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The optimal power flow (OPF) problem, which plays a central role in operating electrical networks is considered. The problem is nonconvex and is in fact NP hard. Therefore, designing efficient algorithms of practical relevance is crucial, though their global optimality is not guaranteed. Existing semi-definite programming relaxation based approaches are restricted to OPF problems where zero duality holds. In this paper, an efficient novel method to address the general nonconvex OPF problem is investigated. The proposed method is based on alternating direction method of multipliers combined with sequential convex approximations. The global OPF problem is decomposed into smaller problems associated to each bus of the network, the solutions of which are coordinated via a light communication protocol. Therefore, the proposed method is highly scalable. The convergence properties of the proposed algorithm are mathematically substantiated. Finally, the proposed algorithm is evaluated on a number of test examples, where the convergence properties of the proposed algorithm are numerically substantiated and the performance is compared with a global optimal method.

A wide variety of resource management problems of recent interest, including power/rate control, link scheduling, cross-layer control, network utility maximization, beamformer design of multiple-input multiple-output networks, and many others are directly or indirectly reliant on the weighted sum-rate maximization (WSRMax) problem. In general, this problem is very difficult to solve and is NP-hard. In this review, we provide a cohesive discussion of the existing solution

Nonconvex and structured optimization problemsarise in many engineering applications that demand scalableand distributed solution methods. The study of the convergenceproperties of these methods is in general difficult due to thenonconvexity of the problem. In this paper, two distributedsolution methods that combine the fast convergence propertiesof augmented Lagrangian-based methods with the separabilityproperties of alternating optimization are investigated. The firstmethod is adapted from the classic quadratic penalty functionmethod and is called the Alternating Direction Penalty Method(ADPM). Unlike the original quadratic penalty function method,in which single-step optimizations are adopted, ADPM uses analternating optimization, which in turn makes it scalable. Thesecond method is the well-known Alternating Direction Methodof Multipliers (ADMM). It is shown that ADPM for nonconvexproblems asymptotically converges to a primal feasible pointunder mild conditions and an additional condition ensuringthat it asymptotically reaches the standard first order necessary conditions for local optimality are introduced. In thecase of the ADMM, novel sufficient conditions under whichthe algorithm asymptotically reaches the standard first ordernecessary conditions are established. Based on this, completeconvergence of ADMM for a class of low dimensional problemsare characterized. Finally, the results are illustrated by applyingADPM and ADMM to a nonconvex localization problem inwireless sensor networks.

QC 20151208