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On semidefinite descriptions for convex hulls of quadratic programs

Alex L. Wang, Fatma Kilinc-Karzan

Abstract

Quadratically constrained quadratic programs (QCQPs) are a highly expressive class of nonconvex optimization problems. While QCQPs are NP-hard in general, they admit a natural convex relaxation via the standard semidefinite program (SDP) relaxation. In this paper we study when the convex hull of the epigraph of a QCQP coincides with the projected epigraph of the SDP relaxation. We present a sufficient condition for convex hull exactness and show that this condition is further necessary under an additional geometric assumption. The sufficient condition is based on geometric properties of $Γ$, the cone of convex Lagrange multipliers, and its relatives $Γ_1$ and $Γ^\circ$.

On semidefinite descriptions for convex hulls of quadratic programs

Abstract

Quadratically constrained quadratic programs (QCQPs) are a highly expressive class of nonconvex optimization problems. While QCQPs are NP-hard in general, they admit a natural convex relaxation via the standard semidefinite program (SDP) relaxation. In this paper we study when the convex hull of the epigraph of a QCQP coincides with the projected epigraph of the SDP relaxation. We present a sufficient condition for convex hull exactness and show that this condition is further necessary under an additional geometric assumption. The sufficient condition is based on geometric properties of , the cone of convex Lagrange multipliers, and its relatives and .
Paper Structure (11 sections, 12 theorems, 40 equations)

This paper contains 11 sections, 12 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Convex hull exactness
  5. Sufficient conditions for convex hull exactness
  6. Necessary conditions for convex hull exactness
  7. Explicit descriptions of ${\cal R}'(x,t)$
  8. Revisiting the setting of polyhedral $\Gamma$
  9. Example applications of \ref{['thm:conv_hull_sufficient', 'thm:conv_nec', 'thm:polyhedral']}
  10. Quadratic matrix programs
  11. Sets defined by two quadratic constraints

Key Result

lemma 1

Suppose as:definite holds. Then,

Theorems & Definitions (33)

  • definition 1
  • remark 1
  • lemma 1
  • proof
  • corollary 1
  • corollary 2
  • remark 2
  • definition 2
  • definition 3
  • definition 4
  • ...and 23 more