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New finite relaxation hierarchies for concavo-convex, disjoint bilinear programs, and facial disjunctions

Mohit Tawarmalani

TL;DR

This paper develops finite, unified relaxation hierarchies for concavo-convex programs, including disjoint bilinear programs and facial disjunctions, by exploiting a double-description approach to compute barycentric coordinates as rational functions. It builds a geometric-algebraic bridge between disjunctive programming (DP) and reformulation-linearization technique (RLT), enabling tighter relaxations through multiple outer-approximations and affine relations across hierarchy levels. The proposed hierarchies terminate finitely in many key cases (e.g., 0-1 FDP) and yield algebraic certificates of optimality, with extensions to FDP and potential connections to sum-of-squares and fractional programming techniques. Overall, the work provides a robust framework that unifies DP and RLT perspectives for continuous optimization problems, while offering practical avenues for constructing stronger, finitely convergent relaxations.

Abstract

This paper introduces novel relaxation hierarchies for concavo-convex programs (CXP), a class of problems that includes disjoint bilinear programming (DBP) and concave minimization (CM) as special cases. At the core of these hierarchies is an algorithm based on double-description (DD) that computes the barycentric coordinates of a polyhedral cone as rational, non-negative functions representing multipliers associated with the cone's rays. These hierarchies combine geometric structure derived from barycentric coordinates with algebraic techniques via rational functions, achieving the convex hull in $m$ iterations, where $m$ is the number of inequalities that a subset of the variables must satisfy. Our framework offers the first unified approach to analyze and tighten relaxations from disjunctive programming (DP) and reformulation-linearization technique (RLT) for CXP. We also demonstrate that our methods extend to facial disjunctive programs (FDP), where solutions are constrained to lie on faces of a Cartesian product of polytopes, generalizing known hierarchies for 0-1 programs.

New finite relaxation hierarchies for concavo-convex, disjoint bilinear programs, and facial disjunctions

TL;DR

This paper develops finite, unified relaxation hierarchies for concavo-convex programs, including disjoint bilinear programs and facial disjunctions, by exploiting a double-description approach to compute barycentric coordinates as rational functions. It builds a geometric-algebraic bridge between disjunctive programming (DP) and reformulation-linearization technique (RLT), enabling tighter relaxations through multiple outer-approximations and affine relations across hierarchy levels. The proposed hierarchies terminate finitely in many key cases (e.g., 0-1 FDP) and yield algebraic certificates of optimality, with extensions to FDP and potential connections to sum-of-squares and fractional programming techniques. Overall, the work provides a robust framework that unifies DP and RLT perspectives for continuous optimization problems, while offering practical avenues for constructing stronger, finitely convergent relaxations.

Abstract

This paper introduces novel relaxation hierarchies for concavo-convex programs (CXP), a class of problems that includes disjoint bilinear programming (DBP) and concave minimization (CM) as special cases. At the core of these hierarchies is an algorithm based on double-description (DD) that computes the barycentric coordinates of a polyhedral cone as rational, non-negative functions representing multipliers associated with the cone's rays. These hierarchies combine geometric structure derived from barycentric coordinates with algebraic techniques via rational functions, achieving the convex hull in iterations, where is the number of inequalities that a subset of the variables must satisfy. Our framework offers the first unified approach to analyze and tighten relaxations from disjunctive programming (DP) and reformulation-linearization technique (RLT) for CXP. We also demonstrate that our methods extend to facial disjunctive programs (FDP), where solutions are constrained to lie on faces of a Cartesian product of polytopes, generalizing known hierarchies for 0-1 programs.
Paper Structure (27 sections, 20 theorems, 109 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 27 sections, 20 theorems, 109 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation:
  3. A Roadmap
  4. Disjunctive Programming and Vertex Representations:
  5. Barycentric coordinates and representations:
  6. Convex hull formulations:
  7. Tighter relaxations:
  8. Inequalities in the hierarchy:
  9. Facial Disjunctions:
  10. Preliminaries: Disjunctive Programming and RLT
  11. Double description and a simple finite hierarchy
  12. Finiteness of the basic hierarchy:
  13. The structure of barycentric coordinates
  14. An algebraic relaxation hierarchy
  15. Hierarchy:
  16. ...and 12 more sections

Key Result

lemma 1

The optimal value of $(\mathop{\mathrm{DDP}}\nolimits_{\mathcal{V}})$ matches that of $(D)$. Consider an optimal solution $(x_*, \lambda_*,y^1_*,\ldots,y^p_*)$ for $\mathop{\mathrm{DDP}}\nolimits_{\mathcal{V}}$. Let $j'\in J := \{j\mid {(\lambda_*)}_j > 0\}$. Then, $\left(V_{\mathord{:}{},j'},y^{j'}

Figures (1)

  • Figure 1: Stages of relaxation construction. Figure \ref{['fig:Polytopea']} introduces \ref{['eq:P5']} on the simplex bounded by \ref{['eq:P1-3']} and \ref{['eq:P4']}. The points obtained as convex combinations from $\dot{x}$ are shown. Figure \ref{['fig:Polytopeb']} shows the effect of introducing \ref{['eq:P6']} and the points $\hat{x}$, $\check{x}$, $\bar{x}$, and $\mathring{x}$ introduced to capture the multiplier of $x'$.

Theorems & Definitions (42)

  • lemma 1
  • proposition 1
  • proof
  • proposition 2
  • proof
  • definition 1
  • definition 2
  • theorem 1
  • proof
  • theorem 2
  • ...and 32 more