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Certifying solutions of degenerate semidefinite programs

Vladimir Kolmogorov, Simone Naldi, Jeferson Zapata

TL;DR

The paper tackles certifying feasibility for degenerate semidefinite programs when feasible sets may contain irrational points. It develops a hybrid symbolic-numeric framework that reduces feasibility to solving a system of polynomial equations whose real solutions include a maximum-rank SDP solution, without requiring rational feasibility. By combining incidence-variety–based facial reduction with rank-revealing numeric steps and a critical-point method for solving fixed-variable polynomial systems, the approach often certifies feasibility where pure symbolic methods fail. The results show improved performance over purely algebraic techniques on several benchmark instances, with potential extensions to refine approximate solutions via numerical algebraic-geometry tools. Practically, this hybrid method broadens the set of SDPs whose feasibility can be reliably certified in settings where irrational solutions are typical.

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.

Certifying solutions of degenerate semidefinite programs

TL;DR

The paper tackles certifying feasibility for degenerate semidefinite programs when feasible sets may contain irrational points. It develops a hybrid symbolic-numeric framework that reduces feasibility to solving a system of polynomial equations whose real solutions include a maximum-rank SDP solution, without requiring rational feasibility. By combining incidence-variety–based facial reduction with rank-revealing numeric steps and a critical-point method for solving fixed-variable polynomial systems, the approach often certifies feasibility where pure symbolic methods fail. The results show improved performance over purely algebraic techniques on several benchmark instances, with potential extensions to refine approximate solutions via numerical algebraic-geometry tools. Practically, this hybrid method broadens the set of SDPs whose feasibility can be reliably certified in settings where irrational solutions are typical.

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.
Paper Structure (12 sections, 9 theorems, 95 equations, 1 table, 1 algorithm)

This paper contains 12 sections, 9 theorems, 95 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Incidence varieties and facial reduction
  4. Description of the method
  5. Numerical algorithms: implementing steps 2 and 4
  6. Symbolic algorithms: implementing step 5
  7. Numerical Results
  8. Numerical example
  9. Hybrid method
  10. HNS method
  11. Conclusions and future work
  12. Description of SDP Problems

Key Result

Theorem 1.1

There exists an algorithm that, given a feasible system $\mathscr{A}(X) = b, \, X \succeq 0$, approximate solution $\tilde{X}$ and parameter $\epsilon>0$, constructs a system of polynomial equations over symmetric matrices $X\in\mathbb R^{n\times n}$ and auxiliary variables $Y\in\mathbb R^{n \times where $\Pi$ is a permutation matrix, $r\in[n]$, and $K$ is a subset of $\{(i,j)\in [n]\times[n]\::\

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 2.1
  • Proof 1
  • Lemma 3.1
  • Proof 2
  • Lemma 4.1
  • Proof 3
  • Definition 4.2
  • Theorem 4.3
  • Proof 4
  • ...and 8 more