The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard, because it contains vector cardinality minimization as a special case. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this pre-existing concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to solving the norm minimization relaxations, and illustrate our results with numerical examples.

In this paper, we present a nonlinear programming algorithm for solving semidefinite programs (SDPs) in standard form. The algorithm's distinguishing feature is a change of variables that replaces the symmetric, positive semidefinite variable X of the SDP with a rectangular variable R according to the factorization X = RR T . The rank of the factorization, i.e., the number of columns of R, is chosen minimally so as to enhance computational speed while maintaining equivalence with the SDP. Fundamental results concerning the convergence of the algorithm are derived, and encouraging computational results on some large-scale test problems are also presented. Keywords: semidefinite programming, low-rank factorization, nonlinear programming, augmented Lagrangian, limited memory BFGS. 1 Introduction In the past few years, the topic of semidefinite programming, or SDP, has received considerable attention in the optimization community, where interest in SDP has included the investigation of...

The optimal power flow (OPF) problem is nonconvex and generally hard to solve. In this paper, we propose a semidefinite programming (SDP) optimization, which is the dual of an equivalent form of the OPF problem. A global optimum solution to the OPF problem can be retrieved from a solution of this convex dual problem whenever the duality gap is zero. A necessary and sufficient condition is provided in this paper to guarantee the existence of no duality gap for the OPF problem. This condition is satisfied by the standard IEEE benchmark systems with 14, 30, 57, 118 and 300 buses as well as several randomly generated systems. Since this condition is hard to study, a sufficient zero-duality-gap condition is also derived. This sufficient condition holds for IEEE systems after small resistance (10 −5 per unit) is added to every transformer that originally assumes zero resistance. We investigate this sufficient condition and justify that it holds widely in practice. The main underlying reason for the successful convexification of the OPF problem can be traced back to the modeling of transformers and transmission lines as well as the non-negativity of physical quantities such as resistance and inductance.

Abstract—In this paper, the design of controllers that incor-porate structural and multiobjective performance requirements is considered. The control structures under study cover nested, chained, hierarchical, delayed interaction and communications, and symmetric systems. Such structures are strongly related to several modern-day and future applications including integrated flight propulsion systems, platoons of vehicles, micro-electro-me-chanical systems, networked control, control of networks, pro-duction lines and chemical processes. It is shown that the system classes presented have the common feature that all stabilizing controllers can be characterized by convex constraints on the Youla–Kucera parameter. Using this feature, a solution to a gen-eral optimal performance problem that incorporates time domain and frequency domain constraints is obtained. A synthesis proce-dure is provided which at every step yields a feasible controller

Let S = {x ∈ R n: g1(x) ≥ 0, · · · , gm(x) ≥ 0} be a semialgebraic set defined by multivariate polynomials gi(x). Assume S is compact, convex and has nonempty interior. Let Si = {x ∈ R n: gi(x) ≥ 0} and ∂Si = {x ∈ R n: gi(x) = 0} be its boundary. This paper, as does the subject of semidefinite programming (SDP), concerns Linear Matrix Inequalities (LMIs). The set S is said to have an LMI representation if it equals the set of solutions to some LMI and it is known that some convex S may not be LMI representable [6]. A question arising from [13], see [6, 14], is: given S ∈ R n, does there exist an LMI representable set ˆ S in some higher dimensional space R n+N whose projection down onto R n equals S. Such S is called semidefinite representable or SDP representable. This paper addresses the SDP representability problem. The following are the main contributions of this paper: (i) Assume gi(x) are all concave on S. If the positive definite Lagrange Hessian (PDLH) condition holds, i.e., the Hessian of the Lagrange function for optimization problem of minimizing any nonzero linear function ℓ T x on S is positive definite at the minimizer, then S is SDP representable. (ii) If each gi(x) is either sos-concave (− ∇ 2 gi(x) = W(x) T W(x) for some matrix polynomial W(x)) or strictly quasi-concave on S, then S is SDP representable. (iii) If each Si is either sos-convex or poscurv-convex (Si is compact, convex and has smooth boundary with positive curvature), then S is SDP representable. This also holds for Si for which ∂Si ∩ S extends smoothly to the boundary of a poscurv-convex set containing S. (iv) We give the complexity of Schmüdgen and Putinar’s matrix Positivstellensatz, which are critical to the proofs of (i)-(iii).

Wireless sensor networks (WSNs) are networks of distributed autonomous devices that can sense or monitor physical or environmental conditions cooperatively. WSNs face many challenges, mainly caused by communication failures, storage and computational constraints and limited power supply. Paradigms of computational intelligence (CI) have been successfully used in recent years to address various challenges such as data aggregation and fusion, energy aware routing, task scheduling, security, optimal deployment and localization. CI provides adaptive mechanisms that exhibit intelligent behavior in complex and dynamic environments like WSNs. CI brings about flexibility, autonomous behavior, and robustness against topology changes, communication failures and scenario changes. However, WSN developers are usually not or not completely aware of the potential CI algorithms offer. On the other side, CI researchers are not familiar with all real problems and subtle requirements of WSNs. This mismatch makes collaboration and development difficult. This paper intends to close this gap and foster collaboration by offering a detailed introduction to WSNs and their properties. An extensive survey of CI applications to various problems in WSNs from various research areas and publication venues is presented in the paper. Besides, a discussion on advantages and disadvantages of CI algorithms over traditional WSN solutions is offered. In addition, a general evaluation of CI algorithms is presented, which will serve as a guide for using CI algorithms for WSNs.

A set S ⊆ R n is called to be Semidefinite (SDP) representable if S equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI). Clearly, if S is SDP representable, then S must be convex and semialgebraic (it is describable by conjunctions and disjunctions of polynomial equalities or inequalities). This paper proves sufficient conditions and necessary conditions for SDP representability of convex sets and convex hulls by proposing a new approach to construct SDP representations. The contributions of this paper are: (i) For bounded SDP representable sets W1, · · · , Wm, we give an explicit construction of an SDP representation for conv( ∪ m k=1 Wk). This provides a technique for building global SDP representations from the local ones. (ii) For the SDP representability of a compact convex semialgebraic set S, we prove sufficient condition: the boundary ∂S is positively curved, and necessary condition: ∂S has nonnegative curvature at smooth points and on nondegenerate corners. This amounts to the strict versus nonstrict quasi-concavity of defining polynomials on those points on ∂S where they vanish. The gaps between them are ∂S having positive versus nonnegative curvature and smooth versus nonsmooth points. A sufficient condition bypassing the gaps is when some defining polynomials of S satisfy an algebraic condition called sos-concavity. (iii) For the SDP representability of the convex hull of a compact nonconvex semialgebraic set T, we find that the critical object is ∂cT, the maximum subset of ∂T contained in ∂conv(T). We prove sufficient conditions for SDP representability: ∂cT is positively curved, and necessary conditions: ∂cT has nonnegative curvature at smooth points and on nondegenerate corners. The gaps between sufficient and necessary conditions are similar to case (ii). The positive definite Lagrange Hessian (PDLH) condition is also discussed.

We formulate the sensor network localization problem as finding the global minimizer of a quartic polynomial. Then sum of squares (SOS) relaxations can be applied to solve it. However, the general SOS relaxations are too expensive to implement for large problems. Exploiting the special features of this polynomial, we propose a new structured SOS relaxation, and discuss its various properties. When distances are given exactly, this SOS relaxation often returns true sensor locations. At each step of interior point methods solving this SOS relaxation, the complexity is O(n 3), where n is the number of sensors. When the distances have small perturbations, we show that the sensor locations given by this SOS relaxation are accurate within a constant factor of the perturbation error under some technical assumptions. The performance of this SOS relaxation is tested on some randomly generated problems.

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This paper discusses how to find the global minimum of functions that are summations of small polynomials (“small ” means involving a small number of variables). Some sparse sum of squares (SOS) techniques are proposed. We compare their computational complexity and lower bounds with prior SOS relaxations. Under certain conditions, we also discuss how to extract the global minimizers from these sparse relaxations. The proposed methods are especially useful in solving sparse polynomial system and nonlinear least squares problems. Numerical experiments are presented, which show that the proposed methods significantly improve the computational performance of prior methods for solving these problems. Lastly, we present applications of this sparsity technique in solving polynomial systems derived from nonlinear differential equations and sensor network localization. Key words: Polynomials, sum of squares (SOS), sparsity, nonlinear least squares, polynomial system, nonlinear differential equations, sensor network localization 1