Generalized Polyhedral DC Optimization Problems

TL;DR

This paper investigates optimization problems involving the difference of convex functions within a generalized polyhedral framework, providing new optimality conditions, solution sets, and duality-based algorithms.

Contribution

It introduces novel results for DC problems with generalized polyhedral convex functions and develops solution methods leveraging duality theory.

Findings

Derived optimality conditions for generalized polyhedral DC problems

Characterized local and global solution sets

Proposed duality-based solution algorithms

Abstract

The problem of minimizing the difference of two lower semicontinuous, proper, convex functions (a DC function) on a nonempty closed convex set in a locally convex Hausdorff topological vector space is studied in this paper. The focus is made on the situations where either the second component of the objective function is a generalized polyhedral convex function or the first component of the objective function is a generalized polyhedral convex function and the constraint set is generalized polyhedral convex. Various results on optimality conditions, the local solution set, the global solution set, and solution algorithms via duality are obtained. Useful illustrative examples are considered.