Cutting plane methods with gradient-based heuristics

TL;DR

This work addresses nonlinear discrete optimization by integrating cutting-plane (outer-approximation) methods with gradient-based local search on binary domains. It proposes a hybrid framework that generates tighter cuts by performing projected gradient steps within the current feasible region and by adding cuts at nearby points, including lower-bound cuts and offset-based tightening. The authors establish convergence properties for the projected gradient steps in this discrete setting and demonstrate, through extensive numerical experiments on maximum diversity problems and Lima–Grossmann BQP instances, that the hybrid approach substantially improves solution quality and convergence speed compared to standard CPM. The results indicate a flexible, scalable toolkit that leverages local structure to enhance global optimization, with potential extensions to more general discrete domains and problem classes.

Abstract

Cutting plane methods, particularly outer approximation, are a well-established approach for solving nonlinear discrete optimization problems without relaxing the integrality of decision variables. While powerful in theory, their computational performance can be highly variable. Recent research has shown that constructing cutting planes at the projection of infeasible points onto the feasible set can significantly improve the performance of cutting plane approaches. Motivated by this, we examine whether constructing cuts at feasible points closer to the optimal solution set could further enhance the effectiveness of cutting plane methods. We propose a hybrid method that combines the global convergence guarantees of cutting plane methods with the local exploration capabilities of first-order optimization techniques. Specifically, we use projected gradient methods as a heuristic to identify promising regions of the solution space and generate tighter, more informative cuts. We focus on binary optimization problems with convex differentiable objective functions, where projection operations can be efficiently computed via mixed-integer linear programming. By constructing cuts at points closer to the optimal solution set and eliminating non-optimal regions, the algorithm achieves better approximation of the feasible region and faster convergence. Numerical experiments confirm that our approach improves both the quality of the solution and computational efficiency across different solver configurations. This framework provides a flexible foundation for further extensions to more general discrete domains and offers a promising heuristic to the toolkit for nonlinear discrete optimization.

Figures (10)

• Figure 1: The tangent plane constructed at $x_1$ (points that are close to the optimal solution $x^*$) provides tighter bounds than the tangent plane constructed at $x_2$. Combining cuts at points $x_1$ and $x_3$, on opposite sides of $x^*$, also generates a tight relaxation.
• Figure 2: Comparison of solver configurations on GKD-c instances: median residue profiles relative to iterations (top left panel) and runtime (top right panel), and residue distributions at fixed runtimes (bottom panels). Residue calculation based on the best known values from MDPLIB.
• Figure 3: Comparison of solver configurations on Lima--Grossmann problems: median residue profiles relative to runtime. Test instances with $n=50$ and $m=10$ (left) and $m=40$ (right). Legend as in \ref{['fig:GKD_c']}.
• Figure 4: Comparison of solver configurations on GKD-c: residue along iterations (left) and runtime (right) for each problem instance.
• Figure 5: Comparison of solver configurations on GKD-a instances: median residue profiles relative to iterations (top left panel) and runtime (top right panel), and residue distributions at fixed runtimes (bottom panels). Legend as in \ref{['fig:GKD_c_logger']}.

Theorems & Definitions (2)

• Theorem 2.1
• proof