Optimization over convex polyhedra via Hadamard parametrizations

TL;DR

This paper introduces a novel approach for linearly constrained optimization by using Hadamard parametrization, transforming the feasible set into an algebraic variety with favorable geometric properties, enabling new optimality verification and solution methods.

Contribution

It develops explicit formulas for tangent cones, a procedure for recovering Lagrangian multipliers, and a hybrid Riemannian projected gradient algorithm with convergence guarantees.

Findings

Effective verification of optimality conditions without extra constraint qualifications

Successful stratification of the variety into Riemannian manifolds

Numerical experiments show competitive performance against state-of-the-art methods

Abstract

In this paper, we study linearly constrained optimization problems (LCP). After applying Hadamard parametrization, the feasible set of the parametrized problem (LCPH) becomes an algebraic variety, with conducive geometric properties which we explore in depth. We derive explicit formulas for the tangent cones and second-order tangent sets associated with the parametrized polyhedra. Based on these formulas, we develop a procedure to recover the Lagrangian multipliers associated with the constraints to verify the optimality conditions of the given primal variable without requiring additional constraint qualifications. Moreover, we develop a systematic way to stratify the variety into a disjoint union of finitely many Riemannian manifolds. This leads us to develop a hybrid algorithm combining Riemannian optimization and projected gradient to solve (LCP) with convergence guarantees. Numerical experiments are conducted to verify the effectiveness of our method compared with various state-of-the-art algorithms.