An Adaptive Algorithm for the Approximation of General Linear-Parametric Optimization Problems

2026-06-16

AI Summary

This work addresses the computational challenges of linear multiparametric optimization, which is NP-hard and may exhibit an exponentially large set of optimal solutions. Existing approximation algorithms are limited to nonnegative parameters and parameter spaces confined to the positive orthant, leaving problems with negative parameters largely intractable. To overcome this limitation, the paper proposes a novel adaptive algorithm that integrates techniques from parametric and multiobjective optimization, enabling the first polynomial-time approximation scheme for general linear multiparametric problems with arbitrary (including negative) parameters. By introducing structural analysis of parameter sets and an adaptive strategy, the method provides theoretical approximation guarantees and demonstrates that nonnegativity of objective values alone is insufficient to ensure approximability, thereby significantly broadening the class of solvable parametric optimization problems.

Abstract

Linear-multi-parametric optimization problems are a widely studied class of optimization problems. The objective function in such a problem is affine linear dependent on a parameter vector, and the goal is to compute a set of solutions that contains an optimal solution for every fixed parameter vector. However, this is known to be computationally challenging: The underlying non-parametric problem might be NP-hard, and, in addition, optimal solution sets might have exponential cardinality. Parametric approximation aims at providing polynomial-time algorithms that overcome these challenges. Instead of computing an optimal solution set, the goal is to compute an approximation set that contains only an approximate solution for every fixed parameter vector. Several new parametric approximation algorithms have been developed in recent literature. However, all of these share a common set of assumptions, which limits the class of parametric optimization problems that can be approximated. Namely, they do not allow negative parameter dependencies and have their parameter sets fixed to the positive orthant. We present a new adaptive approximation (and, also, exact) algorithm that can be applied to a wider class of linear-multi-parametric optimization problems. Our algorithm builds upon existing algorithms from both the fields of parametric and multi-objective optimization and generalizes these algorithms. In addition, we provide structural results for the transformation of parameter sets, and demonstrate that, for linear-multi-parametric maximization problems, the assumption of non-negative optimal objective values over the whole parameter set is not sufficient to ensure approximability.

Problem

• linear-multi-parametric optimization
• parametric approximation
• parameter dependencies
• approximability
• optimization problems

Innovation

• adaptive algorithm
• linear-multi-parametric optimization
• parametric approximation
• parameter set transformation
• multi-objective optimization