当前位置: X-MOL 学术Appl. Math. Optim. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
On the Weak Stationarity Conditions for Mathematical Programs with Cardinality Constraints: A Unified Approach
Applied Mathematics and Optimization ( IF 1.8 ) Pub Date : 2021-02-04 , DOI: 10.1007/s00245-021-09752-0
Evelin H. M. Krulikovski , Ademir A. Ribeiro , Mael Sachine

In this paper, we study a class of optimization problems, called Mathematical Programs with Cardinality Constraints (MPCaC). This kind of problem is generally difficult to deal with, because it involves a constraint that is not continuous neither convex, but provides sparse solutions. Thereby we reformulate MPCaC in a suitable way, by modeling it as mixed-integer problem and then addressing its continuous counterpart, which will be referred to as relaxed problem. We investigate the relaxed problem by analyzing the classical constraints in two cases: linear and nonlinear. In the linear case, we propose a general approach and present a discussion of the Guignard and Abadie constraint qualifications, proving in this case that every minimizer of the relaxed problem satisfies the Karush–Kuhn–Tucker (KKT) conditions. On the other hand, in the nonlinear case, we show that some standard constraint qualifications may be violated. Therefore, we cannot assert about KKT points. Motivated to find a minimizer for the MPCaC problem, we define new and weaker stationarity conditions, by proposing a unified approach.



中文翻译:

具有基数约束的数学程序的弱平稳条件:统一方法

在本文中,我们研究了一类优化问题,称为具有基数约束的数学程序(MPCaC)。这种问题通常很难处理,因为它涉及的约束既不是连续的,也不是凸的,而是提供了稀疏的解决方案。因此,通过将MPCaC建模为混合整数问题,然后解决其连续对应项(称为松弛问题),我们以合适的方式重新制定了MPCaC。我们通过分析两种情况下的经典约束来研究松弛问题:线性和非线性。在线性情况下,我们提出了一种通用方法,并讨论了Guignard和Abadie约束条件,在这种情况下,证明松弛问题的每个最小化条件都满足Karush-Kuhn-Tucker(KKT)条件。另一方面,在非线性情况下,我们表明可能会违反某些标准约束条件。因此,我们无法断言KKT点。为了找到针对MPCaC问题的最小化方法,我们通过提出统一的方法来定义新的和较弱的平稳性条件。

更新日期:2021-02-04
down
wechat
bug