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Proximity results in convex mixed-integer programming

Burak Kocuk, Diego Moran Ramirez

TL;DR

The paper develops a proximity framework for convex mixed-integer programs, extending classical linear-IP results to nonlinear convex constraints. It shows that when the continuous relaxation’s recession cone is full-dimensional, proximity and integrality-gap bounds can be made independent of the RHS or the particular continuous-optimal solution, via a cone-geometry analysis and covering-radius concepts. The authors specialize to simple and non-simple second-order-cone sets, providing explicit bounds for ellipsoids, paraboloids, hyperboloids, and intersections of spheres, and introduce proximity-driven and IG-driven strategies, including lattice-based arguments. They also identify fundamental limitations: in general, with multiple Lorentz cones or irrational cones, RHS-dependent IG may occur, and they supply conditions under which IG stays finite or is bounded by constants. Overall, the work unifies geometric, lattice, and conic-representation techniques to bound proximity and integrality gaps for SOCIPs, with implications for tightening relaxations in MIPs and informing solver design.

Abstract

We study proximity (resp. integrality gap), that is, the distance (resp. difference) between the optimal solutions (resp. optimal values) of convex integer programs (IP) and the optimal solutions (resp. optimal values) of their continuous relaxations. We show that these values can be upper bounded in terms of the recession cone of the feasible region of the continuous relaxation when the recession cone is full-dimensional. If the recession cone is not full-dimensional we give sufficient conditions to obtain a finite integrality gap. We then specialize our analysis to second-order conic IPs. In the case the feasible region is defined by a single Lorentz cone constraint, we give upper bounds on proximity and integrality gap in terms of the data of the problem (the objective function vector, the matrix defining the conic constraint, the right-hand side, and the covering radius of a related lattice). We also give conditions for these bounds to be independent of the right-hand side, akin to the linear IP case. Finally, in the case the feasible region is defined by multiple Lorentz cone constraints, we show that, in general, we cannot give bounds that are independent of the corresponding right-hand side. Although our results are presented for the integer lattice $\mathbb{Z}^n$, the bounds can be easily adapted to work for any general lattice, including the usual mixed-integer lattice $\mathbb{Z}^{n_1}\times\mathbb{R}^{n_2}$, by considering the appropriate covering radius when needed.

Proximity results in convex mixed-integer programming

TL;DR

The paper develops a proximity framework for convex mixed-integer programs, extending classical linear-IP results to nonlinear convex constraints. It shows that when the continuous relaxation’s recession cone is full-dimensional, proximity and integrality-gap bounds can be made independent of the RHS or the particular continuous-optimal solution, via a cone-geometry analysis and covering-radius concepts. The authors specialize to simple and non-simple second-order-cone sets, providing explicit bounds for ellipsoids, paraboloids, hyperboloids, and intersections of spheres, and introduce proximity-driven and IG-driven strategies, including lattice-based arguments. They also identify fundamental limitations: in general, with multiple Lorentz cones or irrational cones, RHS-dependent IG may occur, and they supply conditions under which IG stays finite or is bounded by constants. Overall, the work unifies geometric, lattice, and conic-representation techniques to bound proximity and integrality gaps for SOCIPs, with implications for tightening relaxations in MIPs and informing solver design.

Abstract

We study proximity (resp. integrality gap), that is, the distance (resp. difference) between the optimal solutions (resp. optimal values) of convex integer programs (IP) and the optimal solutions (resp. optimal values) of their continuous relaxations. We show that these values can be upper bounded in terms of the recession cone of the feasible region of the continuous relaxation when the recession cone is full-dimensional. If the recession cone is not full-dimensional we give sufficient conditions to obtain a finite integrality gap. We then specialize our analysis to second-order conic IPs. In the case the feasible region is defined by a single Lorentz cone constraint, we give upper bounds on proximity and integrality gap in terms of the data of the problem (the objective function vector, the matrix defining the conic constraint, the right-hand side, and the covering radius of a related lattice). We also give conditions for these bounds to be independent of the right-hand side, akin to the linear IP case. Finally, in the case the feasible region is defined by multiple Lorentz cone constraints, we show that, in general, we cannot give bounds that are independent of the corresponding right-hand side. Although our results are presented for the integer lattice , the bounds can be easily adapted to work for any general lattice, including the usual mixed-integer lattice , by considering the appropriate covering radius when needed.
Paper Structure (36 sections, 34 theorems, 89 equations, 6 figures)

This paper contains 36 sections, 34 theorems, 89 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and notation
  4. Conic programming
  5. Mixed-integer lattices
  6. Integrality gap and lineality space
  7. Illustrative examples in the nonlinear case
  8. Polyhedral approximations do not suffice
  9. Examples of conic IPs with infinite integrality gap
  10. Example of a simple second-order conic IP with right-hand side dependent integrality gap
  11. Examples of non-simple second-order conic IPs with right-hand side dependent integrality gap
  12. Proximity results for general convex IPs
  13. Convex sets with full-dimensional recession cones
  14. Convex sets with recession cones that are not necessarily full-dimensional
  15. Structural results for simple second-order conic sets
  16. ...and 21 more sections

Key Result

Theorem 1

Let $\Delta$ denote the largest absolute value of any determinant of a square submatrix of ${\textup{A}}$. Assume that ${\mathcal{S}}( \textup{\bf u})\cap {\mathbb{Z}}^n\neq\emptyset$ and that ${\vartheta}( \textup{\bf u})>-\infty$. Then

Figures (6)

  • Figure 1: The feasible region of the continuous relaxation of the conic IP in Example \ref{['ex10']}.
  • Figure 2: The feasible region of the continuous relaxation of the conic IP in Example \ref{['ex11']}. Note that the set of integer feasible points is $\{(0,0), (1,0), (0,1), (1,1)\}$.
  • Figure 3: The main idea behind the ${\textup{\bf IG}}$-driven approach.
  • Figure 4: The feasible region of the continuous relaxation of the conic IP in Example \ref{['ex2']}.
  • Figure 5: The feasible region of the continuous relaxation of the conic IP in Example \ref{['ex3']}.
  • ...and 1 more figures

Theorems & Definitions (88)

  • Theorem 1: cook1986sensitivity
  • Remark 1
  • Definition 1
  • Definition 2
  • Definition 3: Mixed-integer lattice Bertsimas_Weismantel_2005
  • Definition 4: Covering radius
  • Example 1
  • Lemma 1
  • proof
  • Remark 2
  • ...and 78 more