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Axis-Aligned Relaxations for Mixed-Integer Nonlinear Programming

Haisheng Zhu, Taotao He, Mohit Tawarmalani

Abstract

We present a novel relaxation framework for general mixed-integer nonlinear programming (MINLP) grounded in computational geometry. Our approach constructs polyhedral relaxations by convexifying finite sets of strategically chosen points, iteratively refining the approximation to converge toward the simultaneous convex hull of factorable function graphs. The framework is underpinned by three key contributions: (i) a new class of explicit inequalities for products of functions that strictly improve upon standard factorable and composite relaxation schemes; (ii) a proof establishing that the simultaneous convex hull of multilinear functions over axis-aligned regions is fully determined by their values at corner points, thereby generalizing existing results from hypercubes to arbitrary axis-aligned domains; and (iii) the integration of computational geometry tools, specifically voxelization and QuickHull, to efficiently approximate feasible regions and function graphs. We implement this framework and evaluate it on randomly generated polynomial optimization problems and a suite of 619 instances from \texttt{MINLPLib}. Numerical results demonstrate significant improvements over state-of-the-art benchmarks: on polynomial instances, our relaxation closes an additional 20--25\% of the optimality gap relative to standard methods on half the instances. Furthermore, compared against an enhanced factorable programming baseline and Gurobi's root-node bounds, our approach yields superior dual bounds on approximately 30\% of \texttt{MINLPLib} instances, with roughly 10\% of cases exhibiting a gap reduction exceeding 50\%.

Axis-Aligned Relaxations for Mixed-Integer Nonlinear Programming

Abstract

We present a novel relaxation framework for general mixed-integer nonlinear programming (MINLP) grounded in computational geometry. Our approach constructs polyhedral relaxations by convexifying finite sets of strategically chosen points, iteratively refining the approximation to converge toward the simultaneous convex hull of factorable function graphs. The framework is underpinned by three key contributions: (i) a new class of explicit inequalities for products of functions that strictly improve upon standard factorable and composite relaxation schemes; (ii) a proof establishing that the simultaneous convex hull of multilinear functions over axis-aligned regions is fully determined by their values at corner points, thereby generalizing existing results from hypercubes to arbitrary axis-aligned domains; and (iii) the integration of computational geometry tools, specifically voxelization and QuickHull, to efficiently approximate feasible regions and function graphs. We implement this framework and evaluate it on randomly generated polynomial optimization problems and a suite of 619 instances from \texttt{MINLPLib}. Numerical results demonstrate significant improvements over state-of-the-art benchmarks: on polynomial instances, our relaxation closes an additional 20--25\% of the optimality gap relative to standard methods on half the instances. Furthermore, compared against an enhanced factorable programming baseline and Gurobi's root-node bounds, our approach yields superior dual bounds on approximately 30\% of \texttt{MINLPLib} instances, with roughly 10\% of cases exhibiting a gap reduction exceeding 50\%.
Paper Structure (32 sections, 16 theorems, 108 equations, 17 figures, 10 tables, 7 algorithms)

This paper contains 32 sections, 16 theorems, 108 equations, 17 figures, 10 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Tools: Piecewise Approximation and Axis-Aligned Regions
  3. Multiplication of Functions over a Rectangle
  4. Multiplication of Variables over an Arbitrary Domain
  5. Multiplication of Functions over an Arbitrary Domain
  6. Multilinear compositions with Linking Constraints
  7. Axis-aligned approximations and multilinear compositions
  8. Convergence Results
  9. Implementation
  10. Base Relaxation
  11. Treatment of Multiplication Nodes
  12. Voxelization via Approximated Projection
  13. Voxelization via Quadtree
  14. Numerical Results
  15. Polynomial Optimization Problem
  16. ...and 17 more sections

Key Result

Theorem 1

Let $P_i$ be the pentagon defined by the vertices in eq:pentagon for $i = 1, 2$ and let $P = P_1\times P_2$. Define the parameters: The convex envelope of $f_1f_2$ over $P$ is given by $\max_{j=1,\ldots,6}\bigl\{ \bigl\langle \alpha_j, (x,t) \bigr\rangle + c_j \bigr\}$, where $\alpha_j$ denotes the $j^{\text{th}}$ row of the matrix: and the constant term is $c_j = f_1^Uf_2^L - \bigl(\alpha_{j1}

Figures (17)

  • Figure 1: Comparison of relaxation over rectangular bounds versus voxelization
  • Figure 2: Approximation of $x_1^2x_2^2$ via piecewise linear relaxations
  • Figure 3: Refinement of the bilinear relaxation via voxelization. The top row illustrates the domain $D$ (teal) and its axis-aligned outer approximation (red boundary) with increasing levels of accuracy. The bottom row depicts the corresponding three dimensional polyhedral relaxations (green). As the voxelization is refined from left to right, the polyhedral relaxation converges toward the true convex hull of the bilinear term over $D$.
  • Figure 4: Illustration of a cycle (indicated in blue) in the iterative coordinate-wise decomposition. Starting from point $v$, alternating decompositions along the $x_1$ and $x_2$ axes return to $v$ after four iterations, demonstrating that the greedy procedure may fail to yield a finite representation via corner points.
  • Figure 5: Illustration of the convex extension Markov chain: (a) The axis-aligned region $\mathcal{H}$ and (b) the resulting stochastic transitions.
  • ...and 12 more figures

Theorems & Definitions (40)

  • Example 1
  • Theorem 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 1
  • Example 2
  • Definition 1
  • Definition 2
  • ...and 30 more