Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimization over convex polyhedra via Hadamard parametrizations

Tianyun Tang, Kim-Chuan Toh

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.

Optimization over convex polyhedra via Hadamard parametrizations

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.

Paper Structure

This paper contains 30 sections, 17 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Optimization over convex polyhedra
  3. Hadamard parametrization and its limitations
  4. Optimality conditions
  5. Riemannian optimization on algebraic variety
  6. Organization of this paper
  7. Basic notation and definitions
  8. Optimality conditions
  9. Geometric properties of $\mathcal{P}_{A,b}$
  10. Optimality conditions of (\ref{['LCP']}) and (\ref{['LCPH']})
  11. Verifying optimality conditions
  12. Manifold decomposition of $\mathcal{P}_{A,b}$
  13. Manifold decomposition based on regular representation
  14. Some examples
  15. Algorithm
  16. ...and 15 more sections

Key Result

Proposition 2

For any $y\in \mathcal{P}_{A,b}$ with regular representation $(M,N,r),$ the inner and contingent tangent cones of $y$ are given as follows:

Theorems & Definitions (47)

  • Definition 1
  • Proposition 2
  • Proposition 3
  • Remark 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 37 more