Strong duality of a conic optimization problem with a single hyperplane and two cone constraints
Strong duality of a conic optimization problem with a single hyperplane and two cone constraints
复制标题
具有单个超平面和两个圆锥约束的圆锥优化问题的强对偶性
DOI：
10.1080/02331934.2023.2251987
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发表时间：
2021
期刊：
影响因子：
2.2
通讯作者：
M. Kojima
中科院分区：
文献类型：
--
作者：
Sunyoung Kim;M. Kojima
一般锥优化问题的强（拉格朗日）对偶性研究由来已久，其深刻而复杂的结果以不同的形式出现在各种文献中。因此，表征已知和未知结果有时可能很困难。本文的目的是提供一个统一的和几何的观点，强对偶的COP的已知结果。对于我们的框架，我们采用了一个COP最小化一个线性函数在一个向量变量$x$受到一个超平面约束$x \在H$和两个锥约束$x \在K_1$，$x \在K_2$。它可以等同地重新表示为一个更简单的COP与单超平面约束$x \在H$和单锥约束$x \在K_1 \cap K_2$。这种简单的COP及其对偶以及它们之间的对偶关系都可以用几何的方法表示，并且在没有任何约束条件的情况下，它们不存在对偶间隙。如果两个锥K_1 $和K_2 $的乘积的Minkowski和是闭的，或者如果重新表述的COP的对偶满足一定的斯莱特条件，则原始目标COP的对偶等价于重新表述的COP的对偶。因此，这两个条件使得有可能转移所有的对偶结果，包括最优解的存在性和/或有界性，在重新制定的COP上的原始目标COP，并进一步对一个标准的原始对偶对COP的对称性。
Strong (Lagrangian) duality of general conic optimization problems (COPs) has long been studied and its profound and complicated results appear in different forms in a wide range of literatures. As a result, characterizing the known and unknown results can sometimes be difficult. The aim of this article is to provide a unified and geometric view of strong duality of COPs for the known results. For our framework, we employ a COP minimizing a linear function in a vector variable $x$ subject to a single hyperplane constraint $x \in H$ and two cone constraints $x \in K_1$, $x \in K_2$. It can be identically reformulated as a simpler COP with the single hyperplane constraint $x \in H$ and the single cone constraint $x \in K_1 \cap K_2$. This simple COP and its dual as well as their duality relation can be represented geometrically, and they have no duality gap without any constraint qualification. The dual of the original target COP is equivalent to the dual of the reformulated COP if the Minkowski sum of the duals of the two cones $K_1$ and $K_2$ is closed or if the dual of the reformulated COP satisfies a certain Slater condition. Thus, these two conditions make it possible to transfer all duality results, including the existence and/or boundedness of optimal solutions, on the reformulated COP to the ones on the original target COP, and further to the ones on a standard primal-dual pair of COPs with symmetry.