Minimizing the Moreau Envelope of Nonsmooth Convex Functions over the Fixed Point Set of Certain Quasi-Nonexpansive Mappings
Minimizing the Moreau Envelope of Nonsmooth Convex Functions over the Fixed Point Set of Certain Quasi-Nonexpansive Mappings
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DOI：
10.1007/978-1-4419-9569-8_17
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发表时间：
2011
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影响因子：
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通讯作者：
I. Yamada;M. Yukawa;M. Yamagishi
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文献类型：
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作者：
I. Yamada;M. Yukawa;M. Yamagishi
本文的第一个目标是从使用不动点集作为约束的角度，提出一个有用的用于凸优化的准非扩张映射工具箱。许多凸优化问题已经通过优雅地转化为定点问题得到了解决。其基本原理是对某个拟非扩张映射进行迭代运算，生成收敛到其不动点的序列。然而，这样的映射通常具有无限多个固定点，这意味着从固定点集Fix(T)中进行选择应该非常重要。然而，大多数定点方法只能从定点集中返回“未指定”的点，这需要多次迭代。因此，根据常识，希望从定点集中找到一个“最优”的点似乎是不现实的。幸运的是，考虑到准非扩张映射的集合作为工具箱，我们可以简单地通过混合最速下降法来完成这一具有挑战性的任务，只要成本函数是平滑的并且其导数是 Lipschitz 连续的。出现了一个问题：我们如何处理“非平滑”成本函数？第二个目标是提出混合最速下降法和 Moreau-Yosida 正则化思想的重要集成，为 Fix(T) 上的非光滑凸优化这一具有挑战性的问题提供一种有用的方法。关键是通过莫罗-约西达正则化对原始非平滑成本函数进行平滑，其导数始终是 Lipschitz 连续的。混合最速下降法的应用领域可以扩展到理想光滑逼近Fix(T)的最小化。我们提出了所提出方法的数学思想及其在组合优化问题中的应用：高效多输入多输出（MIMO）通信系统的高度非线性容量约束下的最小天线子集选择问题。
The first aim of this paper is to present a useful toolbox of quasi-nonexpansive mappings for convex optimization from the viewpoint of using their fixed point sets as constraints. Many convex optimization problems have been solved through elegant translations into fixed point problems. The underlying principle is to operate a certain quasi-nonexpansive mappingTiteratively and generate a convergent sequence to its fixed point. However, such a mapping often has infinitely many fixed points, meaning that a selection from the fixed point set Fix(T) should be of great importance. Nevertheless, most fixed point methods can only return an “unspecified” point from the fixed point set, which requires many iterations. Therefore, based on common sense, it seems unrealistic to wish for an “optimal” one from the fixed point set. Fortunately, considering the collection of quasi-nonexpansive mappings as a toolbox, we can accomplish this challenging mission simply by thehybrid steepest descent method, provided that the cost function is smooth and its derivative is Lipschitz continuous. A question arises:how can we deal with “nonsmooth” cost functions? The second aim is to propose a nontrivial integration of the ideas of thehybrid steepest descent methodand theMoreau–Yosida regularization, yielding a useful approach to the challenging problem of nonsmooth convex optimization over Fix(T). The key is the use of smoothing of the original nonsmooth cost function by itsMoreau–Yosida regularizationwhose the derivative is always Lipschitz continuous. The field of application of hybrid steepest descent method can be extended to the minimization of the ideal smooth approximation Fix(T). We present the mathematical ideas of the proposed approach together with its application to a combinatorial optimization problem: the minimal antenna-subset selection problem under a highly nonlinear capacity-constraint for efficient multiple input multiple output (MIMO) communication systems.