Inexact Penalty Decomposition Methods for Optimization Problems with Geometric Constraints

This paper provides a theoretical and numerical investigation of a penalty decomposition scheme for the solution of optimization problems with geometric constraints. In particular, we consider someĀ  situations where parts of the constraints are nonconvex and complicated, like cardinality constraints, disjunctive programs, or matrix problems involving rank constraints. By a variable duplication andĀ  decomposition strategy, … Read more

Exterior-point Optimization for Nonconvex Learning

In this paper we present the nonconvex exterior-point optimization solver (NExOS)—a novel first-order algorithm tailored to constrained nonconvex learning problems. We consider the problem of minimizing a convex function over nonconvex constraints, where the projection onto the constraint set is single-valued around local minima. A wide range of nonconvex learning problems have this structure including … Read more

The Direct Extension of ADMM for Multi-block Convex Minimization Problems is Not Necessarily Convergent

The alternating direction method of multipliers (ADMM) is now widely used in many fields, and its convergence was proved when two blocks of variables are alternatively updated. It is strongly desirable and practically valuable to extend ADMM directly to the case of a multi-block convex minimization problem where its objective function is the sum of … Read more

A proximal point algorithm for sequential feature extraction applications

We propose a proximal point algorithm to solve LAROS problem, that is the problem of finding a “large approximately rank-one submatrix”. This LAROS problem is used to sequentially extract features in data. We also develop a new stopping criterion for the proximal point algorithm, which is based on the duality conditions of \eps-optimal solutions of … Read more