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Sparse Quadratically Constrained Quadratic Programming via Semismooth Newton Method

Shuai Li, Shenglong Zhou, Ziyan Luo

TL;DR

The paper tackles sparse quadratically constrained quadratic programming (SQCQP), a challenging NP-hard problem due to the $\

Abstract

Quadratically constrained quadratic programming (QCQP) has long been recognized as a computationally challenging problem, particularly in large-scale or high-dimensional settings where solving it directly becomes intractable. The complexity further escalates when a sparsity constraint is involved, giving rise to the problem of sparse QCQP (SQCQP), which makes conventional solution methods even less effective. Existing approaches for solving SQCQP typically rely on mixed-integer programming formulations, relaxation techniques, or greedy heuristics but often suffer from computational inefficiency and limited accuracy. In this work, we introduce a novel paradigm by designing an efficient algorithm that directly addresses SQCQP. To be more specific, we introduce P-stationarity to establish first- and second-order optimality conditions of the original problem, leading to a system of nonlinear equations whose generalized Jacobian is proven to be nonsingular under mild assumptions. Most importantly, these equations facilitate the development of a semismooth Newton-type method that exhibits significantly low computational complexity due to the sparsity constraint and achieves a locally quadratic convergence rate. Finally, extensive numerical experiments validate the accuracy and computational efficiency of the algorithm compared to several established solvers.

Sparse Quadratically Constrained Quadratic Programming via Semismooth Newton Method

TL;DR

The paper tackles sparse quadratically constrained quadratic programming (SQCQP), a challenging NP-hard problem due to the $\

Abstract

Quadratically constrained quadratic programming (QCQP) has long been recognized as a computationally challenging problem, particularly in large-scale or high-dimensional settings where solving it directly becomes intractable. The complexity further escalates when a sparsity constraint is involved, giving rise to the problem of sparse QCQP (SQCQP), which makes conventional solution methods even less effective. Existing approaches for solving SQCQP typically rely on mixed-integer programming formulations, relaxation techniques, or greedy heuristics but often suffer from computational inefficiency and limited accuracy. In this work, we introduce a novel paradigm by designing an efficient algorithm that directly addresses SQCQP. To be more specific, we introduce P-stationarity to establish first- and second-order optimality conditions of the original problem, leading to a system of nonlinear equations whose generalized Jacobian is proven to be nonsingular under mild assumptions. Most importantly, these equations facilitate the development of a semismooth Newton-type method that exhibits significantly low computational complexity due to the sparsity constraint and achieves a locally quadratic convergence rate. Finally, extensive numerical experiments validate the accuracy and computational efficiency of the algorithm compared to several established solvers.

Paper Structure

This paper contains 29 sections, 12 theorems, 117 equations, 4 figures, 12 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Contribution
  4. Organization
  5. Notation
  6. Optimality Analysis
  7. First-order optimality conditions
  8. Stationary equations
  9. Second-order optimality conditions
  10. Nonsingularity of Generalized Jacobian
  11. A Semi-smooth Newton Algorithm
  12. Algorithm design
  13. Local convergence rate
  14. Numerical Experiments
  15. Recovery problems
  16. ...and 14 more sections

Key Result

Theorem 2.1

Let ${\bf x}^*$ be a local minimizer of (eq: SCQP) and Assumption assum_RLICQ hold at ${\bf x}^*$. Then there is a unique $({\bm \nu }^*,{\bm \mu }^*,{\bm \lambda }^*) \in \mathbb{R}^{n}\times\mathbb{R}^k \times \mathbb{R}^m$ such that ${\bf x}^*$ is a KKT point of (eq: SCQP).

Figures (4)

  • Figure 1: Relationships among different points for problem \ref{['eq: SCQP']}.
  • Figure 2: Comparison of SNSQP and BPG for Example \ref{['example: recovery1']}.
  • Figure 3: Solutions obtained by SNSQP for Example \ref{['example: SCCA1']} with $s=10$.
  • Figure 4: Effect of sparsity level $s$ for Example \ref{['example: SCCA1']}.

Theorems & Definitions (36)

  • Definition 2.1: KKT Points
  • Theorem 2.1
  • proof
  • Example 2.1
  • Remark 2.1
  • Definition 2.2
  • Theorem 2.2: First-order necessary condition
  • proof
  • Theorem 2.3: First-order sufficient condition
  • proof
  • ...and 26 more