Parametric Nonlinear Programming Problems under the Relaxed Constant Rank Condition

Minchenko, L. I.; Stakhovski, Sergey

doi:10.1137/090761318

Cited by 49 publications

(72 citation statements)

References 30 publications

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“…In this section we will prove that, when a limit point x * satisfies some constraint qualifications, the KKT conditions of PA(x * ) hold at x * . In the last few years many weak constraint qualifications were introduced in [4,5,36,37,40,41]. In particular, the Constant Positive Generators (CPG) condition introduced in [5] is one of the weakest constraint qualification that allows one that prove that AKKT implies KKT.…”

Section: Results Under Constraint Qualificationsmentioning

confidence: 99%

Optimality properties of an Augmented Lagrangian method on infeasible problems

2014

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Sometimes, the feasible set of an optimization problem that one aims to solve using a Nonlinear Programming algorithm is empty. In this case, two characteristics of the algorithm are desirable. On the one hand, the algorithm should converge to a minimizer of some infeasibility measure. On the other hand, one may wish to find a point with minimal infeasibility for which some optimality condition, with respect to the objective function, holds. Ideally, the algorithm should converge to a minimizer of the objective function subject to minimal infeasibility. In this paper the behavior of an Augmented Lagrangian algorithm with respect to those properties will be studied.

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Section: Results Under Constraint Qualificationsmentioning

confidence: 99%

Optimality properties of an Augmented Lagrangian method on infeasible problems

2014

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“…For example, it would be interesting to investigate if CRSC can be used to extend results on sensitivity and perturbation analysis that already exist for RCRCQ and CPLD [22,23,24,33]. Other possibility would be to extend CRSC to the context of problems with complementarity or vanishing constraints [18,21], as was done recently for CPLD in [19,20].…”

Section: Resultsmentioning

confidence: 99%

Two New Weak Constraint Qualifications and Applications

Andreani¹,

Haeser²,

Schuverdt³

et al. 2012

SIAM J. Optim.

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We present two new constraint qualifications (CQ) that are weaker than the recently introduced Relaxed Constant Positive Linear Dependence (RCPLD) constraint qualification. RCPLD is based on the assumption that many subsets of the gradients of the active constraints preserve positive linear dependence locally. A major open question was to identify the exact set of gradients whose properties had to be preserved locally and that would still work as a CQ. This is done in the first new constraint qualification, that we call Constant Rank of the Subspace Component (CRSC) CQ. This new CQ also preserves many of the good properties of RCPLD, like local stability and the validity of an error bound. We also introduce an even weaker CQ, called Constant Positive Generator (CPG), that can replace RCPLD in the analysis of the global convergence of algorithms. We close this work extending convergence results of algorithms belonging to all the main classes of nonlinear optimization methods: SQP, augmented Lagrangians, interior point algorithms, and inexact restoration. * This work was supported by PRONEX-Optimization (PRONEX-CNPq/FAPERJ E-26/171.510/2006-APQ1), Fapesp (Grants

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“…Recently, assumptions based on constant rank that have been used to ensure the validity of second-order conditions for every Lagrange multiplier [2,4,20]. In this section we show that such conditions can be naturally replaced by a much weaker condition based on Abadie's CQ.…”

Section: Abadie-type Conditionsmentioning

confidence: 92%

“…It is a direct generalization of [2, Theorem 3.1] and [20,Theorem 6], where we clearly identify the set of gradients that need to be well behaved instead of looking at all the subsets that involve active inequalities. Proof Let d be any non-zero direction in T(x * ), and consider without loss of generality that d = 1.…”

Section: Definition 33 the Index Set Of Positive Inequality Multiplimentioning

confidence: 99%

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On second-order optimality conditions in nonlinear optimization

Andreani

Behling

Haeser

et al. 2016

Optimization Methods and Software

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In this work we present new weak conditions that ensure the validity of necessary second-order optimality conditions (SOC) for nonlinear optimization. We are able to prove that weak and strong SOCs hold for all Lagrange multipliers using Abadie-type assumptions. We also prove weak and strong SOCs for at least one Lagrange multiplier imposing the Mangasarian-Fromovitz constraint qualification and a weak constant rank assumption.

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Parametric Nonlinear Programming Problems under the Relaxed Constant Rank Condition

Cited by 49 publications

References 30 publications

Optimality properties of an Augmented Lagrangian method on infeasible problems

Optimality properties of an Augmented Lagrangian method on infeasible problems

Two New Weak Constraint Qualifications and Applications

On second-order optimality conditions in nonlinear optimization

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