Optimality conditions and constraint qualifications for cardinality constrained optimization problems

IF 1.1 Q2 MATHEMATICS, APPLIED

Numerical Algebra Control and Optimization Pub Date : 2022-09-17 DOI:10.3934/naco.2023011

Zhuo-Feng Xiao, J. Ye

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引用次数: 0

Abstract

The cardinality constrained optimization problem (CCOP) is an optimization problem where the maximum number of nonzero components of any feasible point is bounded. In this paper, we consider CCOP as a mathematical program with disjunctive subspaces constraints (MPDSC). Since a subspace is a special case of a convex polyhedral set, MPDSC is a special case of the mathematical program with disjunctive constraints (MPDC). Using the special structure of subspaces, we are able to obtain more precise formulas for the tangent and (directional) normal cones for the disjunctive set of subspaces. We then obtain first and second order optimality conditions by using the corresponding results from MPDC. Thanks to the special structure of the subspace, we are able to obtain some results for MPDSC that do not hold in general for MPDC. In particular we show that the relaxed constant positive linear dependence (RCPLD) is a sufficient condition for the metric subregularity/error bound property for MPDSC which is not true for MPDC in general. Finally we show that under all constraint qualifications presented in this paper, certain exact penalization holds for CCOP.

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基数约束优化问题的最优性条件和约束条件

基数约束优化问题(CCOP)是任意可行点的非零分量的最大数目有界的优化问题。本文将CCOP看作具有析取子空间约束(MPDSC)的数学规划。由于子空间是凸多面体集的一种特殊情况，因此MPDSC是带析取约束的数学规划的一种特殊情况。利用子空间的特殊结构，我们可以得到子空间的析取集的切锥和(方向)法锥的更精确的公式。然后利用MPDC的相应结果得到了一阶和二阶最优性条件。由于子空间的特殊结构，我们能够得到一些对MPDSC不适用于MPDC的结果。特别地，我们证明了松弛常数正线性相关(RCPLD)是MPDSC的度量子正则性/误差界性质的充分条件，而MPDC一般不成立。最后，我们证明了在本文提出的所有约束条件下，对CCOP存在一定的精确惩罚。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Numerical Algebra Control and Optimization MATHEMATICS, APPLIED-

CiteScore

3.10

自引率

0.00%

发文量

期刊介绍： Numerical Algebra, Control and Optimization (NACO) aims at publishing original papers on any non-trivial interplay between control and optimization, and numerical techniques for their underlying linear and nonlinear algebraic systems. Topics of interest to NACO include the following: original research in theory, algorithms and applications of optimization; numerical methods for linear and nonlinear algebraic systems arising in modelling, control and optimisation; and original theoretical and applied research and development in the control of systems including all facets of control theory and its applications. In the application areas, special interests are on artificial intelligence and data sciences. The journal also welcomes expository submissions on subjects of current relevance to readers of the journal. The publication of papers in NACO is free of charge.