A first-order block-decomposition method for solving two-easy-block structured semidefinite programs

In this paper, we consider a first-order block-decomposition method for minimizing the sum of a convex differentiable function with Lipschitz continuous gradient, and two other proper closed convex (possibly, nonsmooth) functions with easily computable resolvents. The method presented contains two important ingredients from a computational point of view, namely: an adaptive choice of stepsize for performing an extragradient step; and the use of a scaling factor to balance the blocks. We then specialize the method to the context of conic semidefinite programming (SDP) problems consisting of two easy blocks of constraints. Without putting them in standard form, we show that four important classes of graph-related conic SDP problems automatically possess the above two-easy-block structure, namely: SDPs for $\theta $ -functions and $\theta _{+}$ -functions of graph stable set problems, and SDP relaxations of binary integer quadratic and frequency assignment problems. Finally, we present computational results on the aforementioned classes of SDPs showing that our method outperforms the three most competitive codes for large-scale conic semidefinite programs, namely: the boundary point (BP) method introduced by Povh et al., a Newton-CG augmented Lagrangian method, called SDPNAL, by Zhao et al., and a variant of the BP method, called the SPDAD method, by Wen et al.     

Avoiding numerical cancellation in the interior point method for solving semidefinite programs

 The matrix variables in a primal-dual pair of semidefinite programs are getting increasingly ill-conditioned as they approach a complementary solution. Multiplying the primal matrix variable with a vector from the eigenspace of the non-basic part will therefore result in heavy numerical cancellation. This effect is amplified by the scaling operation in interior point methods. A complete example illustrates these numerical issues. In order to avoid numerical problems in interior point methods, we propose to maintain the matrix variables in a Cholesky form. We discuss how the factors of the v-space Cholesky form can be updated after a main iteration of the interior point method with Nesterov-Todd scaling. An analogue for second order cone programming is also developed. Numerical results demonstrate the success of this approach. Received: June 16, 2001 / Accepted: April 5, 2002 Published online: October 9, 2002 Key Words. semidefinite programming – second order cone programming Mathematics Subject Classification (2000): 90C22, 90C20     

A modified nearly exact method for solving low-rank trust region subproblem

In this paper, we first discuss how the nearly exact (NE) method proposed by Moré and Sorensen [14] for solving trust region (TR) subproblems can be modified to solve large-scale “low-rank” TR subproblems efficiently. Our modified algorithm completely avoids computation of Cholesky factorizations by instead relying primarily on the Sherman–Morrison–Woodbury formula for computing inverses of “diagonal plus low-rank” type matrices. We also implement a specific version of the modified log-barrier (MLB) algorithm proposed by Polyak [17] where the generated log-barrier subproblems are solved by a trust region method. The corresponding direction finding TR subproblems are of the low-rank type and are then solved by our modified NE method. We finally discuss the computational results of our implementation of the MLB method and its comparison with a version of LANCELOT [5] based on a collection extracted from CUTEr [12] of nonlinear programming problems with simple bound constraints.      

A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization

 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. Received: March 22, 2001 / Accepted: August 30, 2002 Published online: December 9, 2002 Key Words. semidefinite programming – low-rank factorization – nonlinear programming – augmented Lagrangian – limited memory BFGS This research was supported in part by the National Science Foundation under grants CCR-9902010, INT-9910084, CCR-0203426 and CCR-0203113     

A trust region method for solving linearly constrained locally Lipschitz optimization problems

In this paper, we present a nonsmooth trust region method for solving linearly constrained optimization problems with a locally Lipschitz objective function. Using the approximation of the steepest descent direction, a quadratic approximation of the objective function is constructed. The null space technique is applied to handle the constraints of the quadratic subproblem. Next, the CG-Steihaug method is applied to solve the new approximation quadratic model with only the trust region constraint. Finally, the convergence of presented algorithm is proved. This algorithm is implemented in the MATLAB environment and the numerical results are reported.     

A trust region method for solving the decentralized static output feedback design problem

The shortest-paths problem is a fundamental problem in graph theory and finds diverse applications in various fields. This is why shortest path algorithms have been designed more thoroughly than any other algorithm in graph theory. A large number of optimization problems are mathematically equivalent to the problem of finding shortest paths in a graph. The shortest-path between a pair of vertices is defined as the path with shortest length between the pair of vertices. The shortest path from one vertex to another often gives the best way to route a message between the vertices. This paper presents anO(n ²) time sequential algorithm and anO(n ²/p+logn) time parallel algorithm on EREW PRAM model for solving all pairs shortest paths problem on circular-arc graphs, wherep andn represent respectively the number of processors and the number of vertices of the circular-arc graph.     

A trust region algorithm for solving bilevel programming problems

In this paper, we present a new trust region algorithm for a nonlinear bilevel programming problem by solving a series of its linear or quadratic approximation subproblems. For the nonlinear bilevel programming problem in which the lower level programming problem is a strongly convex programming problem with linear constraints, we show that each accumulation point of the iterative sequence produced by this algorithm is a stationary point of the bilevel programming problem.     

信赖域子问题使用重新开始策略的共轭梯度法

给出了用共轭梯度法解信赖域子问题的重新开始策略，并证明了方法的收敛性，数值结果表明该策略可以大大提高算法的收敛速度．     

A semidefinite framework for trust region subproblems with applications to large scale minimization

Primal-dual pairs of semidefinite programs provide a general framework for the theory and algorithms for the trust region subproblem (TRS). This latter problem consists in minimizing a general quadratic function subject to a convex quadratic constraint and, therefore, it is a generalization of the minimum eigenvalue problem. The importance of (TRS) is due to the fact that it provides the step in trust region minimization algorithms. The semidefinite framework is studied as an interesting instance of semidefinite programming as well as a tool for viewing known algorithms and deriving new algorithms for (TRS). In particular, a dual simplex type method is studied that solves (TRS) as a parametric eigenvalue problem. This method uses the Lanczos algorithm for the smallest eigenvalue as a black box. Therefore, the essential cost of the algorithm is the matrix-vector multiplication and, thus, sparsity can be exploited. A primal simplex type method provides steps for the so-called hard case. Extensive numerical tests for large sparse problems are discussed. These tests show that the cost of the algorithm is 1 +α(n) times the cost of finding a minimum eigenvalue using the Lanczos algorithm, where 0<α(n)<1 is a fraction which decreases as the dimension increases. Research supported by the National Science and Engineering Research Council Canada.     

A nonmonotone conic trust region method based on line search for solving unconstrained optimization

In this paper, we present a nonmonotone conic trust region method based on line search technique for unconstrained optimization. The new algorithm can be regarded as a combination of nonmonotone technique, line search technique and conic trust region method. When a trial step is not accepted, the method does not resolve the trust region subproblem but generates an iterative point whose steplength satisfies some line search condition. The function value can only be allowed to increase when trial steps are not accepted in close succession of iterations. The local and global convergence properties are proved under reasonable assumptions. Numerical experiments are conducted to compare this method with the existing methods.     

A sensitivity result for semidefinite programs

We study the sensitivity of solutions of linear semidefinite programs under small arbitrary perturbations of the data. We present an elementary and self-contained proof of the differentiability of the solutions as functions of the perturbations, and we characterize the derivative as the solution of a system of linear equations.     

A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs

We propose a modified alternating direction method for solving convex quadratically constrained quadratic semidefinite optimization problems. The method is a first-order method, therefore requires much less computational effort per iteration than the second-order approaches such as the interior point methods or the smoothing Newton methods. In fact, only a single inexact metric projection onto the positive semidefinite cone is required at each iteration. We prove global convergence and provide numerical evidence to show the effectiveness of this method.     

A new trust region method for nonlinear equations

In this paper, a new trust region method for the system of nonlinear equations is presented in which the determining of the trust region radius incorporates the information of its natural residual. The global convergence is obtained under mild conditions. Unlike traditional trust region method, the superlinear convergence of the method is proven under the local error bound condition. This condition is weaker than the nondegeneracy assumption which is necessary for superlinear convergence of traditional trust region method. We also propose an approximate algorithm for the trust region subproblem. Preliminary numerical experiments are reported. Acknowledgements.The authors are indebted to our supervisor, Professor Y.-X. Yuan, for his excellent guidance and Jorge J. Moré for his subroutine. And we would like to thank the referees for their valuable suggestions and comments.     

A new trust region method for unconstrained optimization

In this paper, we propose a new trust region method for unconstrained optimization problems. The new trust region method can automatically adjust the trust region radius of related subproblems at each iteration and has strong global convergence under some mild conditions. We also analyze the global linear convergence, local superlinear and quadratic convergence rate of the new method. Numerical results show that the new trust region method is available and efficient in practical computation.     

A quasiconcave minimization method for solving linear two-level programs

In this paper the linear two-level problem is considered. The problem is reformulated to an equivalent quasiconcave minimization problem, via a reverse convex transformation. A branch and bound algorithm is developed which takes the specific structure into account and combines an outer approximation technique with a subdivision procedure.     

A SUCCESSIVE APPROXIMATION METHOD FOR SOLVING PROBABILISTIC CONSTRAINED PROGRAMS

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A regularized penalty method for solving convex semi-infinite programs

This paper is concerned with the stable solution of ill-posed convex semi-infinite problems on the base of their sequential approximation by finite dimensional convex problems on a sequence of grids. These auxiliary programs are constructed by using the iterative Prox-regularization and for solving each of them only one step of a penalty method is applied. A simple deletion procedure of inactive constraints is suggested. The choice of the control parameters secure the convergence of the methods and a linear convergence rate is obtained     

A preliminary set of applications leading to stochastic semidefinite programs and chance-constrained semidefinite programs

Ariyawansa and Zhu have recently introduced (two-stage) stochastic semidefinite programs (with recourse) (SSDPs) [1] and chance-constrained semidefinite programs (CCSDPs) [2] as paradigms for dealing with uncertainty in applications leading to semidefinite programs. Semidefinite programs have been the subject of intense research during the past 15 years, and one of the reasons for this research activity is the novelty and variety of applications of semidefinite programs. This research activity has produced, among other things, efficient interior point algorithms for semidefinite programs. Semidefinite programs however are defined using deterministic data while uncertainty is naturally present in applications. The definitions of SSDPs and CCSDPs in [1] and [2] were formulated with the expectation that they would enhance optimization modeling in applications that lead to semidefinite programs by providing ways to handle uncertainty in data. In this paper, we present results of our attempts to create SSDP and CCSDP models in four such applications. Our results are promising and we hope that the applications presented in this paper would encourage researchers to consider SSDP and CCSDP as new paradigms for stochastic optimization when they formulate optimization models.