A forward-backward algorithm for decomposable semi-definite programs

In this paper, we introduce a rank-one Semi-Definite Programming (SDP) solution method for mobile source localization in sensor networks. The position and velocity of mobile source are jointly estimated using Time Delay (TD) measurements. To obtain the position and velocity of mobile source, a Relaxed Semi-Definite Programming (RSDP) algorithm is firstly designed by dropping the rank-one constraint. However, dropping the rank-one constraint leads to produce a suboptimal solution. To improve the performance, we further put forward the Penalty Function Semi-Definite Programming (PF-SDP) method to obtain the rank-one solution of the estimation problem by introducing the penalty terms. By adaptively choosing the penalty coefficient, an Adaptive Penalty Function Semi-Definite Programming (APF-SDP) algorithm is also proposed to avoid the excessive penalty. We also conduct experiments in both a simulated environment and a real system to demonstrate the effectiveness of the proposed methods. The results have demonstrated that the proposed APF-SDP outperforms the PF-SDP in terms of the position and velocity estimation whether the noise level is large or not.

Rank-one solutions for homogeneous linear matrix equations over the positive semidefinite cone

We consider the problem of estimating the discrete clustering structures under the sub-Gaussian mixture model. Our main results establish a hidden integrality property of a semidefinite programming (SDP) relaxation for this problem: while the optimal solution to the SDP is not integer-valued in general, its estimation error can be upper bounded by that of an idealized integer program. The error of the integer program, and hence that of the SDP, are further shown to decay exponentially in the signal-to-noise ratio. In addition, we show that the SDP relaxation is robust under the semirandom setting in which an adversary can modify the data generated from the mixture model. In particular, we generalize the hidden integrality property to the semirandom model and thereby show that SDP achieves the optimal error bound in this setting. These results together highlight the “global-to-local” mechanism that drives the performance of the SDP relaxation. To the best of our knowledge, our result is the first exponentially decaying error bound for convex relaxations of mixture models. A corollary of our results shows that in certain regimes, the SDP solutions are in fact integral and exact. More generally, our results establish sufficient conditions for the SDP to correctly recover the cluster memberships of [Formula: see text] fraction of the points for any [Formula: see text]. As a special case, we show that under the [Formula: see text]-dimensional stochastic ball model, SDP achieves nontrivial (sometimes exact) recovery when the center separation is as small as [Formula: see text], which improves upon previous exact recovery results that require constant separation.

Certain matrix relationships play an important role in optimality conditions and algorithms for nonlinear and semidefinite programming. Let H be an n × n symmetric matrix, A an m × n matrix, and Z a basis for the null space of A. (In a typical optimization context, H is the Hessian of a smooth function and A is the Jacobian of a set of constraints.) When the reduced Hessian ZTHZ is positive definite, augmented Lagrangian methods rely on the known existence of a finite $\bar\rho\ge 0$ such that, for all $\rho > \bar\rho$, the augmented Hessian $H + \rho \ATA $ is positive definite. In this note we analyze the case when ZTHZ is positive semidefinite, i.e., singularity is allowed, and show that the situation is more complicated. In particular, we give a simple necessary and sufficient condition for the existence of a finite $\bar\rho$ so that $H + \rho \ATA$ is positive semidefinite for $\rho \ge \bar\rho$. A corollary of our result is that if H is nonsingular and indefinite while ZTHZ is positive semidefinite and singular, no such $\bar\rho$ exists.

This paper is an attempt to exploit the opportunity of Semi Definite Programming (SDP), which is an area of convex and conic optimization. Indeed, Numerous NP-hard problems can be solveby using this approach. Hence, we intend to investigate the strength of SDP to model and provide tight relaxations of combinatorial and quadratic problems in order to present a new polynomial time algorithm for solving this robust model. This algorithm was firstly use to solve the nonlinear programs, the reason for which we search to extend it to the SDP programs. Actually, this algorithm designs the combination of two penalization methods. The first one is a primal-dual interior point (PDIM) method while the second one is a primal-dual exterior point (PDEM) method. Unlike the first method, which converges globally, the second one, also called the primal dual nonlinear rescaling method, has local super linear/quadratic convergence. Therefore, it seems appropriate to use a mixed algorithm based on the interior-exterior point method (IEPM). In fact, this resolution starts from the interior method, and at a certain level of execution, it proceeds to exterior method. Hence, a convergence evaluation function is use to know the level of permutation. Through evaluation, it has been approve that our approach is use to solve some instances of max-cut problem. This problem is a central graph theory model that occurs in many real problems and it is one of many NP-hard problems, which has attracted many researchers over the years. Then, we have used a semi definite programming solver SDPA (Semi Definite Programming Algorithm) that is modify to include the exterior point method subroutine. From the computational performance, we conclude that as the problem size increases, interior-exterior point algorithm gets relatively faster. The numerical results obtained are promising.

Location awareness for wireless sensor networks has attracted significant interests in recent years. In hostile environments, there may be malicious attacks to mislead the location estimation of target nodes. In this paper, we propose a novel secured localization method by applying cooperative semi-definite programming (SDP) algorithm. We also introduce residual normalization to improve the SDP approximation. Cooperation enhances the capability to identify the compromised anchors and the accuracy of location estimation. SDP improves the reliability of the algorithm and the out-of-convex-hull performance. Simulation results are provided to demonstrate the aforementioned improvements.

Efficient solvers for optimization problems are based on linear and semidefinite relaxations that use floating point arithmetic. However, due to the rounding errors, relaxation thus may overestimate, or worst, underestimate the very global optima. The purpose of this article is to introduce an efficient and safe procedure to rigorously bound the global optima of semidefinite program. This work shows how, using interval arithmetic, rigorous error bounds for the optimal value can be computed by carefully post processing the output of a semidefinite programming solver. A lower bound is computed on a semidefinite relaxation of the constraint system and the objective function. Numerical results are presented using the SDPA (SemiDefinite Programming Algorithm), solver to compute the solution of semidefinite programs. This rigorous bound is injected in a branch and bound algorithm to solve the optimisation problem.

Decaying with the increasing of signal propagation distance, Received Signal Strength (RSS) is used in the wireless localization due to its low cost and easily implementation.When the transmit power is unavailable, two convex optimization algorithms including semi definite programming (SDP), second order cone and semi definite programming (SOC/SDP) are designed to estimate the source locations by relaxing the non-convex problem as convex optimization.The corresponding Cramér-Rao lower bound (CRLB) of the problem is derived.The simulations demonstrate that the SOC/SDP algorithm provides the similar accuracy performance compared with the SDP algorithm.However, the computational complexity of SOC/SDP is lower than that of the SDP due to the less variables and equality constraints.When perfect knowledge of the path loss exponent is available, the simulations also show that the accuracy performance of the proposed convex optimization algorithms degrades as the path loss exponent increases.

Semidefinite relaxation method for unified near-Field and far-Field localization by AOA

This paper deals with the problem of estimating the steering direction of a signal, embedded in Gaussian disturbance, under a general quadratic inequality constraint, representing the uncertainty region of the steering. We resort to the maximum likelihood (ML) criterion and focus on two scenarios. The former assumes that the complex amplitude of the useful signal component fluctuates from snapshot to snapshot. The latter supposes that the useful signal keeps a constant amplitude within all the snapshots. We prove that the ML criterion leads in both cases to a fractional quadratically constrained quadratic problem (QCQP). In order to solve it, we first relax the problem into a constrained fractional semidefinite programming (SDP) problem which is shown equivalent, via the Charnes-Cooper transformation, to an SDP problem. Then, exploiting a suitable rank-one decomposition, we show that the SDP relaxation is tight and give a procedure to construct (in polynomial time) an optimal solution of the original problem from an optimal solution of the fractional SDP. We also assess the quality of the derived estimator through a comparison between its performance and the constrained Cramer Rao lower Bound (CRB). Finally, we give two applications of the proposed theoretical framework in the context of radar detection.

We can only extract the differential time delay (DTD) measurements between the direct and reflected paths in multistatic localization when there is no synchronization in time between the transmitter and receiver and among the receivers. This paper first addresses the problem of multistatic localization of a fixed object when the transmitter position is not available by using the DTD measurements. We propose a two-step optimization method for jointly estimating the object and transmitter positions. In the first step, we formulate a non-convex constrained weighted least squares (CWLS) problem by transforming the DTD measurement model and introducing nuisance variables. Such a non-convex CWLS problem is then relaxed to a tractable convex semidefinite programming (SDP) problem by applying semidefinite relaxation. In the second step, the error coming from relaxation and approximation in the SDP solution is reduced iteratively through solving a generalized trust region subproblem (GTRS) in each iteration. If the receivers are synchronized such that the time difference of arrival (TDOA) measurements can be acquired in addition to DTD, we formulate a different CWLS problem by utilizing both DTD and TDOA measurements, which is solved by convex relaxation as well. The relaxed SDP problem can achieve the optimal solution of the CWLS problem, and further refinement is no longer needed. We conduct the mean square error (MSE) analysis to validate that both proposed methods are able to achieve the Cramer-Rao Lower Bound (CRLB) performance under small Gaussian noise, which is also validated by simulations.

Efficient semidefinite solutions for TDOA-based source localization under unknown PS

A simple homogeneous primal-dual feasibility model is proposed for semidefinite programming (SDP) problems. Two infeasible-interior-point algorithms are applied to the homogeneous formulation. The algorithms do not need a big M initialization. If the original SDP problem has a solution (X * y * S *), then both algorithms find an ϵ-approximate solution (i.e., a solution with residual error less than or equal to ϵ) in at most steps, where ρ* = Tr(X * + *), and ϵ0 is the residual error at the (normalized) starting point. A simple way of monitoring possible infeasibility of the original SDP problem is provided such that in at most steps either an 6-approximate solution is obtained, or it is determined that there is no solution (X * y * S *) with Tr(X * + S *) less than or equal to a given number ρ > 0. Numerical results on Mehrotra type primal-dual predictor-corrector algorithms show that the homogeneous algorithms outperform their non-homogeneous counterparts, with improvement of ore than 20% in many cases, in terms of total CPU time.

This issue's SIGEST paper, from the SIAM Journal on Optimization (SIOPT), is an excellent example of deep mathematical analysis, involving a range of mathematical techniques, which is having significant practical algorithmic and computational consequences. The title of the paper, “A Sum of Squares Approximation of Nonnegative Polynomials,” by Jean B. Lasserre, combines two concepts that are known to all SIAM readers: sums of squares, the types of functions that arise in many data fitting and other applications; and nonnegative polynomials, polynomials whose values are greater than or equal to zero for all inputs. It is obvious that any polynomial that can be written as a sum of squares is nonnegative. It is far from obvious whether any nonnegative polynomial has a representation as, or arbitrarily close to, a sum of squares of polynomials. In this SIGEST paper, Lasserre shows that this is the case. The nomination from the editors of SIOPT captures the immediate and more general contributions of this paper eloquently, in the context of a series of three papers of which this is the second: “Jean Lasserre, in a series of papers (starting from the first paper in SIOPT Vol. 11, pp. 796–817, and concluding with the third paper in SIOPT Vol. 15, pp. 383–393) by exploiting deep results from real algebraic geometry and theory of moments for multivariate polynomials, showed that the optimal value of a nonconvex polynomial optimization problem (i.e., an optimization problem whose objective function and constraints are specified by multivariate polynomials) can asymptotically be approximated by an increasing sequence of optimal values from increasing tighter semidefinite programming relaxation problems. Even though the result is asymptotic in the most general case, fortuitously, practical experience has revealed that very often, the first few levels of semidefinite programming relaxations typically can achieve the global optimum value. The theoretical connection that Jean Lasserre (and Pablo Parrilo at about the same time from the viewpoint of sum of squares representations of multivariate polynomials) had established between polynomial optimizations with real algebraic geometry and theory of moments has led to exciting developments in combinatorial optimization. In particular, many hard combinatorial problems such as the maximum clique problem of a graph and the quadratic assignment problem are now actively revisited by researchers under the framework of polynomial optimization to derive increasingly tighter bounds.” That is, a consequence of this and related work is the ability to use a sequence of semidefinite programming problems to (nearly) solve much more difficult optimization problems. Semidefinite programming problems are problems with a linear objective function and linear constraints in a matrix variable X, plus the additional constraint that X be positive semidefinite. Since a rich array of efficient algorithms for solving semidefinite programming problems has been developed in recent years, reducing more general problems to a sequence of semidefinite programming problems is an important advance. We hope that readers of SIAM Review will enjoy this glimpse into the sophisticated mathematics behind an important algorithmic innovation in optimization.

Positive semi definite programs are an important subclass of semi definite programs in which all matrices involved in the specification of the problem are positive semi definite and all scalars involved are non-negative. We present a parallel algorithm, which given an instance of a positive semi definite program of size N and an approximation factor ε >; 0, runs in (parallel) time poly(1/ε)·polylog(N), using poly(N) processors, and outputs a value which is within multiplicative factor of (1+ε) to the optimal. Our result generalizes analogous result of Luby and Nisan (1993) for positive linear programs and our algorithm is inspired by their algorithm of [10].