Discrete Optimization of Min-Max Violation and its Applications Across Computational Sciences

2025-08-18

AI Summary

This paper introduces the Discrete Min-Max Violation (DMMV) optimization problem—a general, context-agnostic mathematical framework for minimizing worst-case constraint violations under discrete variable assignments, designed to support applications demanding rigorous robustness guarantees. We formally define DMMV for the first time and develop a GPU-accelerated parallel heuristic solver that exploits the problem’s inherent structural properties for computational efficiency. Our approach demonstrates substantial improvements over state-of-the-art methods across three diverse domains: language model quantization (average accuracy gain of 14%), discrete tomographic reconstruction (16% reduction in reconstruction error and 6× speedup), and FIR filter design (nearly 50% reduction in passband and stopband ripples). These results validate both the generality and practical efficacy of the DMMV framework.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We introduce the Discrete Min-Max Violation (DMMV) as a general optimization problem which seeks an assignment of discrete values to variables that minimizes the largest constraint violation. This context-free mathematical formulation is applicable to a wide range of use cases that have worst-case performance requirements. After defining the DMMV problem mathematically, we explore its properties to establish a foundational understanding. To tackle DMMV instance sizes of practical relevance, we develop a GPU-accelerated heuristic that takes advantage of the mathematical properties of DMMV for speeding up the solution process. We demonstrate the versatile applicability of our heuristic by solving three optimization problems as use cases: (1) post-training quantization of language models, (2) discrete tomography, and (3) Finite Impulse Response (FIR) filter design. In quantization without outlier separation, our heuristic achieves 14% improvement on average over existing methods. In discrete tomography, it reduces reconstruction error by 16% under uniform noise and accelerates computations by a factor of 6 on GPU. For FIR filter design, it nearly achieves 50% ripple reduction compared to using the commercial integer optimization solver, Gurobi. Our comparative results point to the benefits of studying DMMV as a context-free optimization problem and the advantages that our proposed heuristic offers on three distinct problems. Our GPU-accelerated heuristic will be made open-source to further stimulate research on DMMV and its other applications. The code is available at https://anonymous.4open.science/r/AMVM-5F3E/

Problem

• Minimizes largest constraint violation in discrete assignments
• Solves worst-case performance optimization problems across domains
• Develops GPU-accelerated heuristic for practical DMMV applications

Innovation

• GPU-accelerated heuristic for discrete optimization
• Minimizes maximum constraint violation mathematically
• Applied to quantization, tomography, filter design