A penalty barrier framework for nonconvex constrained optimization

TL;DR

This paper introduces a novel penalty barrier framework for nonconvex constrained optimization that combines penalty and interior-point methods, enabling efficient solving of complex problems with theoretical convergence guarantees.

Contribution

It develops a unified approach merging penalty and barrier techniques, with a marginalization step that results in well-conditioned subproblems and broad applicability to nonconvex optimization.

Findings

The framework ensures convergence in nonconvex settings.

Subproblems are well-conditioned and independent of explicit constraints.

Numerical examples demonstrate effectiveness across diverse problems.

Abstract

We consider minimization problems with structured objective function and smooth constraints, and present a flexible framework that combines the beneficial regularization effects of (exact) penalty and interior-point methods. In the fully nonconvex setting, a pure barrier approach requires careful steps when approaching the infeasible set, thus hindering convergence. We show how a tight integration with a penalty scheme mitigates this issue and enables the construction of subproblems whose domain is independent of the explicit constraints. This decoupling allows us to leverage efficient solvers designed for unconstrained or suitably structured optimization tasks. The key behind all this is a marginalization step: closely related to a conjugacy operation, this step effectively merges (exact) penalty and barrier into a smooth, full domain functional object. When the penalty exactness takes effect, the generated subproblems do not suffer the ill-conditioning typical of barrier methods, nor do they exhibit the nonsmoothness of exact penalty terms. We provide a theoretical characterization of the algorithm and its asymptotic properties, deriving convergence results for fully nonconvex problems. Stronger conclusions are available for the convex setting, where optimality can be guaranteed. Illustrative examples and numerical simulations demonstrate the wide range of problems our theory and algorithm are able to cover.