Complexity Analysis of Convex Majorization Schemes for Nonconvex Constrained Optimization

TL;DR

This paper develops and analyzes algorithms for nonconvex constrained optimization using convex surrogate envelopes, providing complexity results and demonstrating promising numerical performance.

Contribution

It introduces new convex majorization schemes for nonconvex problems, extending to second-order information and self-concordant settings, with iteration complexity analysis.

Findings

Algorithms efficiently solve nonconvex constrained problems.

Surrogate envelopes extend to second-order and self-concordant settings.

Numerical results show promising potential of the proposed methods.

Abstract

We introduce and study various algorithms for solving nonconvex minimization with inequality constraints, based on the construction of convex surrogate envelopes that majorize the objective and the constraints. In the case where the objective and constraint functions are gradient H\"{o}lderian continuous, the surrogate functions can be readily constructed and the solution method can be efficiently implemented. The surrogate envelopes are extended to the settings where the second-order information is available, and the convex subproblems are further represented by Dikin ellipsoids using the self-concordance of the convex surrogate constraints. Iteration complexities have been developed for both convex and nonconvex optimization models. The numerical results show promising potential of the proposed approaches.