从透视图到金石投影
From perspective maps to epigraphical projections
摘要：
将闭正常凸函数投影到上图或水平集上，可以通过寻找包含作为邻近参数的函数的邻近算子的标量方程的根来实现。本文发展了这类标量方程的变分分析。该方法是基于一般凸优化问题的变分分析性质的研究，这些问题是（部分）无穷投影的问题中的函数和凸核的透视图。当核是欧几里得范数平方时，解映射对应于邻近映射，因此为一般情况导出的变分性质适用于邻近情况。值函数和相应的解决方案映射的性质-包括局部Lipschitz连续性，方向可微性和半光滑性-推导。建立了一个用于计算epigraphical和level-set投影的SC $^1$优化框架。在1-范数投影上的数值实验表明了该方法的有效性，并与专门的算法进行了比较
英文摘要：
The projection onto the epigraph or a level set of a closed proper convex function can be achieved by finding a root of a scalar equation that involves the proximal operator as a function of the proximal parameter. This paper develops the variational analysis of this scalar equation. The approach is based on a study of the variational-analytic properties of general convex optimization problems that are (partial) infimal projections of the the sum of the function in question and the perspective map of a convex kernel. When the kernel is the Euclidean norm squared, the solution map corresponds to the proximal map, and thus the variational properties derived for the general case apply to the proximal case. Properties of the value function and the corresponding solution map -- including local Lipschitz continuity, directional differentiability, and semismoothness -- are derived. An SC$^1$ optimization framework for computing epigraphical and level-set projections is thus established. Numerical experiments on 1-norm projection illustrate the effectiveness of the approach as compared with specialized algorithms
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