论文标题
关于具有基数限制的数学程序的弱平稳性条件：统一方法
On the weak stationarity conditions for Mathematical Programs with Cardinality Constraints: a unified approach
论文作者
论文摘要
在本文中，我们研究了一类优化问题，称为具有基数限制的数学程序（MPCAC）。这种问题通常很难解决，因为它涉及的约束并非连续凸，而是提供稀疏的解决方案。因此，我们通过将其建模为混合企业问题，然后解决其持续的对应物，以合适的方式重新重新重新进行MPCAC，将其称为轻松的问题。我们通过在两种情况下分析经典约束来研究放松的问题：线性和非线性。在线性案例中，我们提出了一种一般方法，并进行了对吉尼亚德和阿巴迪约束资格的讨论，在这种情况下证明，轻松问题的每个最小化都可以满足Karush-Kuhn-Tucker（KKT）条件。另一方面，在非线性情况下，我们表明可能违反了一些标准的约束资格。因此，我们不能断言KKT点。通过提出一种从最弱到最强的平稳性，我们定义了用于MPCAC问题最小化的最小化器，我们定义了新的和较弱的平稳性条件。
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 that goes from the weakest to the strongest stationarity.
