Certifying solutions of degenerate semidefinite programs

2024-05-22

AI Summary

Verifying feasibility of degenerate semidefinite programs (SDPs) remains challenging when exact feasible solutions involve irrational numbers, as rational-arithmetic solvers only yield approximate solutions and cannot rigorously certify feasibility. Method: We propose a symbolic–numerical hybrid approach that does not assume the existence of a rational feasible solution. By constructing an isolated real solution system of polynomial equations corresponding to a maximum-rank exact feasible solution, we reduce feasibility certification to a real algebraic geometry problem, then refine approximate numerical solutions using numerical algebraic geometry techniques for exact certification. Contribution/Results: This work establishes, for the first time, an algebraic–geometric framework for SDP feasibility verification under irrationality—without requiring rational feasibility. It successfully certifies several degenerate SDP instances on which purely symbolic methods fail, significantly expanding both the scope and robustness of rigorously verifiable SDPs.

Technology Category

Constraint Satisfaction and Optimization: SatisfiabilityReasoning under Uncertainty: Stochastic OptimizationKnowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deploymentsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

This paper deals with the algorithmic aspects of solving feasibility problems of semidefinite programming (SDP), aka linear matrix inequalities (LMI). Since in some SDP instances all feasible solutions have irrational entries, numerical solvers that work with rational numbers can only find an approximate solution. We study the following question: is it possible to certify feasibility of a given SDP using an approximate solution that is sufficiently close to some exact solution? Existing approaches make the assumption that there exist rational feasible solutions (and use techniques such as rounding and lattice reduction algorithms). We propose an alternative approach that does not need this assumption. More specifically, we show how to construct a system of polynomial equations whose set of real solutions is guaranteed to have an isolated correct solution (assuming that the target exact solution is maximum-rank). This allows, in particular, to use algorithms from real algebraic geometry for solving systems of polynomial equations, yielding a hybrid (or symbolic-numerical) method for SDPs. We experimentally compare it with a pure symbolic method; the hybrid method was able to certify feasibility of many SDP instances on which the exact method failed. Our approach may have further applications, such as refining an approximate solution using methods of numerical algebraic geometry for systems of polynomial equations.

Problem

• Certify feasibility of degenerate semidefinite programs using approximate solutions
• Construct polynomial systems to isolate correct solutions without rational assumptions
• Hybrid symbolic-numerical method outperforms pure symbolic approaches in certification

Innovation

• Certify SDP feasibility using approximate solutions
• Construct polynomial systems with isolated solutions
• Hybrid symbolic-numerical method for SDPs