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Nonsmooth convex-concave saddle point problems with cardinality penalties

Wei Bian, Xiaojun Chen

TL;DR

This paper defines a class of strong local saddle points based on the lower bound properties for stability of variable selection in a class of convexly constrained nonsmooth convex–concave saddle point problems with cardinality penalties.

Abstract

In this paper, we focus on a class of convexly constrained nonsmooth convex-concave saddle point problems with cardinality penalties. Although such nonsmooth nonconvex-nonconcave and discontinuous min-max problems may not have a saddle point, we show that they have a local saddle point and a global minimax point, and some local saddle points have the lower bound properties. We define a class of strong local saddle points based on the lower bound properties for stability of variable selection. Moreover, we give a framework to construct continuous relaxations of the discontinuous min-max problems based on the convolution, such that they have the same saddle points with the original problem. We also establish the relations between the continuous relaxation problems and the original problems regarding local saddle points, global minimax points, local minimax points and stationary points. Finally, we illustrate our results with distributionally robust sparse convex regression, sparse robust bond portfolio construction and sparse convex-concave logistic regression saddle point problems.

Nonsmooth convex-concave saddle point problems with cardinality penalties

TL;DR

This paper defines a class of strong local saddle points based on the lower bound properties for stability of variable selection in a class of convexly constrained nonsmooth convex–concave saddle point problems with cardinality penalties.

Abstract

In this paper, we focus on a class of convexly constrained nonsmooth convex-concave saddle point problems with cardinality penalties. Although such nonsmooth nonconvex-nonconcave and discontinuous min-max problems may not have a saddle point, we show that they have a local saddle point and a global minimax point, and some local saddle points have the lower bound properties. We define a class of strong local saddle points based on the lower bound properties for stability of variable selection. Moreover, we give a framework to construct continuous relaxations of the discontinuous min-max problems based on the convolution, such that they have the same saddle points with the original problem. We also establish the relations between the continuous relaxation problems and the original problems regarding local saddle points, global minimax points, local minimax points and stationary points. Finally, we illustrate our results with distributionally robust sparse convex regression, sparse robust bond portfolio construction and sparse convex-concave logistic regression saddle point problems.
Paper Structure (20 sections, 16 theorems, 182 equations, 2 figures, 1 table)

This paper contains 20 sections, 16 theorems, 182 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Existence of local saddle points of problem \ref{['obb']}
  3. Continuous relaxations
  4. Density functions
  5. Continuous relaxation models to \ref{['obb']}
  6. Theoretical analysis on exact continuous relaxations
  7. Relations on saddle points
  8. Relations on minimax points
  9. Smoothing functions to a nonsmooth convex-concave function $c$
  10. A particular case of problem \ref{['obb']}
  11. Density function $\rho$ under Assumption \ref{['ass7']}
  12. Relations on saddle points and minimax points
  13. Stationary points of \ref{['obb-3r']} with $\rho$ in Example \ref{['example4.1']}
  14. Density function $\rho$ under Assumption \ref{['ass8']}
  15. Continuous relaxations defined by different density functions
  16. ...and 5 more sections

Key Result

Proposition 2.1

Min-max problem obb always has a global minimax point.

Figures (2)

  • Figure 1: Relations between problems \ref{['obb-3-1']} and \ref{['obb-3r']} with different relaxations
  • Figure 2: Convergence of $\mathbf{x}^k$, $\mathbf{y}^k$, $p({\mathbf{x}}^k)$ and $q({\mathbf{y}}^k)$ generated by PGDA and convergence of $\mathbf{x}^k$, $\mathbf{y}^k$, $\tilde{p}({\mathbf{x}}^k)$ and $\tilde{q}({\mathbf{y}}^k)$ generated by APGDA

Theorems & Definitions (48)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Example 2.1
  • Definition 2.3
  • Example 2.2
  • Theorem 2.1
  • ...and 38 more