Enhanced Karush–Kuhn–Tucker condition and weaker constraint qualifications

Ye, Jane J.; Jin, Zhang

doi:10.1007/s10107-013-0667-7

Cited by 26 publications

(45 citation statements)

References 21 publications

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“…Hence if x * is D * -quasinormal for problem (33), then (x * , Ψ (x * )) is quasinormal for problem (34). Moreover, gph Ψ is a closed subset in ℜ d+1 by the continuity of Ψ .…”

Section: Corollarymentioning

confidence: 99%

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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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“…Hence if x * is D * -quasinormal for problem (33), then (x * , Ψ (x * )) is quasinormal for problem (34). Moreover, gph Ψ is a closed subset in ℜ d+1 by the continuity of Ψ .…”

Section: Corollarymentioning

confidence: 99%

“…Theorem 3 Let x * be a local minimizer of problem (33). If D * -quasi-normality holds at x * , then there exists ρ 0 > 0 such that for any ρ ≥ ρ 0 , x * is also a local minimizer of the exact penalization problem min x∈Ω f (x) + Ψ (x) + ρ ( g(x) + + h(x) ) .…”

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confidence: 99%

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

Guo

2017

Math. Program.

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“…stands for the existence of sequences which connect the sign of the multiplier with the sign of the associated constraint in a neighborhood of the stationary point. Enhanced Fritz-John conditions were used previously to generalize some classical results [13,31]. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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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“…Since the enhanced FJ condition is stronger than the classical FJ condition, it results in constraint qualifications such as quasi-normality that are weaker than the NNAMCQ. Very recently, the enhanced KKT conditions for problem (NLP) with locally Lipschitzian data based on the limiting subdifferential and limiting normal cone were derived in [34] and the sensitivity of the value function was established in terms of the set of the enhanced KKT multipliers which may be smaller than the set of the classical KKT multipliers and hence provides sharper results.…”

Section: Introductionmentioning

confidence: 99%

“…There are two technical difficulties involved when the space X is not finite dimensional. First, unlike in the finite dimensional case, the quadratic penalization approach in [34] cannot be employed anymore because the compactness of the closed unit ball is possibly invalid in a Banach space X, which plays a key role in guaranteeing the existence of enhanced sequential approximating solution by using the Weierstrass theorem in [34]. Nevertheless, by virtue of the optimization process, for any > 0, a problem in the form of (MPGC) always possesses an -optimal solution (see [4] for a definition), provided that the optimal value of the problem is finite.…”

Section: Introductionmentioning

confidence: 99%

Mathematical Programs with Geometric Constraints in Banach Spaces: Enhanced Optimality, Exact Penalty, and Sensitivity

Guo¹,

Ye²,

Jin³

2013

SIAM J. Optim.

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In this paper we study the mathematical program with geometric constraints such that the image of a mapping from a Banach space is included in a nonempty and closed subset of a finite dimensional space. We obtain the nonsmooth enhanced Fritz John necessary optimality conditions in terms of the approximate subdifferential. In the case where the Banach space is a weakly compactly generated Asplund space, the optimality condition obtained can be expressed in terms of the limiting subdifferential, while in the general case it can be expressed in terms of the Clarke subdifferential. One of the technical difficulties in obtaining such a result in an infinite dimensional space is that no compactness result can be used to show the existence of local minimizers of a perturbed problem. In this paper we employ the celebrated Ekeland's variational principle to obtain the results instead. The enhanced Fritz John condition allows us to obtain the enhanced Karush-Kuhn-Tucker condition under the pseudo-normality and the quasi-normality conditions which are weaker than the classical normality conditions. We then prove that the quasi-normality is a sufficient condition for the existence of local error bounds of the constraint system. Finally we obtain a tighter upper estimate for the subdifferentials of the value function of the perturbed problem in terms of the enhanced multipliers.

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Enhanced Karush–Kuhn–Tucker condition and weaker constraint qualifications

Cited by 26 publications

References 21 publications

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

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

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

Mathematical Programs with Geometric Constraints in Banach Spaces: Enhanced Optimality, Exact Penalty, and Sensitivity

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