Strict Constraint Qualifications and Sequential Optimality Conditions for Constrained Optimization

Andreani, Roberto; Martínez, José Mário; Ramos, Alberto; Silva, Paulo J. S.

doi:10.1287/moor.2017.0879

Cited by 58 publications

(37 citation statements)

References 34 publications

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“…As we have already mentioned in the introduction, sequential optimality conditions have been used to strengthen the theoretical convergence results of several methods. See [2,3,4,7,8,9,15,25] and references therein. The goal is to prove that a specific method achieves, say, AKKT points.…”

Section: Strength Of the Pakkt Condition: Akkt Vs Pakkt Methodsmentioning

confidence: 99%

“…In [8], the authors presented the weakest strict constraint qualification associated with AKKT, called Cone Continuity Property (CCP). Recently [9], CCP was renamed to "AKKT-regular" and the weakest SCQs related to SAKKT [21], CAKKT [7] and AGP [25] conditions were established.…”

Section: The Positive Approximate Karush-kuhn-tucker Conditionmentioning

confidence: 99%

“…This is the central issue in this work. In the last years, special attention has been devoted to so-called sequential optimality conditions for nonlinear constrained optimization (see for example [3,7,9,8,15,25]). They are related to the stopping criteria of algorithms, and aim to unify their theoretical convergence results.…”

Section: Introductionmentioning

confidence: 99%

“…During the last years, it has been proved that AKKT points are stationary points under different constraint qualifications such as the constant positive linear dependence (CPLD) [6,27], the relaxed constant positive linear dependence (RCPLD) [4] and the constant positive generator (CPG) [5]. Finally, it was shown that the cone continuity property (CCP) (also called AKKT-regular CQ [9]) is the least stringent constraint qualification with this property [8]. All these CQs deal with rank and/or positive linear dependence assumptions.…”

Section: Introductionmentioning

confidence: 99%

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A Sequential Optimality Condition Related to the Quasi-normality Constraint Qualification and Its Algorithmic Consequences

Andreani¹,

Fazzio²,

Schuverdt³

et al. 2019

SIAM J. Optim.

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In the present paper, we prove that the augmented Lagrangian method converges to KKT points under the quasinormality constraint qualification, which is associated with the external penalty theory. An interesting consequence is that the Lagrange multipliers estimates computed by the method remain bounded in the presence of the quasinormality condition. In order to establish a more general convergence result, a new sequential optimality condition for smooth constrained optimization, called PAKKT, is defined. The new condition takes into account the sign of the dual sequence, constituting an adequate sequential counterpart to the (enhanced) Fritz-John necessary optimality conditions proposed by Hestenes, and later extensively treated by Bertsekas. PAKKT points are substantially better than points obtained by the classical Approximate KKT (AKKT) condition, which has been used to establish theoretical convergence results for several methods. In particular, we present a simple problem with complementarity constraints such that all its feasible points are AKKT, while only the solutions and a pathological point are PAKKT. This shows the efficiency of the methods that reach PAKKT points, particularly the augmented Lagrangian algorithm, in such problems. We also provided the appropriate strict constraint qualification associated with the PAKKT sequential optimality condition, called PAKKT-regular, and we prove that it is strictly weaker than both quasinormality and cone continuity property. PAKKT-regular connects both branches of these independent constraint qualifications, generalizing all previous theoretical convergence results for the augmented Lagrangian method in the literature.where f : R n → R and X is the feasible set composed of equality and inequality constraints of the form X = {x | h(x) = 0, g(x) ≤ 0}, * This work has been partially supported by CEPID-CeMEAI (FAPESP 2013/07375-0), FAPESP (Grant 2013/05475-7) and CNPq (Grant 303013/2013-3).

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Section: Strength Of the Pakkt Condition: Akkt Vs Pakkt Methodsmentioning

confidence: 99%

Section: The Positive Approximate Karush-kuhn-tucker Conditionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Sequential Optimality Condition Related to the Quasi-normality Constraint Qualification and Its Algorithmic Consequences

Andreani¹,

Fazzio²,

Schuverdt³

et al. 2019

SIAM J. Optim.

Self Cite

View full text Add to dashboard Cite

show abstract

“…This property makes sequential optimality conditions useful tools for naturally providing a perturbed optimality condition, which is suitable for the definition of stopping criteria and complexity analysis for several algorithms. Also, a careful study of the relation of sequential optimality conditions with classical stationarity measures under a constraint qualification, yields global convergence results under weak assumptions [AFSS19,AMRS16,AMRS18].…”

Section: Introductionmentioning

confidence: 99%

Condições de otimalidade para otimização cônica

Viana¹

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In this work, we perform an extension of the so-called Approximate Karush-Kuhn-Tucker (AKKT) condition, initially introduced in nonlinear programming [AHM11], for nonlinear symmetric cone programming. A new condition, which we call Trace AKKT (TAKKT), was also presented for the nonlinear semidefinite programming problem. TAKKT proved to be more practical than AKKT for nonlinear semidefinite programming. We prove that both the AKKT condition and the TAKKT condition are optimality conditions. Results of global convergence for the augmented Lagrangian method were obtained. Strict qualification conditions were introduced to measure the strength of the overall convergence results presented. Through these strict qualification conditions, it was possible to verify that our results of global convergence proved to be better than those known in the literature. We also present a proof for a particular case of the conjecture made in [AMS07].

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Necessary optimality conditions and exact penalization for non-Lipschitz nonlinear programs

Guo

2017

Math. Program.

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When the objective function is not locally Lipschitz, constraint qualifications are no longer sufficient for Karush-Kuhn-Tucker (KKT) conditions to hold at a local minimizer, let alone ensuring an exact penalization. In this paper, we extend quasi-normality and relaxed constant positive linear dependence (RCPLD) condition to allow the non-Lipschitzness of the objective function and show that they are sufficient for KKT conditions to be necessary for optimality. Moreover, we derive exact penalization results for the following two special cases. When the non-Lipschitz term in the objective function is the sum of a composite function of a separable lower semi-continuous function with a continuous function and an indicator function of a closed subset, we show that a local minimizer of our problem is also a local minimizer of an exact penalization problem under a local error bound condition for a restricted constraint region and a suitable assumption on the outer separable function. When the non-Lipschitz term is the sum of a continuous function and an indicator function of a closed subset, we also show that our problem admits an exact penalization under an extended quasi-normality involving the coderivative of the continuous function.Keywords Non-Lipschitz program · necessary optimality · exact penalization · error bound Mathematics Subject Classification (2010) 90C26 · 90C30 · 90C46

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Strict Constraint Qualifications and Sequential Optimality Conditions for Constrained Optimization

Cited by 58 publications

References 34 publications

A Sequential Optimality Condition Related to the Quasi-normality Constraint Qualification and Its Algorithmic Consequences

A Sequential Optimality Condition Related to the Quasi-normality Constraint Qualification and Its Algorithmic Consequences

Condições de otimalidade para otimização cônica

Necessary optimality conditions and exact penalization for non-Lipschitz nonlinear programs

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