A Survey of the S-Lemma

Abstract

In this survey we review the many faces of the S-lemma, a result about the correctness of the S-procedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the S-lemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry.

Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint

“…This theorem shows that by congruence transformation, a symmetric matrix pair (A, B) can be block diagonalized with three types of diagonal blocks (27), (28) and (29). Each block corresponds to an eigenvalue or singular part of the pencil A + λB as summarized below.…”

“…The blocks in the right-hand side of (27) correspond to a singular part; any matrix pair that possess these blocks is singular, that is, det(A + λB) = 0 for every λ. The main message of this section is that the canonical form under congruence contains full information about QCQP (2).…”

“…Remarkably, assuming Slater's condition is satisfied, the SDP (3) is a dual problem with no duality gap, that is, the solution γ to (3) is equal to the optimal value of the QCQP (2), even when (2) is nonconvex; see [4,App. B] for details and a proof, which relies on the S-lemma [27].…”

Improving the Performance of MIQP Solvers for Quadratic Programs with Cardinality and Minimum Threshold Constraints: A Semidefinite Program Approach

“…where µ i ≥ 0 is the multiplier for the constraint: x 2 i − (α i + u i )x i + α i u i ≤ 0 and the last equality holds due to S-Lemma (see, e.g., Pólik and Terlaky (2007)). Thus, the constraint π + q + se ≥ 0 can be rewritten as…”

On the Low Rank Solutions for Linear Matrix Inequalities

“…The S-Lemma has played an instrumental role in optimization and control theory (see [16] for a recent survey on the S-Lemma). In fact, Theorem 2.6 and Theorem 3.2 lead to the following extension of the S-Lemma.…”