A Triangle Algorithm for Semidefinite Version of Convex Hull Membership Problem

Given a subset $\mathbf{S}=\{A_1, \dots, A_m\}$ of $\mathbb{S}^n$, the set of $n \times n$ real symmetric matrices, we define its {\it spectrahull} as the set $SH(\mathbf{S}) = \{p(X) \equiv (Tr(A_1 X), \dots, Tr(A_m X))^T : X \in \mathbf{\Delta}_n\}$, where ${\bf \Delta}_n$ is the {\it spectraplex}, $\{ X \in \mathbb{S}^n : Tr(X)=1, X \succeq 0 \}$. We let {\it spectrahull membership} (SHM) to be the problem of testing if a given $b \in \mathbb{R}^m$ lies in $SH(\mathbf{S})$. On the one hand when $A_i$'s are diagonal matrices, SHM reduces to the {\it convex hull membership} (CHM), a fundamental problem in LP. On the other hand, a bounded SDP feasibility is reducible to SHM. By building on the {\it Triangle Algorithm} (TA) \cite{kalchar,kalsep}, developed for CHM and its generalization, we design a TA for SHM, where given $\varepsilon$, in $O(1/\varepsilon^2)$ iterations it either computes a hyperplane separating $b$ from $SH(\mathbf{S})$, or $X_\varepsilon \in \mathbf{\Delta}_n$ such that $\Vert p(X_\varepsilon) - b \Vert \leq \varepsilon R$, $R$ maximum error over $\mathbf{\Delta}_n$. Under certain conditions iteration complexity improves to $O(1/\varepsilon)$ or even $O(\ln 1/\varepsilon)$. The worst-case complexity of each iteration is $O(mn^2)$, plus testing the existence of a pivot, shown to be equivalent to estimating the least eigenvalue of a symmetric matrix. This together with a semidefinite version of Carath\'eodory theorem allow implementing TA as if solving a CHM, resorting to the {\it power method} only as needed, thereby improving the complexity of iterations. The proposed Triangle Algorithm for SHM is simple, practical and applicable to general SDP feasibility and optimization. Also, it extends to a spectral analogue of SVM for separation of two spectrahulls.Comment: 18 page

On the Equivalence of SDP Feasibility and a Convex Hull Relaxation for System of Quadratic Equations

We show {\it semidefinite programming} (SDP) feasibility problem is equivalent to solving a {\it convex hull relaxation} (CHR) for a finite system of quadratic equations. On the one hand, this offers a simple description of SDP. On the other hand, this equivalence makes it possible to describe a version of the {\it Triangle Algorithm} for SDP feasibility based on solving CHR. Specifically, the Triangle Algorithm either computes an approximation to the least-norm feasible solution of SDP, or using its {\it distance duality}, provides a separation when no solution within a prescribed norm exists. The worst-case complexity of each iteration is computing the largest eigenvalue of a symmetric matrix arising in that iteration. Alternate complexity bounds on the total number of iterations can be derived. The Triangle Algorithm thus provides an alternative to the existing interior-point algorithms for SDP feasibility and SDP optimization. In particular, based on a preliminary computational result, we can efficiently solve SDP relaxation of {\it binary quadratic} feasibility via the Triangle Algorithm. This finds application in solving SDP relaxation of MAX-CUT. We also show in the case of testing the feasibility of a system of convex quadratic inequalities, the problem is reducible to a corresponding CHR, where the worst-case complexity of each iteration via the Triangle Algorithm is solving a {\it trust region subproblem}. Gaining from these results, we discuss potential extension of CHR and the Triangle Algorithm to solving general system of polynomial equations.Comment: 9 page