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Solving convex QPs with structured sparsity under indicator conditions

Daniel Bienstock, Tongtong Chen

TL;DR

A family of polynomial-time approximation algorithms and negative complexity results for convex optimization problems where disjoint blocks of variables are controlled by binary indicator variables that are also subject to conditions, e.g., cardinality.

Abstract

We study convex optimization problems where disjoint blocks of variables are controlled by binary indicator variables that are also subject to conditions, e.g., cardinality. Several classes of important examples can be formulated in such a way that both the objective and the constraints are separable convex quadratics. We describe a family of polynomial-time approximation algorithms and negative complexity results.

Solving convex QPs with structured sparsity under indicator conditions

TL;DR

A family of polynomial-time approximation algorithms and negative complexity results for convex optimization problems where disjoint blocks of variables are controlled by binary indicator variables that are also subject to conditions, e.g., cardinality.

Abstract

We study convex optimization problems where disjoint blocks of variables are controlled by binary indicator variables that are also subject to conditions, e.g., cardinality. Several classes of important examples can be formulated in such a way that both the objective and the constraints are separable convex quadratics. We describe a family of polynomial-time approximation algorithms and negative complexity results.

Paper Structure

This paper contains 11 sections, 16 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Tree-decompositions, treewidth and pathwidth
  3. Applications and Examples
  4. Cardinality-constrained portfolio optimization
  5. Long-short problems with orthogonality constraints.
  6. Convex quadratic optimization problems with banded matrices and indicator variables
  7. Truss topology design problems
  8. Algorithm
  9. Dynamic programming algorithm for minimum-value sketch ensemble problem
  10. NP-hardness
  11. Glossary

Key Result

Theorem 1.4

Let $0 < \epsilon < 1$ and let $\omega$ denote the treewidth of the constraint-block intersection graph. Let $\kappa$ denote the largest support of any combinatorial constraint, and let $m_c$ denote the number of combinatorial constraints. There is an algorithm that runs in time $O( (n + m)(2/\epsil

Theorems & Definitions (26)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.4
  • Lemma 1.5
  • Lemma 1.6
  • Definition 1.7
  • Definition 1.8
  • Lemma 2.1
  • Lemma 2.2
  • Definition 3.1
  • ...and 16 more