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Convex Analysisの二,三の進展について

On robust estimation of the location parameter

Convex Analysis: (pms-28)

In this paper, we analyze the convergence of the alternating direction method of multipliers (ADMM) for minimizing a nonconvex and possibly nonsmooth objective function, $$\phi (x_0,\ldots ,x_p,y)$$
 , subject to coupled linear equality constraints. Our ADMM updates each of the primal variables $$x_0,\ldots ,x_p,y$$
 , followed by updating the dual variable. We separate the variable y from $$x_i$$
 ’s as it has a special role in our analysis. The developed convergence guarantee covers a variety of nonconvex functions such as piecewise linear functions, $$\ell _q$$
 quasi-norm, Schatten-q quasi-norm (
 $$0<q<1$$
 ), minimax concave penalty (MCP), and smoothly clipped absolute deviation penalty. It also allows nonconvex constraints such as compact manifolds (e.g., spherical, Stiefel, and Grassman manifolds) and linear complementarity constraints. Also, the $$x_0$$
 -block can be almost any lower semi-continuous function. By applying our analysis, we show, for the first time, that several ADMM algorithms applied to solve nonconvex models in statistical learning, optimization on manifold, and matrix decomposition are guaranteed to converge. Our results provide sufficient conditions for ADMM to converge on (convex or nonconvex) monotropic programs with three or more blocks, as they are special cases of our model. ADMM has been regarded as a variant to the augmented Lagrangian method (ALM). We present a simple example to illustrate how ADMM converges but ALM diverges with bounded penalty parameter $$\beta $$
 . Indicated by this example and other analysis in this paper, ADMM might be a better choice than ALM for some nonconvex nonsmooth problems, because ADMM is not only easier to implement, it is also more likely to converge for the concerned scenarios.

http://www.optimization-online.org/DB_FILE/2015/11/5222.pdf

Global Convergence of ADMM in Nonconvex Nonsmooth Optimization

In this paper we consider a mathematical program with semidefinite cone complementarity constraints (SDCMPCC). Such a problem is a matrix analogue of the mathematical program with (vector) complementarity constraints (MPCC) and includes MPCC as a special case. We first derive explicit formulas for the proximal and limiting normal cone of the graph of the normal cone to the positive semidefinite cone. Using these formulas and classical nonsmooth first order necessary optimality conditions we derive explicit expressions for the strong-, Mordukhovich- and Clarke- (S-, M- and C-)stationary conditions. Moreover we give constraint qualifications under which a local solution of SDCMPCC is a S-, M- and C-stationary point. Moreover we show that applying these results to MPCC produces new and weaker necessary optimality conditions.

https://www.math.uvic.ca/faculty/janeye/publications/papers/SDCMPCC_Revised_Nov30_13.pdf

First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints

In this paper, we define a class of linear conic programming (which we call matrix cone programming or MCP) involving the epigraphs of five commonly used matrix norms and the well studied symmetric cone. MCP has recently been found to have many important applications, for example, in nuclear norm relaxations of affine rank minimization problems. In order to make the defined MCP tractable and meaningful, we must first understand the structure of these epigraphs. So far, only the epigraph of the Frobenius matrix norm, which can be regarded as a second order cone, has been well studied. Here, we take an initial step to study several important properties, including its closed form solution, calm Bouligand-differentiability and strong semismoothness, of the metric projection operator over the epigraph of the $$l_1,\,l_\infty $$, spectral or operator, and nuclear matrix norm, respectively. These properties make it possible to apply augmented Lagrangian methods, which have recently received a great deal of interests due to their high efficiency in solving large scale semidefinite programming, to this class of MCP problems. The work done in this paper is far from comprehensive. Rather it is intended as a starting point to call for more insightful research on MCP so that it can serve as a basic tool to solve more challenging convex matrix optimization problems in years to come.

An introduction to a class of matrix cone programming

Matrix optimization problems (MOPs) involving the Ky Fan $k$-norm arise frequently from many applications. In order to design algorithms to solve large scale MOPs involving the Ky Fan $k$-norm, we need to understand the first and second order properties of the Moreau--Yosida regularization of the Ky Fan $k$-norm function and the indicator function of the Ky Fan $k$-norm ball. According to the general theory on spectral functions, in this paper we shall conduct a thorough study on the Moreau--Yosida regularization of the vector $k$-norm function and the indicator function of the vector $k$-norm ball. In particular, we show that the proximal mappings associated with these two vector $k$-norm related functions both admit fast and analytically computable solutions. Moreover, we propose algorithms of low computational cost to compute the directional derivatives of these two proximal mappings and then completely characterize their Frechet differentiability. The work here thus builds the fundamental tools needed...

http://www.optimization-online.org/DB_FILE/2011/03/2978.pdf

On the Moreau--Yosida Regularization of the Vector $k$-Norm Related Functions

This paper is devoted to studying the robust isolated calmness of the Karush--Kuhn--Tucker (KKT) solution mapping for a large class of interesting conic programming problems (including most commonly known ones arising from applications) at a locally optimal solution. Under the Robinson constraint qualification, we show that the KKT solution mapping is robustly isolated calm if and only if both the strict Robinson constraint qualification and the second order sufficient condition hold. This implies, among others, that at a locally optimal solution the second order sufficient condition is needed for the KKT solution mapping to have the Aubin property.

Characterization of the Robust Isolated Calmness for a Class of Conic Programming Problems

This paper is devoted to studying the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) solution mapping for a large class of interesting conic programming problems (including most commonly known ones arising from applications) at a locally optimal solution. Under the Robinson constraint qualification, we show that the KKT solution mapping is robustly isolated calm if and only if both the strict Robinson constraint qualification and the second order sufficient condition hold. This implies, among others, that at a locally optimal solution the second order sufficient condition is needed for the KKT solution mapping to have the Aubin property.

Chao Ding

Papers

First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints

An introduction to a class of matrix cone programming

On the Moreau--Yosida Regularization of the Vector $k$-Norm Related Functions

Characterization of the Robust Isolated Calmness for a Class of Conic Programming Problems

Characterization of the Robust Isolated Calmness for a Class of Conic Programming Problems