Goldfarb-Idnani Revisited:Invariants, Certificates, and the Limits of Guessing

Abstract

The dual active-set method of Goldfarb and Idnani solves the strictly convex quadratic program by adding and dropping one constraint at a time, requiring no phase one. Primal--dual active set and block principal pivoting skip the walk altogether: they guess an entire active set at once and repair it from returning signs. For bound constraints the guess is safe. The system solved on a candidate set is a principal submatrix of $G^{-1}$, positive definite regardless of the guess; this $P$-matrix property guarantees finite termination. For $C^\top x \ge b$ that hypothesis fails, which is our central focus. The working-set system is $C_{\mathcal{A}}^\top G^{-1} C_{\mathcal{A}}$, positive definite only when $C_{\mathcal{A}}$ has full column rank, a property of the guess, not the data. The $P$-matrix property is lost and the structural obstruction invalidates the guarantee. Strict convexity keeps the method usable: KKT conditions make candidate sets certified rather than trusted. Over 600 random instances across four constraint families this failure never occurs. However, duplicating columns of $C$ raises failure rates to 93\%, dropping certified fraction from 100\% to zero. On real long-only portfolio data, the active set reaches 442 of $494 + 1$ constraints.