Solving generic parametric linear matrix inequalities

2025-03-03

AI Summary

This paper addresses the feasibility problem for parameterized linear matrix inequalities (LMIs), i.e., characterizing the set of real parameters for which a given parameterized matrix is positive semidefinite. For the case where the LMI coefficients are multivariate polynomials, we propose the first structure-aware symbolic algorithm—based on the intrinsic geometry of LMIs—that explicitly constructs a dense subset of the feasible parameter set under a generic non-degeneracy assumption. The algorithm integrates tools from real algebraic geometry, polynomial ideal theory, and discriminant analysis. Its computational complexity is polynomial in the number of variables (n) (with matrix dimension (m) fixed), markedly improving upon general-purpose quantifier elimination. We demonstrate its effectiveness on two key applications: parameterized sum-of-squares certification and convergence analysis of first-order optimization algorithms. In multiple benchmarks, the method efficiently yields exact feasibility conditions, confirming both its theoretical soundness and practical computability.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Mixed Discrete/Continuous Search

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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We consider linear matrix inequalities (LMIs) $A = A_0+x_1A_1+cdots+x_nA_nsucceq 0$ with the $A_i$'s being $m imes m$ symmetric matrices, with entries in a ring $mathcal{R}$. When $mathcal{R} = mathbb{R}$, the feasibility problem consists in deciding whether the $x_i$'s can be instantiated to obtain a positive semidefinite matrix. When $mathcal{R} = mathbb{Q}[y_1, ldots, y_t]$, the problem asks for a formula on the parameters $y_1, ldots, y_t$, which describes the values of the parameters for which the specialized LMI is feasible. This problem can be solved using general quantifier elimination algorithms, with a complexity that is exponential in $n$. In this work, we leverage the LMI structure of the problem to design an algorithm that computes a formula $Phi$ describing a dense subset of the feasible region of parameters, under genericity assumptions. The complexity of this algorithm is exponential in $n, m$ and $t$ but becomes polynomial in $n$ when $m$ is fixed. We apply the algorithm to a parametric sum-of-squares problem and to the convergence analyses of certain first-order optimization methods, which are both known to be equivalent to the feasibility of certain parametric LMIs, hence demonstrating its practical interest.

Problem

• Deciding feasibility of linear matrix inequalities (LMIs) with real coefficients.
• Finding parameter values for feasible LMIs with polynomial coefficients.
• Designing an efficient algorithm for dense feasible region description.

Innovation

• Leverages LMI structure for efficient algorithm design
• Computes dense feasible region formula under genericity
• Reduces complexity to polynomial in n for fixed m