bibkey: boydvandenberghe2004convex
authors: Stephen Boyd and Lieven Vandenberghe
year: 2004
title: Convex Optimization
doi: null
url: https://web.stanford.edu/~boyd/cvxbook/
claim: The one-constraint S-procedure is exact for real quadratic implications under a strictly feasible constraint, and is equivalent to a nonnegative-multiplier matrix inequality.
strata_touched: []
license: citation-only
triage: anchor

Convex Optimization

Author-hosted primary PDF: https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf .

Appendix B.2, printed p. 655 (PDF p. 669), equations (B.6)–(B.7), states the exact quadratic alternative and the equivalent S-procedure implication. Its necessary hypothesis is a point at which the constraint quadratic is strictly negative. It allows a nonnegative multiplier and a positive-semidefinite, potentially singular, coefficient matrix.

Recovery volume II, Proposition 87.3 uses the unit ball in the eight-dimensional real Frobenius space ${\mathcal{L}}_2$. The origin is strictly feasible. In that application, the constraint coefficients are $(I,0,-1)$ and the conclusion coefficients are $(H,d/2,A-c)$; the difference after multiplication is the displayed $9\times 9$ PSD certificate. The S-procedure proves equivalence for each fixed common triple. It supplies no universal feasible multiplier or the numerical constant for the full family.