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Convex quadratic sets and the complexity of mixed integer convex quadratic programming

Alberto Del Pia

TL;DR

The paper develops a unified, polynomial-time framework to reduce non-full-dimensional mixed-integer linear and convex quadratic sets to lower-dimensional full-dimensional isomorphic counterparts while preserving mixed-integer vectors, by introducing and leveraging integer reflexive generalized inverses. It extends known polyhedral reduction techniques to convex quadratic sets and proves that mixed integer convex quadratic programming (MICQP) is fixed-parameter tractable (FPT) with respect to the number of integer variables, via a Löwner-John-type projection and Lenstra-type feasibility and optimization schemes. The key contributions include a polynomial-time construction of integer reflexive generalized inverses, a full-dimensionality characterization and reduction for convex quadratic sets, and an FPT algorithm for MICQP that avoids the ellipsoid method, unifying and extending results for pure integer and convex quadratic programming. The results provide a robust toolkit for designing FPT algorithms and subroutines applicable to broader mixed-integer nonlinear programming problems, with potential impact on ellipsoid rounding and approximation approaches in MICQP and related domains.

Abstract

In pure integer linear programming it is often desirable to work with polyhedra that are full-dimensional, and it is well known that it is possible to reduce any polyhedron to a full-dimensional one in polynomial time. More precisely, using the Hermite normal form, it is possible to map a non full-dimensional polyhedron to a full-dimensional isomorphic one in a lower-dimensional space, while preserving integer vectors. In this paper, we extend the above result simultaneously in two directions. First, we consider mixed integer vectors instead of integer vectors, by leveraging on the concept of "integer reflexive generalized inverse." Second, we replace polyhedra with convex quadratic sets, which are sets obtained from polyhedra by enforcing one additional convex quadratic inequality. We study structural properties of convex quadratic sets, and utilize them to obtain polynomial time algorithms to recognize full-dimensional convex quadratic sets, and to find an affine function that maps a non full-dimensional convex quadratic set to a full-dimensional isomorphic one in a lower-dimensional space, while preserving mixed integer vectors. We showcase the applicability and the potential impact of these results by showing that they can be used to prove that mixed integer convex quadratic programming is fixed parameter tractable with parameter the number of integer variables. Our algorithm unifies and extends the known polynomial time solvability of pure integer convex quadratic programming in fixed dimension and of convex quadratic programming.

Convex quadratic sets and the complexity of mixed integer convex quadratic programming

TL;DR

The paper develops a unified, polynomial-time framework to reduce non-full-dimensional mixed-integer linear and convex quadratic sets to lower-dimensional full-dimensional isomorphic counterparts while preserving mixed-integer vectors, by introducing and leveraging integer reflexive generalized inverses. It extends known polyhedral reduction techniques to convex quadratic sets and proves that mixed integer convex quadratic programming (MICQP) is fixed-parameter tractable (FPT) with respect to the number of integer variables, via a Löwner-John-type projection and Lenstra-type feasibility and optimization schemes. The key contributions include a polynomial-time construction of integer reflexive generalized inverses, a full-dimensionality characterization and reduction for convex quadratic sets, and an FPT algorithm for MICQP that avoids the ellipsoid method, unifying and extending results for pure integer and convex quadratic programming. The results provide a robust toolkit for designing FPT algorithms and subroutines applicable to broader mixed-integer nonlinear programming problems, with potential impact on ellipsoid rounding and approximation approaches in MICQP and related domains.

Abstract

In pure integer linear programming it is often desirable to work with polyhedra that are full-dimensional, and it is well known that it is possible to reduce any polyhedron to a full-dimensional one in polynomial time. More precisely, using the Hermite normal form, it is possible to map a non full-dimensional polyhedron to a full-dimensional isomorphic one in a lower-dimensional space, while preserving integer vectors. In this paper, we extend the above result simultaneously in two directions. First, we consider mixed integer vectors instead of integer vectors, by leveraging on the concept of "integer reflexive generalized inverse." Second, we replace polyhedra with convex quadratic sets, which are sets obtained from polyhedra by enforcing one additional convex quadratic inequality. We study structural properties of convex quadratic sets, and utilize them to obtain polynomial time algorithms to recognize full-dimensional convex quadratic sets, and to find an affine function that maps a non full-dimensional convex quadratic set to a full-dimensional isomorphic one in a lower-dimensional space, while preserving mixed integer vectors. We showcase the applicability and the potential impact of these results by showing that they can be used to prove that mixed integer convex quadratic programming is fixed parameter tractable with parameter the number of integer variables. Our algorithm unifies and extends the known polynomial time solvability of pure integer convex quadratic programming in fixed dimension and of convex quadratic programming.
Paper Structure (12 sections, 13 theorems, 98 equations)

This paper contains 12 sections, 13 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Systems of linear equations and polyhedra
  3. Characterization of mixed integer solutions to systems of linear equations
  4. Reduction to full-dimensional polyhedra
  5. Convex Quadratic Sets
  6. Characterization of full-dimensional convex quadratic sets
  7. Reduction to full-dimensional convex quadratic sets
  8. Complexity of Mixed Integer Convex Quadratic Programming
  9. Löwner-John-type algorithm
  10. The feasibility problem
  11. Boundedness of \ref{['prob MICQP']}
  12. The optimization problem

Key Result

lemma 1

Let $A \in \mathbb Q^{m \times n}$. There is a polynomial time algorithm that finds an integer reflexive generalized inverse $A^{\#}_I \in \mathbb Q^{n \times m}$ of $A$. The algorithm also finds a unimodular matrix $U \in \mathbb Z^{n \times n}$ such that $A^{\#}_I A = U U^{-1}$, where $r$ is the

Theorems & Definitions (29)

  • lemma 1
  • proof
  • proposition 1
  • proof
  • theorem 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof
  • ...and 19 more