The Use of Squared Slack Variables in Nonlinear Second-Order Cone Programming
The Use of Squared Slack Variables in Nonlinear Second-Order Cone Programming
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DOI：
10.1007/s10957-016-0904-3
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发表时间：
2016-02
影响因子：
1.9
通讯作者：
E. H. Fukuda;M. Fukushima
中科院分区：
文献类型：
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作者：
E. H. Fukuda;M. Fukushima
在传统的非线性规划中，通过增加平方松弛变量，将具有不等式约束的问题转化为仅包含等式约束的问题的技术是众所周知的。不幸的是，它被认为是一个避免的技术在优化社区，因为优点通常不能弥补缺点，如增加的问题，数值不稳定性，和奇异性的维度。然而，在非线性二阶锥规划的背景下，情况发生了变化，因为具有平方松弛变量的重新表述的问题不再具有锥约束。这一事实使我们能够通过使用通用非线性规划求解器来解决问题。本文的目的是通过二阶充分条件和正则性条件建立原问题和重构问题的Karush-Kuhn-Tucker点之间的关系。我们还提出了一些初步的数值实验。
In traditional nonlinear programming, the technique of converting a problem with inequality constraints into a problem containing only equality constraints, by the addition of squared slack variables, is well known. Unfortunately, it is considered to be an avoided technique in the optimization community, since the advantages usually do not compensate for the disadvantages, like the increase in the dimension of the problem, the numerical instabilities, and the singularities. However, in the context of nonlinear second-order cone programming, the situation changes, because the reformulated problem with squared slack variables has no longer conic constraints. This fact allows us to solve the problem by using a general-purpose nonlinear programming solver. The objective of this work is to establish the relation between Karush–Kuhn–Tucker points of the original and the reformulated problems by means of the second-order sufficient conditions and regularity conditions. We also present some preliminary numerical experiments.