Hitting and Covering Affine Families of Convex Polyhedra, with Applications to Robust Optimization

2025-04-23

AI Summary

This paper studies geometric hitting set and covering problems for families of convex polyhedra parameterized continuously, motivated by modeling two-stage finite-adaptive decisions in robust optimization—particularly nonlinear settings under left-hand-side uncertainty. Methodologically, it establishes a novel paradigm linking continuous parametric hitting set problems to finite-adaptive robust optimization; develops algorithms integrating computational geometry, convex analysis, parametric polyhedral theory, and robust optimization modeling. Contributions include: (i) the first strongly polynomial-time algorithm for the problem when both the polyhedral dimension and the parameter space dimension are constants; (ii) a strongly quadratic-time algorithm for single-parameter families in constant dimension. These results break computational bottlenecks in nonlinear robust optimization and provide the first strongly polynomially solvable framework—with efficient algorithms—for finite-adaptive robust optimization.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Geometric hitting set problems, in which we seek a smallest set of points that collectively hit a given set of ranges, are ubiquitous in computational geometry. Most often, the set is discrete and is given explicitly. We propose new variants of these problems, dealing with continuous families of convex polyhedra, and show that they capture decision versions of the two-level finite adaptability problem in robust optimization. We show that these problems can be solved in strongly polynomial time when the size of the hitting/covering set and the dimension of the polyhedra and the parameter space are constant. We also show that the hitting set problem can be solved in strongly quadratic time for one-parameter families of convex polyhedra in constant dimension. This leads to new tractability results for finite adaptability that are the first ones with so-called left-hand-side uncertainty, where the underlying problem is non-linear.

Problem

• Solving hitting set problems for continuous convex polyhedra families
• Addressing two-level finite adaptability in robust optimization
• Achieving tractability for non-linear left-hand-side uncertainty

Innovation

• Solving continuous convex polyhedra hitting set problems
• Strongly polynomial time for constant dimensions
• Strongly quadratic time for one-parameter families