A Triangle Algorithm for Semidefinite Version of Convex Hull Membership Problem
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Given a subset S={A_1, ..., A_m} of S^n, the set of n × n real symmetric matrices, we define its spectrahull as the set SH(S) = {p(X) ≡ (Tr(A_1 X), ..., Tr(A_m X))^T : X ∈Δ_n}, where Δ_n is the spectraplex, { X ∈S^n : Tr(X)=1, X ≽ 0 }. We let spectrahull membership (SHM) to be the problem of testing if a given b ∈R^m lies in SH(S). On the one hand when A_i's are diagonal matrices, SHM reduces to the 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 Triangle Algorithm (TA) kalchar,kalsep, developed for CHM and its generalization, we design a TA for SHM, where given ε, in O(1/ε^2) iterations it either computes a hyperplane separating b from SH(S), or X_ε∈Δ_n such that p(X_ε) - b ≤ε R, R maximum error over Δ_n. Under certain conditions iteration complexity improves to O(1/ε) or even O( 1/ε). 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 Carathéodory theorem allow implementing TA as if solving a CHM, resorting to the 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.
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