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Finite convergence of the Moment-SOS hierarchy under hidden convexity

Srećko Ðurašinović, Jean B. Lasserre

Abstract

One considers polynomial optimization problems with compact feasible set $\mathbfΩ$ defined by SOS-concave polynomials $g_j$, and with a globally non-convex polynomial objective $f$. We show that if $f$ is strongly convex on $\mathbfΩ$, or SOS-convex on $\mathbfΩ$ when the constraints $g_j$ are at most quadratic, then the associated Moment-SOS hierarchy converges in finitely many steps, without à priori knowledge of this hidden (local) convexity. In addition, in the latter case, the exact order for which the relaxation is exact is provided by the degree of a Putinar-like certificate of convexity. This demonstrates that a general-purpose hierarchy can adapt to favorable hidden properties of a specific instance without being informed of them, yielding certified global minimizers.

Finite convergence of the Moment-SOS hierarchy under hidden convexity

Abstract

One considers polynomial optimization problems with compact feasible set defined by SOS-concave polynomials , and with a globally non-convex polynomial objective . We show that if is strongly convex on , or SOS-convex on when the constraints are at most quadratic, then the associated Moment-SOS hierarchy converges in finitely many steps, without à priori knowledge of this hidden (local) convexity. In addition, in the latter case, the exact order for which the relaxation is exact is provided by the degree of a Putinar-like certificate of convexity. This demonstrates that a general-purpose hierarchy can adapt to favorable hidden properties of a specific instance without being informed of them, yielding certified global minimizers.
Paper Structure (6 sections, 2 theorems, 19 equations)

This paper contains 6 sections, 2 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Notation, definitions and preliminary result
  4. Main result
  5. Finite convergence of the Moment-SOS hierarchy
  6. Conclusion

Key Result

Lemma 2.5

Let Assumption ass-1 hold, and let ${\boldsymbol{\phi}}=(\phi_{\boldsymbol{\alpha}})_{{\boldsymbol{\alpha}}\in\mathbb{N}^d_{2n}}$ be such that

Theorems & Definitions (9)

  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • Example 1: Illustration when $d=2$