Semidefinite Programming Algorithm Research Articles

Safe Bounds in Semidefinite Programming by Using Interval Arithmetic

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.

Joint Beamforming and User Selection in Multicast Downlink Channel under Secrecy-Outage Constraint

In this letter, we study the beamforming and user selection problem in multicast MISO wiretap-channel, where a base station (BS) with Nt antennas broadcasts a common confidential message to M legitimate users (Bobs) and N unauthorized users (Eves) attempt to eavesdrop the message. We target at maximizing the minimum secrecy-outage rate of Bobs and propose an iterative successive convex approximation (SCA) algorithm to tackle the original nonconvex problem. We further develop joint semidefinite programming (SDP) and SCA algorithm to find a feasible subset where legitimate users are guaranteed to their requirements. Simulation results are provided to evaluate the proposed approaches.

Orthogonal Invariance and Identifiability

Matrix variables are ubiquitous in modern optimization, in part because variational properties of useful matrix functions often expedite standard optimization algorithms. Convexity is one important such property: permutation-invariant convex functions of the eigenvalues of a symmetric matrix are convex, leading to the wide applicability of semidefinite programming algorithms. We prove the analogous result for the property of “identifiability,” a notion central to many active-set-type optimization algorithms.

A Stochastic Smoothing Algorithm for Semidefinite Programming

We use rank one Gaussian perturbations to derive a smooth stochastic approximation of the maximum eigenvalue function. We then combine this smoothing result with an optimal smooth stochastic optimization algorithm to produce an efficient method for solving maximum eigenvalue minimization problems, and detail a variant of this stochastic algorithm with monotonic line search. Overall, compared to classical smooth algorithms, this method runs a larger number of significantly cheaper iterations and, in certain precision/dimension regimes, its total complexity is lower than that of deterministic smoothing algorithms.

Homogeneous Self-dual Algorithms for Stochastic Second-Order Cone Programming

Jin et al. (in J. Optim. Theory Appl. 155:1073---1083, 2012) proposed homogeneous self-dual algorithms for stochastic semidefinite programs with finite event space. In this paper, we utilize their work to derive homogeneous self-dual algorithms for stochastic second-order cone programs with finite event space. We also show how the structure in the stochastic second-order cone programming problems may be exploited so that the algorithms developed for these problems have less complexity than the algorithms developed for stochastic semidefinite programs mentioned above.

BILGO: Bilateral greedy optimization for large scale semidefinite programming

Many machine learning tasks (e.g. metric and manifold learning problems) can be formulated as convex semidefinite programs. To enable the application of these tasks on a large-scale, scalability and computational efficiency are considered as desirable properties for a practical semidefinite programming algorithm. In this paper, we theoretically analyze a new bilateral greedy optimization (denoted BILGO) strategy in solving general semidefinite programs on large-scale datasets. As compared to existing methods, BILGO employs a bilateral search strategy during each optimization iteration. In such an iteration, the current semidefinite matrix solution is updated as a bilateral linear combination of the previous solution and a suitable rank-1 matrix, which can be efficiently computed from the leading eigenvector of the descent direction at this iteration. By optimizing for the coefficients of the bilateral combination, BILGO reduces the cost function in every iteration until the KKT conditions are fully satisfied, thus, it tends to converge to a global optimum. In fact, we prove that BILGO converges to the global optimal solution at a rate of O(1/k), where k is the iteration counter. The algorithm thus successfully combines the efficiency of conventional rank-1 update algorithms and the effectiveness of gradient descent. Moreover, BILGO can be easily extended to handle low rank constraints. To validate the effectiveness and efficiency of BILGO, we apply it to two important machine learning tasks, namely Mahalanobis metric learning and maximum variance unfolding. Extensive experimental results clearly demonstrate that BILGO can solve large-scale semidefinite programs efficiently.

A second-order Mehrotra-type predictor-corrector algorithm with a new wide neighbourhood for semi-definite programming

In this paper, we present a new second-order Mehrotra-type predictor–corrector algorithm for semi-definite programming (SDP). The proposed algorithm is based on a new wide neighbourhood. We are particularly concerned with an important inequality. Based on the inequality, the convergence is shown for a specific class of search directions. In particular, the complexity bound is O(√nlogϵ−1) for the Nesterov–Todd search direction and O(n logϵ−1) for the Helmberg-Kojima-Monteiro search direction. The derived complexity bounds coincide with the currently best known theoretical complexity bounds obtained so far for SDP. We provide some preliminary numerical results as well.

Approximating the Lovász θ Function with the Subgradient Method

Abstract The famous Lovasz theta number θ ( G ) is expressed as the optimal solution of a semidefinite program. As such, it can be computed in polynomial time to an arbitrary precision. Nevertheless, computing it in practice yields some difficulties as the size of the graph gets larger and larger, despite recent significant advances of semidefinite programming (SDP) solvers. We present a way around SDP which exploits a well-known equivalence between SDP and lagrangian relaxations of non-convex quadratic programs. This allows us to design a subgradient algorithm which is shown to be competitive with SDP algorithms in terms of efficiency, while being preferable as far as memory requirements, flexibility and stability are concerned.

Eigenvalue‐optimisation‐based optimal power flow with small‐signal stability constraints

The occasional oscillation in large interconnected power system can cause the small-signal stability problem. As a complement to the damping controllers, the small-signal stability constrained-optimal power flow (SSSC-OPF) model has been used to obtain the required stability margin. Applying the approximate technique to SSSC-OPF may not only increase the value of the objective function, but also suffer the oscillation of the iterations during the solving process. In this study, an eigenvalue-optimisation-based non-linear semi-definite programming (NLSDP) model and algorithm is proposed for the small-signal stability constraints. It is a significant challenge to model the SSSC-OPF directly because of the implicit and non-Lipschitz property for the spectral abscissa of the system state matrix. Based on the Lyapunov theorem, the positive definite constraints can express the small-signal stability accurately and equivalently, so that SSSC-OPF can be modelled as NLSDP. Afterward, the NLSDP model is transformed into a non-linear programming problem by formulating the positive definite constraints into non-linear ones, which can be solved by the interior point method finally. Numerical simulations for two systems confirm the validity of the model and the robustness of the algorithm.

Cooperative Received Signal Strength-Based Sensor Localization With Unknown Transmit Powers

Cooperative localization (also known as sensor network localization) using received signal strength (RSS) measurements when the source transmit powers are different and unknown is investigated. Previous studies were based on the assumption that the transmit powers of source nodes are the same and perfectly known which is not practical. In this paper, the source transmit powers are considered as nuisance parameters and estimated along with the source locations. The corresponding Cramer-Rao lower bound (CRLB) of the problem is derived. To find the maximum likelihood (ML) estimator, it is necessary to solve a nonlinear and nonconvex optimization problem, which is computationally complex. To avoid the difficulty in solving the ML estimator, we derive a novel semidefinite programming (SDP) relaxation technique by converting the ML minimization problem into a convex problem which can be solved efficiently. The algorithm requires only an estimate of the path loss exponent (PLE). We initially assume that perfect knowledge of the PLE is available, but we then examine the effect of imperfect knowledge of the PLE on the proposed SDP algorithm. The complexity analyses of the proposed algorithms are also studied in detail. Computer simulations showing the remarkable performance of the proposed SDP algorithm are presented.

Simplified analysis for full-Newton step infeasible interior-point algorithm for semidefinite programming

We present an analysis of the full-Newton step infeasible interior-point algorithm for semidefinite optimization, which is an extension of the algorithm introduced by Roos [C. Roos, A full-Newton step 𝒪(n) infeasible interior-point algorithm for linear optimization, SIAM J. Optim. 16 (2006), pp. 1110–1136] for the linear optimization case. We use the proximity measure σ(V) = ‖I − V 2‖ to overcome the difficulty of obtaining an upper bound of updated proximity after one full-Newton step, where I is an identity matrix and V is a symmetric positive definite matrix. It turns out that the complexity analysis of the algorithm is simplified and the iteration bound obtained is improved slightly.

Algorithm 925

A SemiDefinite Programming (SDP) problem is one of the most central problems in mathematical optimization. SDP provides an effective computation framework for many research fields. Some applications, however, require solving a large-scale SDP whose size exceeds the capacity of a single processor both in terms of computation time and available memory. SDPARA (SemiDefinite Programming Algorithm paRAllel package) [Yamashita et al. 2003b] was designed to solve such large-scale SDPs. Its parallel performance is outstanding for general SDPs in most cases. However, the parallel implementation is less successful for some sparse SDPs obtained from applications such as Polynomial Optimization Problems (POPs) or Sensor Network Localization (SNL) problems, since this version of SDPARA cannot directly handle sparse Schur Complement Matrices (SCMs). In this article we improve SDPARA by focusing on the sparsity of the SCM and we propose a new parallel implementation using the formula-cost-based distribution along with a replacement of the dense Cholesky factorization. We verify numerically that these features are key to solving SDPs with sparse SCMs more quickly on parallel computing systems. The performance is further enhanced by multithreading and the new SDPARA attains considerable scalability in general. It also finds solutions for extremely large-scale SDPs arising from POPs which cannot be obtained by other solvers.

A robust algorithm for semidefinite programming

Current successful methods for solving semidefinite programs (SDPs) are based on primal–dual interior-point approaches. These usually involve a symmetrization step to allow for application of Newton's method followed by block elimination to reduce the size of the Newton equation. Both these steps create ill-conditioning in the Newton equation and singularity of the Jacobian of the optimality conditions at the optimum. In order to avoid the ill-conditioning, we derive and test a backwards stable primal–dual interior-point method for SDP. Relative to current public domain software, we realize both a distinct improvement in the accuracy of the optimum and a reduction in the number of iterations. This is true for random problems as well as for problems of special structure. Our algorithm is based on a Gauss–Newton approach applied to a single bilinear form of the optimality conditions. The well-conditioned Jacobian allows for a preconditioned (matrix-free) iterative method for finding the search direction at each iteration.

Extensive v2DM study of the one-dimensional Hubbard model for large lattice sizes: Exploiting translational invariance and parity

Using variational density matrix optimization with two- and three-index conditions we study the one-dimensional Hubbard model with periodic boundary conditions at various filling factors. Special attention is directed to the full exploitation of the available symmetries, more specifically the combination of translational invariance and space-inversion parity, which allows for the study of large lattice sizes. We compare the computational scaling of three different semidefinite programming algorithms with increasing lattice size, and find the boundary point method to be the most suited for this type of problem. Several physical properties, such as the two-particle correlation functions, are extracted to check the physical content of the variationally determined density matrix. It is found that the three-index conditions are needed to correctly describe the full phase diagram of the Hubbard model. We also show that even in the case of half filling, where the ground-state energy is close to the exact value, other properties such as the spin-correlation function can be flawed.

An Extended Sequential Quadratically Constrained Quadratic Programming Algorithm for Nonlinear, Semidefinite, and Second-Order Cone Programming

This paper is concerned with nonlinear, semidefinite, and second-order cone programs. A general algorithm, which includes sequential quadratic programming and sequential quadratically constrained quadratic programming methods, is presented for solving these problems. In the particular case of standard nonlinear programs, the algorithm can be interpreted as a prox-regularization of the Solodov sequential quadratically constrained quadratic programming method presented in Mathematics of Operations Research (2004). For such type of methods, the main cost of computation amounts to solve a linear cone program for which efficient solvers are available. Usually, “global convergence results” for these methods require, as for the Solodov method, the boundedness of the primal sequence generated by the algorithm. The other purpose of this paper is to establish global convergence results without boundedness assumptions on any of the iterative sequences built by the algorithm.

Homogeneous Self-dual Algorithms for Stochastic Semidefinite Programming

Ariyawansa and Zhu have proposed a new class of optimization problems termed stochastic semidefinite programs to handle data uncertainty in applications leading to (deterministic) semidefinite programs. For stochastic semidefinite programs with finite event space, they have also derived a class of volumetric barrier decomposition algorithms, and proved polynomial complexity of certain members of the class. In this paper, we consider homogeneous self-dual algorithms for stochastic semidefinite programs with finite event space. We show how the structure in such problems may be exploited so that the algorithms developed in this paper have complexity similar to those of the decomposition algorithms mentioned above.

New complexity analysis of a Mehrotra-type predictor–corrector algorithm for semidefinite programming

In this paper, we propose a new Mehrotra-type predictor–corrector interior-point algorithm for semidefinite programming. This algorithm is an extension of the variant of Mehrotra-type algorithm that was proposed by Salahi et al. [On Mehrotra-type predictor–corrector algorithms, SIAM J. Optim. 18 (2007), pp. 1377–1397] for linear programming problems. We modify the step sizes lightly in the predictor step of Koulaei and Terlaky [On the complexity analysis of a Mehrotra-type primal–dual feasible algorithm for semidefinite optimization, Optim. Methods Softw. 25 (2010), pp. 467–485]. In such a way, we obtain O(nlog(Tr(X 0 S 0)/ϵ)) iteration complexity of the algorithm, where (X 0, y 0, S 0) is the initial feasible point and ϵ is the required precision.

Computation of Correlated Equilibrium with Global-Optimal Expected Social Welfare

In this paper, we propose an algorithm which computes the correlated equilibrium with global-optimal (i.e., maximum) expected social welfare for single stage polynomial games. We first derive tractable primal/dual semidefinite programming (SDP) relaxations for an infinite-dimensional formulation of correlated equilibria. We give an asymptotic convergence proof, which ensures solving the sequence of relaxations leads to solutions that converge to the correlated equilibrium with the highest expected social welfare. Finally, we give a dedicated sequential SDP algorithm and demonstrate it in a wireless application with numerical results.

A new second-order corrector interior-point algorithm for semidefinite programming

In this paper, we propose a second-order corrector interior-point algorithm for semidefinite programming (SDP). This algorithm is based on the wide neighborhood. The complexity bound is \({O(\sqrt{n}L)}\) for the Nesterov-Todd direction, which coincides with the best known complexity results for SDP. To our best knowledge, this is the first wide neighborhood second-order corrector algorithm with the same complexity as small neighborhood interior-point methods for SDP. Some numerical results are provided as well.

A Decomposition Algorithm for KYP-SDPs

In this paper, a structure exploiting algorithm for semidefinite programs derived from the Kalmatz-Yakubovich-Popov lemma, where some of the constraints appear as complicating constraints is presented. A decomposition algorithm is proposed, where the structure of the problem can be utilized. In a numerical example, where a controller that minimizes the stun of the H-2-norm and the H-infinity-norm is designed, the algorithm, is shown to be faster than SeDuMi and the special purpose solver KYPD.