A new method for reducing algebraic programs to polynomial programs

2025-02-12

AI Summary

Optimization and feasibility checking for algebraic functions involving radicals, rational expressions, and similar non-polynomial terms pose significant challenges due to the “curse of dimensionality” inherent in conventional layered auxiliary-variable lifting. Method: This paper introduces a compact polynomial reconstruction framework that replaces each algebraic function with a single new variable, coupled with an integrated algorithmic pipeline grounded in real algebraic geometry: implicitization, branch-isolating inequality generation, and Positivstellensatz/SOS-based feasibility verification. Contribution/Results: By avoiding redundant variable proliferation, the method substantially improves modeling efficiency and scalability of polynomial programming formulations. On enhanced classical benchmarks, it achieves up to 50× speedup over naive reconstruction. The approach provides both theoretical guarantees and a practical toolkit for efficiently translating algebraic programs into polynomial optimization problems.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Optimization
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We consider a generalization of polynomial programs: algebraic programs, which are optimization or feasibility problems with algebraic objectives or constraints. Algebraic functions are defined as zeros of multivariate polynomials. They are a rich set of functions that includes polynomials themselves, but also ratios and radicals, and finite compositions thereof. When an algebraic program is given in terms of radical expressions, a straightforward way of reformulating into a polynomial program is to introduce a new variable for each distinct radical that appears. Hence, the rich theory and algorithms for polynomial programs, including satisfiability via cylindrical algebraic decomposition, infeasibility certificates via Positivstellensatz theorems, and optimization with sum-of-squares programming directly apply to algebraic programs. We propose a different reformulation, that in many cases introduces significantly fewer new variables, and thus produces polynomial programs that are easier to solve. First, we exhibit an algorithm that finds a defining polynomial of an algebraic function given as a radical expression. As a polynomial does not in general define a unique algebraic function, additional constraints need to be added that isolate the algebraic function from others defined by the same polynomial. Using results from real algebraic geometry, we develop an algorithm that generates polynomial inequalities that isolate an algebraic function. This allows us to reformulate an algebraic program into a polynomial one, by introducing only a single new variable for each algebraic function. On modified versions of classic optimization benchmarks with added algebraic terms, our formulation achieves speedups of up to 50x compared to the straightforward reformulation.

Problem

• Reduces algebraic to polynomial programs
• Minimizes new variables in reformulation
• Enhances solving speed by 50x

Innovation

• Reduces algebraic to polynomial programs
• Introduces fewer new variables
• Achieves up to 50x speedup