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Convex mixed-integer optimization with Frank-Wolfe methods

Deborah Hendrych, Hannah Troppens, Mathieu Besançon, Sebastian Pokutta

TL;DR

This work tackles convex mixed-integer optimization by embedding a branch-and-bound framework inside the convex hull of integer-feasible points, solving FW-based subproblems where the LMO is implemented via a MIP with node-specific bounds, thus obtaining vertices of the integer hull. By optimizing over $\mathrm{conv}({\mathcal{X}})$ rather than the standard continuous relaxation, the method yields stronger relaxations and multiple feasible solutions per node, while preserving the original polyhedral structure and avoiding epigraph formulations. The authors introduce Blended Pairwise Conditional Gradient (BPCG) with active/shadow sets to dramatically reduce MIP calls, and incorporate lazy updates, warm starts, and dual tightening to prune the search tree. They also adapt strong-branching ideas to estimate bound improvements using LP/LMO subproblems and provide error-adaptive termination and tightening strategies to balance accuracy and computational effort. Empirical results on diverse MINLP instances demonstrate competitive performance, with the approach implemented in the open-source Boscia.jl package and showing notable improvements when leveraging strong convexity, tightening, and warm-starts.

Abstract

Mixed-integer nonlinear optimization encompasses a broad class of problems that present both theoretical and computational challenges. We propose a new type of method to solve these problems based on a branch-and-bound algorithm with convex node relaxations. These relaxations are solved with a Frank-Wolfe algorithm over the convex hull of mixed-integer feasible points instead of the continuous relaxation via calls to a mixed-integer linear solver as the linear minimization oracle. The proposed method computes feasible solutions while working on a single representation of the polyhedral constraints, leveraging the full extent of mixed-integer linear solvers without an outer approximation scheme and can exploit inexact solutions of node subproblems.

Convex mixed-integer optimization with Frank-Wolfe methods

TL;DR

This work tackles convex mixed-integer optimization by embedding a branch-and-bound framework inside the convex hull of integer-feasible points, solving FW-based subproblems where the LMO is implemented via a MIP with node-specific bounds, thus obtaining vertices of the integer hull. By optimizing over rather than the standard continuous relaxation, the method yields stronger relaxations and multiple feasible solutions per node, while preserving the original polyhedral structure and avoiding epigraph formulations. The authors introduce Blended Pairwise Conditional Gradient (BPCG) with active/shadow sets to dramatically reduce MIP calls, and incorporate lazy updates, warm starts, and dual tightening to prune the search tree. They also adapt strong-branching ideas to estimate bound improvements using LP/LMO subproblems and provide error-adaptive termination and tightening strategies to balance accuracy and computational effort. Empirical results on diverse MINLP instances demonstrate competitive performance, with the approach implemented in the open-source Boscia.jl package and showing notable improvements when leveraging strong convexity, tightening, and warm-starts.

Abstract

Mixed-integer nonlinear optimization encompasses a broad class of problems that present both theoretical and computational challenges. We propose a new type of method to solve these problems based on a branch-and-bound algorithm with convex node relaxations. These relaxations are solved with a Frank-Wolfe algorithm over the convex hull of mixed-integer feasible points instead of the continuous relaxation via calls to a mixed-integer linear solver as the linear minimization oracle. The proposed method computes feasible solutions while working on a single representation of the polyhedral constraints, leveraging the full extent of mixed-integer linear solvers without an outer approximation scheme and can exploit inexact solutions of node subproblems.
Paper Structure (12 sections, 2 theorems, 30 equations, 34 figures, 1 table, 2 algorithms)

This paper contains 12 sections, 2 theorems, 30 equations, 34 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Nonlinear branch-and-bound over integer hulls with linear oracles
  3. Constraint representability with integer hulls.
  4. Blended Pairwise Conditional Gradient (BPCG).
  5. Active and shadow set branching.
  6. Branch-and-bound techniques for error-adaptive convex relaxations
  7. Strong branching.
  8. Computational experiments
  9. Conclusion
  10. Blended Pairwise Conditional Gradient
  11. Tighter lower bound and dual tightening inequalities
  12. Detailed and extended experiments results

Key Result

Theorem 1

Let us assume that the bounds are $\left[\mathbf{l},\mathbf{u}\right] \supseteq {\mathcal{X}}\xspace$ and that we have a relaxed solution $\mathbf{x}^{(t)}$ and a variable $j \in J$ such that $\mathbf{x}^{(t)}_j = \mathbf{l}_j$ and $\nabla f(\mathbf{x}^{(t)})_j \geq 0$. Then, if there exists $M \in with $g(\cdot)$ the Frank-Wolfe gap: then the upper bound can be tightened to $\mathbf{x}^{(t)}_j

Figures (34)

  • Figure 1: Three main algorithmic frameworks for MINLPs. Diamond blocks represent the nodal relaxations in the given framework. \ref{['fig:branchnlp']} corresponds to a classic BnB framework on top of NLP relaxations, \ref{['fig:lpbased']} represents the mechanism of LP-based MINLP frameworks and outer approximations, \ref{['fig:branchwolfe']} is our proposed approach with the linearized models solved as MIPs within the Frank-Wolfe algorithm on top of which we branch.
  • Figure 2: Branching over the convex hull. \ref{['fig:baselineprob']} shows the baseline problem with the level curves of the objective, polyhedron, the optimum over the relaxation $\tilde{x}$, and examples of potentially active vertices for a near-optimal solution (the top vertex is dropped in an optimal solution). Branching over the continuous relaxation is shown in \ref{['fig:branchcontinuous']}. Our approach is shown in \ref{['fig:convexhull']} and \ref{['fig:convexhull2']}, optimizing over the convex hull with the help of the MIP solver. Branching only once results in an optimal solution in the left part in \ref{['fig:convexhull2']}, with the right part being pruned once $\tilde{x}_l$ is found.
  • Figure 3: Positive and negative illustrations for Assumption \ref{['A-CONV']}. The blue solid curve represents the boundary of $\overline{X}$, the red dashed shaded area is $\mathrm{conv}({\mathcal{X}}\xspace)$.
  • Figure 4: Progress of the lower bound and the incumbent of Boscia.jl over the number of nodes as well as the accumulated number of BLMO calls.
  • Figure 5: Primal-dual convergence comparison between Boscia.jl with and without using strong convexity on MIPLIB neos5 instances.
  • ...and 29 more figures

Theorems & Definitions (5)

  • Theorem 1
  • proof
  • Definition 1: Hölder Error Bound
  • Theorem 2
  • proof