Solving Conic Optimization Problems via Self-Dual Embedding and Facial Reduction: A Unified Approach
Solving Conic Optimization Problems via Self-Dual Embedding and Facial Reduction: A Unified Approach
复制标题
通过自对偶嵌入和面部缩减解决圆锥优化问题：统一方法
DOI：
10.1137/15m1049415
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
E. Andersen
中科院分区：
文献类型：
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作者：
Frank Permenter;Henrik A. Friberg;E. Andersen
当应用于圆锥优化问题时，我们建立了Borwein和Wolkowicz的面部还原算法与Goldman和Tucker的自以为是的同质模型之间的联系。具体而言，我们表明，自偶偶有的均匀模型在未能返回原始偶偶式最佳解决方案或不可行证书时返回面部还原证书。使用此观察结果，我们提供了一种基于面部减少的算法，以解决原则上始终成功的原始问题。 （对于双重问题，很容易说出一种类似的算法。）该算法具有吸引人的属性，即仅在需要时才进行面部减少，而不是在可能的情况下进行。例如，如果存在原始的双重最佳解决方案，即使Slater的状况失败，它也可以代替面部减少证书。对于线性，二阶和半芬特优化的情况，我们表明可以通过假设Oracle访问Central-Path Limit POI ...来实现该算法...
We establish connections between the facial reduction algorithm of Borwein and Wolkowicz and the self-dual homogeneous model of Goldman and Tucker when applied to conic optimization problems. Specifically, we show that the self-dual homogeneous model returns facial reduction certificates when it fails to return a primal-dual optimal solution or a certificate of infeasibility. Using this observation, we give an algorithm based on facial reduction for solving the primal problem that, in principle, always succeeds. (An analogous algorithm is easily stated for the dual problem.) This algorithm has the appealing property that it only performs facial reduction when it is required, not when it is possible; e.g., if a primal-dual optimal solution exists, it will be found in lieu of a facial reduction certificate even if Slater's condition fails. For the case of linear, second-order, and semidefinite optimization, we show that the algorithm can be implemented by assuming oracle access to the central-path limit poi...