Directional Quasi-/Pseudo-Normality as Sufficient Conditions for Metric Subregularity

Bai, Kuang; Ye, Jane J.; Jin, Zhang

doi:10.1137/18m1232498

Cited by 25 publications

(17 citation statements)

References 59 publications

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“…The relation between MPSC piecewise RCPLD and MPSC-RCPLD can be checked easily by using definitions. To obtain all other relationships, we use definitions and the results presented here together with the results from [2,7,9,23]. From the diagram, we can see that directional conditions in a nonzero critical direction d are weaker than the corresponding nondirectional ones.…”

Section: Discussionmentioning

confidence: 99%

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Optimality Conditions and Exact Penalty for Mathematical Programs with Switching Constraints

Liang¹,

Ye²

2020

Preprint

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In this paper, we study optimality conditions and exact penalization for the mathematical program with switching constraints (MPSC). We give new optimality conditions for MPSC including MPSC S-stationary/strongly M-stationary/M-stationary condition in critical directions. The strongly M-stationarity in critical directions builds a bridge between the S-stationarity in critical directions and the M-stationarity in critical directions. We propose some sufficient conditions for the local error bound property and obtain exact penalty results for MPSC. Finally we apply our results to the mathematical program with either-or-constraints.

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Section: Discussionmentioning

confidence: 99%

“…Based on the directional M-stationary condition (8) and directional S-stationary condition (9), we now define the directional version of the W, S, M-stationarity for MPSC.…”

Section: New Optimality Conditions For Mpscmentioning

confidence: 99%

Optimality Conditions and Exact Penalty for Mathematical Programs with Switching Constraints

Liang¹,

Ye²

2020

Preprint

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“…Recently, weaker sufficient conditions than FOSCMS such as the directional quasi/pseudonormality was introduced in [1]. More sufficient conditions based on directional normal cones or/and for specific systems can be found e.g.…”

Section: Is Also Established and The Proof Is Complete ✷mentioning

confidence: 99%

“…[8]). Unlike the first-order optimality conditions for which much research works have been appeared, there is very little research done with the second-order optimality conditions for MPCC, SOC-MPCC and SDC-MPCC, let alone the general non-convex set-constrained problem (1). The classical second-order necessary optimality condition for MPCCs was given in [32,Theorem 7(1)] under the MPCC strict Mangasarian-Fromovitz constraint qualification (SMFCQ).…”

Section: Introductionmentioning

confidence: 99%

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Second-order optimality conditions for non-convex set-constrained optimization problems

Gfrerer¹,

Ye²,

Zhou³

2019

Preprint

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In this paper we study second-order optimality conditions for non-convex setconstrained optimization problems. For a convex set-constrained optimization problem, it is well-known that second-order optimality conditions involve the support function of the second-order tangent set. In this paper we propose two approaches for establishing secondorder optimality conditions for the non-convex case. In the first approach we extend the concept of the support function so that it is applicable to general non-convex set-constrained problems, whereas in the second approach we introduce the notion of the directional regular tangent cone and apply classical results of convex duality theory. Besides the second-order optimality conditions, the novelty of our approach lies in the systematic introduction and use, respectively, of directional versions of well-known concepts from variational analysis.

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On the directional asymptotic approach in optimization theory

Benko,

Mehlitz

2024

Math. Program.

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As a starting point of our research, we show that, for a fixed order $$\gamma \ge 1$$ γ ≥ 1 , each local minimizer of a rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary in the classical sense (corresponding to stationarity of order 1), satisfies stationarity conditions in terms of a coderivative construction of order $$\gamma $$ γ , or is asymptotically stationary with respect to a critical direction as well as order $$\gamma $$ γ in a certain sense. By ruling out the latter case with a constraint qualification not stronger than directional metric subregularity, we end up with new necessary optimality conditions comprising a mixture of limiting variational tools of orders 1 and $$\gamma $$ γ . These abstract findings are carved out for the broad class of geometric constraints and $$\gamma :=2$$ γ : = 2 , and visualized by examples from complementarity-constrained and nonlinear semidefinite optimization. As a byproduct of the particular setting $$\gamma :=1$$ γ : = 1 , our general approach yields new so-called directional asymptotic regularity conditions which serve as constraint qualifications guaranteeing M-stationarity of local minimizers. We compare these new regularity conditions with standard constraint qualifications from nonsmooth optimization. Further, we extend directional concepts of pseudo- and quasi-normality to arbitrary set-valued mappings. It is shown that these properties provide sufficient conditions for the validity of directional asymptotic regularity. Finally, a novel coderivative-like variational tool is used to construct sufficient conditions for the presence of directional asymptotic regularity. For geometric constraints, it is illustrated that all appearing objects can be calculated in terms of initial problem data.

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Directional Quasi-/Pseudo-Normality as Sufficient Conditions for Metric Subregularity

Cited by 25 publications

References 59 publications

Optimality Conditions and Exact Penalty for Mathematical Programs with Switching Constraints

Optimality Conditions and Exact Penalty for Mathematical Programs with Switching Constraints

Second-order optimality conditions for non-convex set-constrained optimization problems

On the directional asymptotic approach in optimization theory

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