach to balancing feasibility and optimality has been quickly picked upby other researchers, spurring the analysis and development of a number ofoptimization algorithms in such diverse contexts as constrained andunconstrained nonlinear optimization, solving systems of nonlinearequations, and derivative-free optimization. The generality of the filter ideaallows its use, for example, in trust region and line search methods, as wellas in active set and interior point frameworks. Currently, some of the mosteffective nonlinear optimization codes are based on filter methods. Theimportance of the work cited here will continue to grow as more algorithmsand codes are developed.”
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A Recent Survey
R. Fletcher, S. Leyffer, and Ph. L. Toint, A brief history of filtermethods, SIAM SIAG/OPT Views-and-News, 18 (1) (2006) 2–12.
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Description of the Problem
Nonlinear programming (NLP) problems:
minx∈IRn
f (x) s.t. h(x) = 0 and x ≥ 0,
where f : IRn −→ IR and h : IRn −→ IRm are twice continuously differentiablefunctions on an open set Ω ⊂ IRn.
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Karush-Kuhn-Tucker (KKT) Conditions
Lagrangian: `(x , y , z) = f (x) + h(x)>y − x>z.
First-order necessary conditions:
∇x`(x , y , z) = 0 ⇐⇒ ∇f (x) +∇h(x)y − z = 0h(x) = 0
Xz = 0x ≥ 0 e z ≥ 0,
where y ∈ IRm and z ∈ IRn are the Lagrange multipliers.
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Karush-Kuhn-Tucker (KKT) Conditions
Lagrangian: `(x , y , z) = f (x) + h(x)>y − x>z.
First-order necessary conditions:
∇x`(x , y , z) = 0 ⇐⇒ ∇f (x) +∇h(x)y − z = 0h(x) = 0
Xz = 0x ≥ 0 e z ≥ 0,
where y ∈ IRm and z ∈ IRn are the Lagrange multipliers.
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Perturbed KKT System
Perturbing the KKT system, we obtain:
∇x`(x , y , z) = 0h(x) = 0
Xz = µe.
The step ∆w = (∆x ,∆y ,∆z) is the solution of the Newton system: ∇2xx`(x , y , z) ∇h(x) −I∇h(x)> 0 0
Z 0 X
∆x∆y∆z
= −
∇x`(x , y , z)h(x)
Xz − µe
.
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Perturbed KKT System
Perturbing the KKT system, we obtain:
∇x`(x , y , z) = 0h(x) = 0
Xz = µe.
The step ∆w = (∆x ,∆y ,∆z) is the solution of the Newton system: ∇2xx`(x , y , z) ∇h(x) −I∇h(x)> 0 0
Z 0 X
∆x∆y∆z
= −
∇x`(x , y , z)h(x)
Xz − µe
.
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Central Path – LP
Linear Programming (LP): f (x) = c>x and h(x) = Ax − b.
The central path is formed by all (x , y , z) such that (x , z) > 0 and
∇x`(x , y , z) = 0h(x) = 0
Xz = µe,
for all µ > 0.
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Central Path – LP
Linear Programming (LP): f (x) = c>x and h(x) = Ax − b.
The central path is formed by all (x , y , z) such that (x , z) > 0 and
∇x`(x , y , z) = 0h(x) = 0
Xz = µe,
for all µ > 0.
OPTEC, December 02, 2008 22/ 58
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Central Path – LP
OPTEC, December 02, 2008 23/ 58
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Central Path – LP
OPTEC, December 02, 2008 24/ 58
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Central Path – NLP
The central path only exists locally in NLP!
OPTEC, December 02, 2008 25/ 58
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Central Path – NLP
OPTEC, December 02, 2008 26/ 58
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Quasi-Central Path – NLP
The quasi-central path is formed by all (x , z) > 0 such that
h(x) = 0Xz = µe,
for all µ > 0.
The quasi-central path has a global existence.
OPTEC, December 02, 2008 27/ 58
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Quasi-Central Path – NLP
The quasi-central path is formed by all (x , z) > 0 such that
h(x) = 0Xz = µe,
for all µ > 0.
The quasi-central path has a global existence.
OPTEC, December 02, 2008 27/ 58
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Perturbed KKT System
Perturbing the KKT system, we obtain:
∇x`(x , y , z) = 0h(x) = 0
Xz = µe.
The step ∆w = (∆x ,∆y ,∆z) is the solution of the Newton system: ∇2xx`(x , y , z) ∇h(x) −I∇h(x)> 0 0
Z 0 X
∆x∆y∆z
= −
∇x`(x , y , z)h(x)
Xz − µe
.
We set µ = σµ, with µ = x>zn a measure of centrality and σ ∈ (0, 1) a
centering parameter.
OPTEC, December 02, 2008 28/ 58
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Perturbed KKT System
Perturbing the KKT system, we obtain:
∇x`(x , y , z) = 0h(x) = 0
Xz = µe.
The step ∆w = (∆x ,∆y ,∆z) is the solution of the Newton system: ∇2xx`(x , y , z) ∇h(x) −I∇h(x)> 0 0
Z 0 X
∆x∆y∆z
= −
∇x`(x , y , z)h(x)
Xz − µe
.
We set µ = σµ, with µ = x>zn a measure of centrality and σ ∈ (0, 1) a
centering parameter.
OPTEC, December 02, 2008 28/ 58
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Perturbed KKT System
Perturbing the KKT system, we obtain:
∇x`(x , y , z) = 0h(x) = 0
Xz = µe.
The step ∆w = (∆x ,∆y ,∆z) is the solution of the Newton system: ∇2xx`(x , y , z) ∇h(x) −I∇h(x)> 0 0
Z 0 X
∆x∆y∆z
= −
∇x`(x , y , z)h(x)
Xz − µe
.
We set µ = σµ, with µ = x>zn a measure of centrality and σ ∈ (0, 1) a
centering parameter.
OPTEC, December 02, 2008 28/ 58
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Motivation – in the NLP Context
Find optimality and feasibility measures.
Find a decomposition of the primal-dual step ∆w (normal andtangential) yielding decrease in both measures, respectively.
OPTEC, December 02, 2008 29/ 58
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Filter Measures
Feasibility and centrality measure:
θ(w) = ‖h(x)‖+ ‖Xz − µe‖
Optimality measure:
θg(w) = µ + ‖∇x`(w)‖2
w = (x , y , z), with (x , z) ≥ 0, satisfies the KKT conditions if and only if
θ(w) = θg(w) = 0.
OPTEC, December 02, 2008 30/ 58
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Filter Measures
Feasibility and centrality measure:
θ(w) = ‖h(x)‖+ ‖Xz − µe‖
Optimality measure:
θg(w) = µ + ‖∇x`(w)‖2
w = (x , y , z), with (x , z) ≥ 0, satisfies the KKT conditions if and only if
θ(w) = θg(w) = 0.
OPTEC, December 02, 2008 30/ 58
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Filter Measures
Feasibility and centrality measure:
θ(w) = ‖h(x)‖+ ‖Xz − µe‖
Optimality measure:
θg(w) = µ + ‖∇x`(w)‖2
w = (x , y , z), with (x , z) ≥ 0, satisfies the KKT conditions if and only if
θ(w) = θg(w) = 0.
OPTEC, December 02, 2008 30/ 58
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Step Decomposition
Normal step: ∇2xx`(w) ∇h(x) −I
∇h(x)> 0 0Z 0 X
sn = −
0h(x)
Xz − µe
,
“towards” the quasi-central path.
Tangential step: ∇2xx`(w) ∇h(x) −I
∇h(x)> 0 0Z 0 X
st = −
∇x`(w)0
(1− σ)µe
.
Adding both steps: ∆w = sn + st .
OPTEC, December 02, 2008 31/ 58
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Step Decomposition
Normal step: ∇2xx`(w) ∇h(x) −I
∇h(x)> 0 0Z 0 X
sn = −
0h(x)
Xz − µe
,
“towards” the quasi-central path.
Tangential step: ∇2xx`(w) ∇h(x) −I
∇h(x)> 0 0Z 0 X
st = −
∇x`(w)0
(1− σ)µe
.
Adding both steps: ∆w = sn + st .
OPTEC, December 02, 2008 31/ 58
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Step Decomposition
Normal step: ∇2xx`(w) ∇h(x) −I
∇h(x)> 0 0Z 0 X
sn = −
0h(x)
Xz − µe
,
“towards” the quasi-central path.
Tangential step: ∇2xx`(w) ∇h(x) −I
∇h(x)> 0 0Z 0 X
st = −
∇x`(w)0
(1− σ)µe
.
Adding both steps: ∆w = sn + st .
OPTEC, December 02, 2008 31/ 58
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Step Sizes
Using a trust-region type parameter:
‖αn(∆)sn‖ ≤ ∆, ‖αt(∆)st‖ ≤ ∆
⇓‖s(∆)‖ = αn(∆)sn + αt(∆)st ≤ 2∆
αn(∆) = min
1,∆
‖sn‖
, αt(∆) = min
αn(∆),
∆
‖st‖
w(∆) = w + s(∆) = (x(∆), y(∆), z(∆)).
OPTEC, December 02, 2008 32/ 58
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Step Sizes
Using a trust-region type parameter:
‖αn(∆)sn‖ ≤ ∆, ‖αt(∆)st‖ ≤ ∆
⇓‖s(∆)‖ = αn(∆)sn + αt(∆)st ≤ 2∆
αn(∆) = min
1,∆
‖sn‖
, αt(∆) = min
αn(∆),
∆
‖st‖
w(∆) = w + s(∆) = (x(∆), y(∆), z(∆)).
OPTEC, December 02, 2008 32/ 58
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Step Sizes
Using a trust-region type parameter:
‖αn(∆)sn‖ ≤ ∆, ‖αt(∆)st‖ ≤ ∆
⇓‖s(∆)‖ = αn(∆)sn + αt(∆)st ≤ 2∆
αn(∆) = min
1,∆
‖sn‖
, αt(∆) = min
αn(∆),
∆
‖st‖
w(∆) = w + s(∆) = (x(∆), y(∆), z(∆)).
OPTEC, December 02, 2008 32/ 58
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Step Sizes
Using a trust-region type parameter:
‖αn(∆)sn‖ ≤ ∆, ‖αt(∆)st‖ ≤ ∆
⇓‖s(∆)‖ = αn(∆)sn + αt(∆)st ≤ 2∆
αn(∆) = min
1,∆
‖sn‖
, αt(∆) = min
αn(∆),
∆
‖st‖
w(∆) = w + s(∆) = (x(∆), y(∆), z(∆)).
OPTEC, December 02, 2008 32/ 58
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Centrality Neighborhood
The iterates will be kept in:
N (γ, M) = w : (x , z) > 0; Xz ≥ γµe; ‖h(w)‖+ ‖∇x`(w)‖ ≤ Mµ .
Lemma
If ‖F ′(w)−1‖ ≤ C, γ ∈ (0, 1), M > 0, then
w ∈ N (γ, M)
⇓∃∆min > 0 : w(∆) ∈ N (γ, M)
for all ∆ ∈]0,∆min].
F ′(w) is the KKT matrix (the Jacobian of the KKT residual).
OPTEC, December 02, 2008 33/ 58
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Centrality Neighborhood
The iterates will be kept in:
N (γ, M) = w : (x , z) > 0; Xz ≥ γµe; ‖h(w)‖+ ‖∇x`(w)‖ ≤ Mµ .
Lemma
If ‖F ′(w)−1‖ ≤ C, γ ∈ (0, 1), M > 0, then
w ∈ N (γ, M)
⇓∃∆min > 0 : w(∆) ∈ N (γ, M)
for all ∆ ∈]0,∆min].
F ′(w) is the KKT matrix (the Jacobian of the KKT residual).
OPTEC, December 02, 2008 33/ 58
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Primal-Dual Interior-Point Filter Method
Decomposition of the primal-dual step (normal and tangential)yielding decrease in a feasibility and an optimality measures(associated with the filter components).
Different step lengths for the normal and the tangential steps.
The iterates are kept in a central neighborhood.
Possible use of a feasibility restoration procedure.
The new iterates are always acceptable to the filter.
Imposition of a sufficient decrease if the feasibility measure issmall enough.
OPTEC, December 02, 2008 34/ 58
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Primal-Dual Interior-Point Filter Method
Decomposition of the primal-dual step (normal and tangential)yielding decrease in a feasibility and an optimality measures(associated with the filter components).
Different step lengths for the normal and the tangential steps.
The iterates are kept in a central neighborhood.
Possible use of a feasibility restoration procedure.
The new iterates are always acceptable to the filter.
Imposition of a sufficient decrease if the feasibility measure issmall enough.
OPTEC, December 02, 2008 34/ 58
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Primal-Dual Interior-Point Filter Method
Decomposition of the primal-dual step (normal and tangential)yielding decrease in a feasibility and an optimality measures(associated with the filter components).
Different step lengths for the normal and the tangential steps.
The iterates are kept in a central neighborhood.
Possible use of a feasibility restoration procedure.
The new iterates are always acceptable to the filter.
Imposition of a sufficient decrease if the feasibility measure issmall enough.
OPTEC, December 02, 2008 34/ 58
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Primal-Dual Interior-Point Filter Method
Decomposition of the primal-dual step (normal and tangential)yielding decrease in a feasibility and an optimality measures(associated with the filter components).
Different step lengths for the normal and the tangential steps.
The iterates are kept in a central neighborhood.
Possible use of a feasibility restoration procedure.
The new iterates are always acceptable to the filter.
Imposition of a sufficient decrease if the feasibility measure issmall enough.
OPTEC, December 02, 2008 34/ 58
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Primal-Dual Interior-Point Filter Method
Decomposition of the primal-dual step (normal and tangential)yielding decrease in a feasibility and an optimality measures(associated with the filter components).
Different step lengths for the normal and the tangential steps.
The iterates are kept in a central neighborhood.
Possible use of a feasibility restoration procedure.
The new iterates are always acceptable to the filter.
Imposition of a sufficient decrease if the feasibility measure issmall enough.
OPTEC, December 02, 2008 34/ 58
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Primal-Dual Interior-Point Filter Method
Decomposition of the primal-dual step (normal and tangential)yielding decrease in a feasibility and an optimality measures(associated with the filter components).
Different step lengths for the normal and the tangential steps.
The iterates are kept in a central neighborhood.
Possible use of a feasibility restoration procedure.
The new iterates are always acceptable to the filter.
Imposition of a sufficient decrease if the feasibility measure issmall enough.
OPTEC, December 02, 2008 34/ 58
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Primal-Dual Interior-Point Filter Method
0. Choose (x0, z0) > 0, y0, ∆0 > 0, k = 0. Choose also γ, M suchthat (x0, y0, z0) ∈ N (γ, M).
1. Stop if θ(wk ) + θg(wk ) < εtol . Otherwise compute snk and st
k .
2/3. Compute ∆k such that wk (∆k ) ∈ N (γ, M).
OPTEC, December 02, 2008 35/ 58
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Primal-Dual Interior-Point Filter Method
0. Choose (x0, z0) > 0, y0, ∆0 > 0, k = 0. Choose also γ, M suchthat (x0, y0, z0) ∈ N (γ, M).
1. Stop if θ(wk ) + θg(wk ) < εtol . Otherwise compute snk and st
k .
2/3. Compute ∆k such that wk (∆k ) ∈ N (γ, M).
OPTEC, December 02, 2008 35/ 58
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Primal-Dual Interior-Point Filter Method
0. Choose (x0, z0) > 0, y0, ∆0 > 0, k = 0. Choose also γ, M suchthat (x0, y0, z0) ∈ N (γ, M).
1. Stop if θ(wk ) + θg(wk ) < εtol . Otherwise compute snk and st
k .
2/3. Compute ∆k such that wk (∆k ) ∈ N (γ, M).
OPTEC, December 02, 2008 35/ 58
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Primal-Dual Interior-Point Filter Method – Trial Tests
Sufficient reduction criterion: ρk ≥ η
η ∈ (0, 1) and ρkdef= actual reduction
predicted reductiondef=
θg(wk )−θg(wk (∆k ))mk (wk )−mk (wk (∆k ))
where
m def= linearization of x>z
n+
squared norm of the linearization of ∇x`(w)
is a quadratic model of θg .
OPTEC, December 02, 2008 36/ 58
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Primal-Dual Interior-Point Filter Method – Trial Tests
Sufficient reduction criterion: ρk ≥ η
η ∈ (0, 1) and ρkdef= actual reduction
predicted reductiondef=
θg(wk )−θg(wk (∆k ))mk (wk )−mk (wk (∆k ))
where
m def= linearization of x>z
n+
squared norm of the linearization of ∇x`(w)
is a quadratic model of θg .
OPTEC, December 02, 2008 36/ 58
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Primal-Dual Interior-Point Filter Method – Trial Tests
Sufficient reduction criterion: ρk ≥ η
η ∈ (0, 1) and ρkdef= actual reduction
predicted reductiondef=
θg(wk )−θg(wk (∆k ))mk (wk )−mk (wk (∆k ))
where
m def= linearization of x>z
n+
squared norm of the linearization of ∇x`(w)
is a quadratic model of θg .
OPTEC, December 02, 2008 36/ 58
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Primal-Dual Interior-Point Filter Method – Trial Tests
Sufficient reduction criterion: ρk ≥ η
η ∈ (0, 1) and ρkdef= actual reduction
predicted reductiondef=
θg(wk )−θg(wk (∆k ))mk (wk )−mk (wk (∆k ))
where
m def= linearization of x>z
n+
squared norm of the linearization of ∇x`(w)
is a quadratic model of θg .
OPTEC, December 02, 2008 36/ 58
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Primal-Dual Interior-Point Filter Method
θ(wk ) ≤ ∆k minγ1, γ2∆βk ?
true
HHHHjfalse
wk (∆k ) is acceptable to filter?(with wk considered in filter
if predk < κθ(wk )2)
add wk to filterenter restoration
compute wk+1 such that:wk+1 is acceptable to filter,
θ(wk+1) ≤ ∆k+1 minγ1, γ2(∆k+1)β,
and return to step 1 with ∆k+1 = ∆k
?true
HHHHHj
false
∆k+1 = ∆k /2return to step 3
ρk ≥ η or predk < κθ(wk )2?
*false
?true
add wk to filterif predk < κθ(wk )2 ∆k+1 ≥ ∆k
wk+1 = wk (∆k )return to step 1
OPTEC, December 02, 2008 37/ 58
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Global Convergence – Assumptions
(A1) (xk , yk , zk is bounded.
(A2) ∇h and ∇2xw` are Lipschitz continuous in an open set D such
that wk ∈ D and [wk , wk + s(∆k )] ∈ D.
(A3) ∃C > 0 such that ‖F ′(w)−1‖ ≤ C for all k .
OPTEC, December 02, 2008 38/ 58
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Global Convergence – Main Result
Theorem
lim infk→+∞
θ(wk ) + θg(wk ) = 0
(there exists a limit point that is a KKT point).
OPTEC, December 02, 2008 39/ 58
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New Filter Measures
Previous filter optimality measure:
θg(w) = µ + ‖∇x`(w)‖2
Choice made in Ulbrich, Ulbrich, and Vicente [2004].Does not distinguish minima from maxima!
OPTEC, December 02, 2008 40/ 58
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New Filter Measures
New possibilities (c > 0):
θg(w) = f (x) + cµ
or
θg(w) = f (x) + h(x)>y + cµ = `(x , y , z) + (c + n)µ
Reflect better the minimization goal.f is the driven force (when h(x) and µ = x>z/n are small).
OPTEC, December 02, 2008 41/ 58
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New Filter Measures
Let w ∈ N (γ, M) and c ≥ constant(σ, γ, n).
We can derive an estimate of the type:
θg(w(∆))− θg(w) ≤ −(Kµ)∆ +O(θ(w)) +O(∆2).
When θ(w) is small, the primal-dual step is a descent direction forθg(w).
OPTEC, December 02, 2008 42/ 58
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New Filter Measures
Let w ∈ N (γ, M) and c ≥ constant(σ, γ, n).
We can derive an estimate of the type:
θg(w(∆))− θg(w) ≤ −(Kµ)∆ +O(θ(w)) +O(∆2).
When θ(w) is small, the primal-dual step is a descent direction forθg(w).
OPTEC, December 02, 2008 42/ 58
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New Filter Measures
Let w ∈ N (γ, M) and c ≥ constant(σ, γ, n).
We can derive an estimate of the type:
θg(w(∆))− θg(w) ≤ −(Kµ)∆ +O(θ(w)) +O(∆2).
When θ(w) is small, the primal-dual step is a descent direction forθg(w).
OPTEC, December 02, 2008 42/ 58
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New Filter Measures
One can prove as before:
Thorem
lim infk→+∞
‖h(xk )‖+x>k zk
n+ ‖∇x`(wk )‖ = 0
(there exists a limit point that is a KKT point).
...under the assumption that ∇2xx`(wk ) + (1/2)X− 1
2k Zk is positive
semi-definite on the null space of ∇g(xk )> for all k ...
...same algorithm (the only modification is in the model m for θg(w))...
OPTEC, December 02, 2008 43/ 58
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New Centrality Neighborhoods
Consider a more general scenario that allows the use of differenttypes of approximations H 6= ∇2
xx`(w):
N (γ, M, p) = w : (x , z) > 0, Xz ≥ γµe, ‖h(x)‖+ ‖∇x`(w)‖p ≤ Mµ
is the family of centrality neighborhoods parameterized by p ∈ [1, 2].
When p = 1: N (γ, M, 1) = N (γ, M) (old centrality neighborhood).
As before, we can prove that
w ∈ N (γ, M, p) =⇒ w(∆) ∈ N (γ, M, p), for ∆ > 0 sufficiently small.
OPTEC, December 02, 2008 44/ 58
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New Centrality Neighborhoods
Consider a more general scenario that allows the use of differenttypes of approximations H 6= ∇2
xx`(w):
N (γ, M, p) = w : (x , z) > 0, Xz ≥ γµe, ‖h(x)‖+ ‖∇x`(w)‖p ≤ Mµ
is the family of centrality neighborhoods parameterized by p ∈ [1, 2].
When p = 1: N (γ, M, 1) = N (γ, M) (old centrality neighborhood).
As before, we can prove that
w ∈ N (γ, M, p) =⇒ w(∆) ∈ N (γ, M, p), for ∆ > 0 sufficiently small.
OPTEC, December 02, 2008 44/ 58
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New Centrality Neighborhoods
Consider a more general scenario that allows the use of differenttypes of approximations H 6= ∇2
xx`(w):
N (γ, M, p) = w : (x , z) > 0, Xz ≥ γµe, ‖h(x)‖+ ‖∇x`(w)‖p ≤ Mµ
is the family of centrality neighborhoods parameterized by p ∈ [1, 2].
When p = 1: N (γ, M, 1) = N (γ, M) (old centrality neighborhood).
As before, we can prove that
w ∈ N (γ, M, p) =⇒ w(∆) ∈ N (γ, M, p), for ∆ > 0 sufficiently small.
OPTEC, December 02, 2008 44/ 58
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New Centrality Neighborhoods
Consider a more general scenario that allows the use of differenttypes of approximations H 6= ∇2
xx`(w):
N (γ, M, p) = w : (x , z) > 0, Xz ≥ γµe, ‖h(x)‖+ ‖∇x`(w)‖p ≤ Mµ
is the family of centrality neighborhoods parameterized by p ∈ [1, 2].
When p = 1: N (γ, M, 1) = N (γ, M) (old centrality neighborhood).
As before, we can prove that
w ∈ N (γ, M, p) =⇒ w(∆) ∈ N (γ, M, p), for ∆ > 0 sufficiently small.
OPTEC, December 02, 2008 44/ 58
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Use of Second-Order Derivatives
A sufficient decrease condition on ‖∇x`(w)‖p must be satisfied:
‖∇x`(w(∆))‖p ≤ (1− p αt(∆))‖∇x`(w)‖p + M` max∆q ,∆2,
where q ∈ (1, 2].
TRUE for p = 1, q = 2, and H = ∇2xx`(w) — UUV 2004
When p = 1, TRUE if
‖H −∇2xx`(w)‖ ≤ N∆q−1.
When p = 2, TRUE if
‖[H −∇2xx`(w)]∇x`(w)‖ ≤ N‖∇x`(w)‖∆q−1.
OPTEC, December 02, 2008 45/ 58
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Use of Second-Order Derivatives
A sufficient decrease condition on ‖∇x`(w)‖p must be satisfied:
‖∇x`(w(∆))‖p ≤ (1− p αt(∆))‖∇x`(w)‖p + M` max∆q ,∆2,
where q ∈ (1, 2].
TRUE for p = 1, q = 2, and H = ∇2xx`(w) — UUV 2004
When p = 1, TRUE if
‖H −∇2xx`(w)‖ ≤ N∆q−1.
When p = 2, TRUE if
‖[H −∇2xx`(w)]∇x`(w)‖ ≤ N‖∇x`(w)‖∆q−1.
OPTEC, December 02, 2008 45/ 58
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Use of Second-Order Derivatives
A sufficient decrease condition on ‖∇x`(w)‖p must be satisfied:
‖∇x`(w(∆))‖p ≤ (1− p αt(∆))‖∇x`(w)‖p + M` max∆q ,∆2,
where q ∈ (1, 2].
TRUE for p = 1, q = 2, and H = ∇2xx`(w) — UUV 2004
When p = 1, TRUE if
‖H −∇2xx`(w)‖ ≤ N∆q−1.
When p = 2, TRUE if
‖[H −∇2xx`(w)]∇x`(w)‖ ≤ N‖∇x`(w)‖∆q−1.
OPTEC, December 02, 2008 45/ 58
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Use of Second-Order Derivatives
A sufficient decrease condition on ‖∇x`(w)‖p must be satisfied:
‖∇x`(w(∆))‖p ≤ (1− p αt(∆))‖∇x`(w)‖p + M` max∆q ,∆2,
where q ∈ (1, 2].
TRUE for p = 1, q = 2, and H = ∇2xx`(w) — UUV 2004
When p = 1, TRUE if
‖H −∇2xx`(w)‖ ≤ N∆q−1.
When p = 2, TRUE if
‖[H −∇2xx`(w)]∇x`(w)‖ ≤ N‖∇x`(w)‖∆q−1.
OPTEC, December 02, 2008 45/ 58
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Use of Second-Order Derivatives
A sufficient decrease condition on ‖∇x`(w)‖p must be satisfied:
‖∇x`(w(∆))‖p ≤ (1− p αt(∆))‖∇x`(w)‖p + M` max∆q ,∆2,
where q ∈ (1, 2].
TRUE for p = 1, q = 2, and H = ∇2xx`(w) — UUV 2004
When p = 1, TRUE if
‖H −∇2xx`(w)‖ ≤ N∆q−1.
When p = 2, TRUE if
‖[H −∇2xx`(w)]∇x`(w)‖ ≤ N‖∇x`(w)‖∆q−1.
OPTEC, December 02, 2008 45/ 58
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Restoration Phase
Purpose:
Find wk+1 such thatwk+1 ∈ N (γ, M)
wk+1 is acceptable to the filterθ(wk+1) ≤ ∆k minγ1, γ2∆
βk .
The restoration algorithm must terminate in a finite number ofiterations.
OPTEC, December 02, 2008 46/ 58
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Restoration Phase
Purpose:
Find wk+1 such thatwk+1 ∈ N (γ, M)
wk+1 is acceptable to the filterθ(wk+1) ≤ ∆k minγ1, γ2∆
βk .
The restoration algorithm must terminate in a finite number ofiterations.
OPTEC, December 02, 2008 46/ 58
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Restoration Phase
The current restoration algorithm uses our step framework:
s(∆) = αn(∆)sn + αt(∆)st , w(∆) = w + s(∆),
imposing descent on: θ2(w) = 12
(‖h(x)‖2 + ‖Xz − µe‖2
).
Note: ∇θ2(w)>sn = −θ2(w) and ∇θ2(w)>st = 0.
This restoration algorithm is proved to terminate successfully in afinite number of iterations.
OPTEC, December 02, 2008 47/ 58
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Restoration Phase
The current restoration algorithm uses our step framework:
s(∆) = αn(∆)sn + αt(∆)st , w(∆) = w + s(∆),
imposing descent on: θ2(w) = 12
(‖h(x)‖2 + ‖Xz − µe‖2
).
Note: ∇θ2(w)>sn = −θ2(w) and ∇θ2(w)>st = 0.
This restoration algorithm is proved to terminate successfully in afinite number of iterations.
OPTEC, December 02, 2008 47/ 58
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Restoration Phase
The current restoration algorithm uses our step framework:
s(∆) = αn(∆)sn + αt(∆)st , w(∆) = w + s(∆),
imposing descent on: θ2(w) = 12
(‖h(x)‖2 + ‖Xz − µe‖2
).
Note: ∇θ2(w)>sn = −θ2(w) and ∇θ2(w)>st = 0.
This restoration algorithm is proved to terminate successfully in afinite number of iterations.
OPTEC, December 02, 2008 47/ 58
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Restoration Phase
The current restoration algorithm uses our step framework:
s(∆) = αn(∆)sn + αt(∆)st , w(∆) = w + s(∆),
imposing descent on: θ2(w) = 12
(‖h(x)‖2 + ‖Xz − µe‖2
).
Note: ∇θ2(w)>sn = −θ2(w) and ∇θ2(w)>st = 0.
This restoration algorithm is proved to terminate successfully in afinite number of iterations.
OPTEC, December 02, 2008 47/ 58
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Restoration Phase
0. Choose parameters, w0k = wk , ∆0
k = ∆k , j = 0
1. Test the stop criterion
2. Compute snk,j and st
k,j
3. Compute ∆jk such that, for all ∆ ∈ [0,∆j
k ],
X jk (∆)z j
k (∆) ≥ γµjk (∆) and (x j
k (∆), z jk (∆)) > 0
4.
AREDjk
PREDjk
≥ ξ and
‖h(w jk (∆j
k ))‖+ ‖∇x `(w jk (∆j
k ))‖ ≤ Mµjk (∆j
k )?
False: ∆jk+1 = ∆j
k /2repeat step 4
True: w j+1k = w j
k (∆jk )
go to step 1
OPTEC, December 02, 2008 48/ 58
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Restoration Phase
0. Choose parameters, w0k = wk , ∆0
k = ∆k , j = 0
1. Test the stop criterion
2. Compute snk,j and st
k,j
3. Compute ∆jk such that, for all ∆ ∈ [0,∆j
k ],
X jk (∆)z j
k (∆) ≥ γµjk (∆) and (x j
k (∆), z jk (∆)) > 0
4.
AREDjk
PREDjk
≥ ξ and
‖h(w jk (∆j
k ))‖+ ‖∇x `(w jk (∆j
k ))‖ ≤ Mµjk (∆j
k )?
False: ∆jk+1 = ∆j
k /2repeat step 4
True: w j+1k = w j
k (∆jk )
go to step 1
OPTEC, December 02, 2008 48/ 58
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Restoration Phase
0. Choose parameters, w0k = wk , ∆0
k = ∆k , j = 0
1. Test the stop criterion
2. Compute snk,j and st
k,j
3. Compute ∆jk such that, for all ∆ ∈ [0,∆j
k ],
X jk (∆)z j
k (∆) ≥ γµjk (∆) and (x j
k (∆), z jk (∆)) > 0
4.
AREDjk
PREDjk
≥ ξ and
‖h(w jk (∆j
k ))‖+ ‖∇x `(w jk (∆j
k ))‖ ≤ Mµjk (∆j
k )?
False: ∆jk+1 = ∆j
k /2repeat step 4
True: w j+1k = w j
k (∆jk )
go to step 1
OPTEC, December 02, 2008 48/ 58
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Restoration Phase
0. Choose parameters, w0k = wk , ∆0
k = ∆k , j = 0
1. Test the stop criterion
2. Compute snk,j and st
k,j
3. Compute ∆jk such that, for all ∆ ∈ [0,∆j
k ],
X jk (∆)z j
k (∆) ≥ γµjk (∆) and (x j
k (∆), z jk (∆)) > 0
4.
AREDjk
PREDjk
≥ ξ and
‖h(w jk (∆j
k ))‖+ ‖∇x `(w jk (∆j
k ))‖ ≤ Mµjk (∆j
k )?
False: ∆jk+1 = ∆j
k /2repeat step 4
True: w j+1k = w j
k (∆jk )
go to step 1
OPTEC, December 02, 2008 48/ 58
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Restoration Phase
0. Choose parameters, w0k = wk , ∆0
k = ∆k , j = 0
1. Test the stop criterion
2. Compute snk,j and st
k,j
3. Compute ∆jk such that, for all ∆ ∈ [0,∆j
k ],
X jk (∆)z j
k (∆) ≥ γµjk (∆) and (x j
k (∆), z jk (∆)) > 0
4.
AREDjk
PREDjk
≥ ξ and
‖h(w jk (∆j
k ))‖+ ‖∇x `(w jk (∆j
k ))‖ ≤ Mµjk (∆j
k )?
False: ∆jk+1 = ∆j
k /2repeat step 4
True: w j+1k = w j
k (∆jk )
go to step 1
OPTEC, December 02, 2008 48/ 58
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Restoration Phase
0. Choose parameters, w0k = wk , ∆0
k = ∆k , j = 0
1. Test the stop criterion
2. Compute snk,j and st
k,j
3. Compute ∆jk such that, for all ∆ ∈ [0,∆j
k ],
X jk (∆)z j
k (∆) ≥ γµjk (∆) and (x j
k (∆), z jk (∆)) > 0
4.
AREDjk
PREDjk
≥ ξ and
‖h(w jk (∆j
k ))‖+ ‖∇x `(w jk (∆j
k ))‖ ≤ Mµjk (∆j
k )?
False: ∆jk+1 = ∆j
k /2repeat step 4
True: w j+1k = w j
k (∆jk )
go to step 1
OPTEC, December 02, 2008 48/ 58
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Restoration Phase
0. Choose parameters, w0k = wk , ∆0
k = ∆k , j = 0
1. Test the stop criterion
2. Compute snk,j and st
k,j
3. Compute ∆jk such that, for all ∆ ∈ [0,∆j
k ],
X jk (∆)z j
k (∆) ≥ γµjk (∆) and (x j
k (∆), z jk (∆)) > 0
4.
AREDjk
PREDjk
≥ ξ and
‖h(w jk (∆j
k ))‖+ ‖∇x `(w jk (∆j
k ))‖ ≤ Mµjk (∆j
k )?
False: ∆jk+1 = ∆j
k /2repeat step 4
True: w j+1k = w j
k (∆jk )
go to step 1
OPTEC, December 02, 2008 48/ 58
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Code – ipfilter 0.2
Written in FORTRAN 90/95.
Handles all kinds of NLP (including unconstrained problems andproblems with only simple bounds).
Includes standard, SIF and AMPL interfaces.
Symmetrizes the primal-dual systems (avoiding the inversionof Xk ): [
X 1/2k ∇2
xx`(wk )X 1/2k + Zk X 1/2
k ∇h(wk )
(X 1/2k ∇h(wk ))> 0
].
Sparse linear algebra handled by HSL routine MA27 (symmetricsystems).
The formulation handles ` ≤ x ≤ u explicitly.
OPTEC, December 02, 2008 49/ 58
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Code – ipfilter 0.2
Written in FORTRAN 90/95.
Handles all kinds of NLP (including unconstrained problems andproblems with only simple bounds).
Includes standard, SIF and AMPL interfaces.
Symmetrizes the primal-dual systems (avoiding the inversionof Xk ): [
X 1/2k ∇2
xx`(wk )X 1/2k + Zk X 1/2
k ∇h(wk )
(X 1/2k ∇h(wk ))> 0
].
Sparse linear algebra handled by HSL routine MA27 (symmetricsystems).
The formulation handles ` ≤ x ≤ u explicitly.
OPTEC, December 02, 2008 49/ 58
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Code – ipfilter 0.2
Written in FORTRAN 90/95.
Handles all kinds of NLP (including unconstrained problems andproblems with only simple bounds).
Includes standard, SIF and AMPL interfaces.
Symmetrizes the primal-dual systems (avoiding the inversionof Xk ): [
X 1/2k ∇2
xx`(wk )X 1/2k + Zk X 1/2
k ∇h(wk )
(X 1/2k ∇h(wk ))> 0
].
Sparse linear algebra handled by HSL routine MA27 (symmetricsystems).
The formulation handles ` ≤ x ≤ u explicitly.
OPTEC, December 02, 2008 49/ 58
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Code – ipfilter 0.2
Written in FORTRAN 90/95.
Handles all kinds of NLP (including unconstrained problems andproblems with only simple bounds).
Includes standard, SIF and AMPL interfaces.
Symmetrizes the primal-dual systems (avoiding the inversionof Xk ): [
X 1/2k ∇2
xx`(wk )X 1/2k + Zk X 1/2
k ∇h(wk )
(X 1/2k ∇h(wk ))> 0
].
Sparse linear algebra handled by HSL routine MA27 (symmetricsystems).
The formulation handles ` ≤ x ≤ u explicitly.
OPTEC, December 02, 2008 49/ 58
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Code – ipfilter 0.2
Written in FORTRAN 90/95.
Handles all kinds of NLP (including unconstrained problems andproblems with only simple bounds).
Includes standard, SIF and AMPL interfaces.
Symmetrizes the primal-dual systems (avoiding the inversionof Xk ): [
X 1/2k ∇2
xx`(wk )X 1/2k + Zk X 1/2
k ∇h(wk )
(X 1/2k ∇h(wk ))> 0
].
Sparse linear algebra handled by HSL routine MA27 (symmetricsystems).
The formulation handles ` ≤ x ≤ u explicitly.
OPTEC, December 02, 2008 49/ 58
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Code – ipfilter 0.2
Incorporates a restoration phase based on the same stepdecomposition.
Performs a warm start by carrying 5 iterations of
minx
f (x) + ρ h(x)>h(x)− µ0
n∑i=1
log xi , ρ > 0,
starting from the initial given point x0.
Perturbs the systems using inertia control and regularization
Scales the system matrix, variables, objective function andconstraints.
OPTEC, December 02, 2008 50/ 58
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Code – ipfilter 0.2
Incorporates a restoration phase based on the same stepdecomposition.
Performs a warm start by carrying 5 iterations of
minx
f (x) + ρ h(x)>h(x)− µ0
n∑i=1
log xi , ρ > 0,
starting from the initial given point x0.
Perturbs the systems using inertia control and regularization
Scales the system matrix, variables, objective function andconstraints.
OPTEC, December 02, 2008 50/ 58
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Code – ipfilter 0.2
Incorporates a restoration phase based on the same stepdecomposition.
Performs a warm start by carrying 5 iterations of
minx
f (x) + ρ h(x)>h(x)− µ0
n∑i=1
log xi , ρ > 0,
starting from the initial given point x0.
Perturbs the systems using inertia control and regularization
Scales the system matrix, variables, objective function andconstraints.
OPTEC, December 02, 2008 50/ 58
![Page 115: A globally convergent primal-dual interior-point filter method for …lnv/talks/optec.pdf · 2009-02-04 · programming, Mathematical Programming, 100 (2004) 379-410. R. S ILVA ,](https://reader034.vdocuments.mx/reader034/viewer/2022050110/5f47a0b4e520606dcd745912/html5/thumbnails/115.jpg)
Code – ipfilter 0.2
Incorporates a restoration phase based on the same stepdecomposition.
Performs a warm start by carrying 5 iterations of
minx
f (x) + ρ h(x)>h(x)− µ0
n∑i=1
log xi , ρ > 0,
starting from the initial given point x0.
Perturbs the systems using inertia control and regularization
Scales the system matrix, variables, objective function andconstraints.
OPTEC, December 02, 2008 50/ 58
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Testing – ipfilter 0.2
We tunned ipfilter with an old version of the CUTE collection(469 problems) and we also tested 631 problems from the recentCUTER collection (Sept. 2008).
All problems have at least one equality or inequality constraint(different from bounds) and satisfy n ≥ m.
The tests were run on a Fujitsu-Siemens Celsius V810workstation (4G RAM, 2 processors AMD 2.2GHz).
We made a comparison with ipopt (C++, version 3.5.1), aBarrier-Filter code from IBM developed by Andreas Wächter.
We use the same stopping criterion as in ipfilter.Stopping tolerance: 10−8; Maximum # of iterations: 1000.
OPTEC, December 02, 2008 51/ 58
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Testing – ipfilter 0.2
We tunned ipfilter with an old version of the CUTE collection(469 problems) and we also tested 631 problems from the recentCUTER collection (Sept. 2008).
All problems have at least one equality or inequality constraint(different from bounds) and satisfy n ≥ m.
The tests were run on a Fujitsu-Siemens Celsius V810workstation (4G RAM, 2 processors AMD 2.2GHz).
We made a comparison with ipopt (C++, version 3.5.1), aBarrier-Filter code from IBM developed by Andreas Wächter.
We use the same stopping criterion as in ipfilter.Stopping tolerance: 10−8; Maximum # of iterations: 1000.
OPTEC, December 02, 2008 51/ 58
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Testing – ipfilter 0.2
We tunned ipfilter with an old version of the CUTE collection(469 problems) and we also tested 631 problems from the recentCUTER collection (Sept. 2008).
All problems have at least one equality or inequality constraint(different from bounds) and satisfy n ≥ m.
The tests were run on a Fujitsu-Siemens Celsius V810workstation (4G RAM, 2 processors AMD 2.2GHz).
We made a comparison with ipopt (C++, version 3.5.1), aBarrier-Filter code from IBM developed by Andreas Wächter.
We use the same stopping criterion as in ipfilter.Stopping tolerance: 10−8; Maximum # of iterations: 1000.
OPTEC, December 02, 2008 51/ 58
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Testing – ipfilter 0.2
We tunned ipfilter with an old version of the CUTE collection(469 problems) and we also tested 631 problems from the recentCUTER collection (Sept. 2008).
All problems have at least one equality or inequality constraint(different from bounds) and satisfy n ≥ m.
The tests were run on a Fujitsu-Siemens Celsius V810workstation (4G RAM, 2 processors AMD 2.2GHz).
We made a comparison with ipopt (C++, version 3.5.1), aBarrier-Filter code from IBM developed by Andreas Wächter.
We use the same stopping criterion as in ipfilter.Stopping tolerance: 10−8; Maximum # of iterations: 1000.
OPTEC, December 02, 2008 51/ 58
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Constrained Tested Problems
number of problemsdimensions an old CUTE set CUTER (Sept. 2008)n < 1000 388 (45) 390 (56)
1000 ≤ n < 10000 76 (1544) 182 (3194)n ≥ 10000 5 (7979) 59 (6354)
total 469 631problem class an old CUTE set CUTER (Sept. 2008)equality constrained 245 327inequality constrained 177 226mixed (equalities and inequalities) 48 78linearly constrained 171 205nonlinearly constrained 298 426quadratic programming 91 103
OPTEC, December 02, 2008 52/ 58
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Numerical Results – All Constrained Problems
an old CUTE set CUTER (Sept. 2008)ipfilter ipopt ipfilter ipopt
# problems solved 449 448 532 549% robustness 95.74% 95.52% 84.34% 87.00%# average iterations 27.55 27.19 47.44 38.58# problems solved (< 500 iter.) 449 447 525 545% robustness (< 500 iter.) 95.74% 95.31% 83.20% 86.37%# average iterations (< 500 iter.) 27.55 25.78 37.51 34.14
OPTEC, December 02, 2008 53/ 58
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Numerical Results – QP Problems
an old CUTE set CUTER (Sept. 2008)ipfilter ipopt ipfilter ipopt
# problems solved 91 88 97 93% robustness 100.00% 96.70% 94.17% 90.29%# average iterations 26.74 36.45 42.35 47.09# problems solved (< 500 iter.) 91 88 96 92% robustness (< 500 iter.) 100.00% 96.70% 93.20% 89.32%# average iterations (< 500 iter.) 26.74 36.45 33.90 40.96
OPTEC, December 02, 2008 54/ 58
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eNLP Solver – ESA Project
Prime Contractor: Astos Solutions (Germany).
Consortium partners: University of Birmingham, University ofBremen, University of Coimbra, and Skysoft (Portugal).
Objective: Produce a general purpose European sparse NLPSolver, especially for NLPs associated with trajectoryoptimization problems.
Problems: Ascent/reentry of a spacecraft, trajectories planning ofsatellite missions, launcher test case, low thrust orbit transfer, ...
OPTEC, December 02, 2008 55/ 58
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Numerical Results – COPS Problems
COPS colection: difficult nonlinearly constrained optimizationproblems arising from optimal design, fluid dynamics, meshsmoothing, optimal control,...
15 problems from COPS 0.2 are included in the CUTEr collection (Sept. 2008).
ipfilter 0.2 ipopt 3.5.4# problems solved 13 13% robustness 86.67% 86.67%# average iterations 64.54 98.23# problems solved (< 500 iter.) 13 12% robustness (< 500 iter.) 86.67% 80.00%# average iterations (< 500 iter.) 64.54 53.42
OPTEC, December 02, 2008 56/ 58
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Numerical Results – COPS Problems
COPS colection: difficult nonlinearly constrained optimizationproblems arising from optimal design, fluid dynamics, meshsmoothing, optimal control,...
15 problems from COPS 0.2 are included in the CUTEr collection (Sept. 2008).
ipfilter 0.2 ipopt 3.5.4# problems solved 13 13% robustness 86.67% 86.67%# average iterations 64.54 98.23# problems solved (< 500 iter.) 13 12% robustness (< 500 iter.) 86.67% 80.00%# average iterations (< 500 iter.) 64.54 53.42
OPTEC, December 02, 2008 56/ 58
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Numerical Results – COPS Problems
COPS colection: difficult nonlinearly constrained optimizationproblems arising from optimal design, fluid dynamics, meshsmoothing, optimal control,...
15 problems from COPS 0.2 are included in the CUTEr collection (Sept. 2008).
ipfilter 0.2 ipopt 3.5.4# problems solved 13 13% robustness 86.67% 86.67%# average iterations 64.54 98.23# problems solved (< 500 iter.) 13 12% robustness (< 500 iter.) 86.67% 80.00%# average iterations (< 500 iter.) 64.54 53.42
OPTEC, December 02, 2008 56/ 58
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Future Work
Increase the number of problems solved (robustness).
Improve the CPU time.
Use iterative solvers (linear systems) and Hessian secantupdates.
OPTEC, December 02, 2008 57/ 58
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Future Work
Increase the number of problems solved (robustness).
Improve the CPU time.
Use iterative solvers (linear systems) and Hessian secantupdates.
OPTEC, December 02, 2008 57/ 58
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Future Work
Increase the number of problems solved (robustness).
Improve the CPU time.
Use iterative solvers (linear systems) and Hessian secantupdates.
OPTEC, December 02, 2008 57/ 58
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ipfilter Software
The current ipfilter version (0.2) requires first andsecond-order derivatives and is only available for problemswhere n ≥ m.
ipfilter is freely available for academic and researchpurposes.
ipfilter web page: http://www.mat.uc.pt/ipfilter.
OPTEC, December 02, 2008 58/ 58