Stochastic Semidefinite Programming Research Articles

Asymptotic analysis for a stochastic semidefinite programming

Stochastic semidefinite programming (SSDP) is a new class of optimization problems with a wide variety of applications. In this article, asymptotic analysis results of sample average approximation estimator for SSDP are established. Asymptotic analysis result already existing for stochastic nonlinear programming is extended to SSDP, that is, the conditions ensuring the convergence in distribution of sample average approximation estimator for SSDP to a multivariate normal are obtained and the corresponding covariance matrix is described in a closed form.

On Risk-Averse Stochastic Semidefinite Programs with Continuous Recourse

The vast majority of the literature on stochastic semidefinite programs (stochastic SDPs) with recourse is concerned with risk-neutral models. In this paper, we introduce mean-risk models for stochastic SDPs and study structural properties as convexity and (Lipschitz) continuity. Special emphasis is placed on stability with respect to changes of the underlying probability distribution. Perturbations of the true distribution may arise from incomplete information or working with (finite discrete) approximations for the sake of computational efficiency. We discuss extended formulations for stochastic SDPs under finite discrete distributions, which turn out to be deterministic (mixed-integer) SDPs that are (almost) block-structured for many popular risk measures.

Inexact SA Method for Constrained Stochastic Convex SDP and Application in Chinese Stock Market

We propose stochastic convex semidefinite programs (SCSDPs) to handle uncertain data in applications. For these models, we design an efficient inexact stochastic approximation (SA) method and prove the convergence, complexity, and robust treatment of the algorithm. We apply the inexact method for solving SCSDPs where the subproblem in each iteration is only solved approximately and show that it enjoys the similar iteration complexity as the exact counterpart if the subproblems are progressively solved to sufficient accuracy. Numerical experiments show that the method we proposed was effective for uncertain problem.

HOMOGENEOUS SELF-DUAL METHODS FOR SYMMETRIC CONES UNDER UNCERTAINTY

Homogeneous self-dual algorithms for stochastic semidefinite programs with finite event space have been proposed by Jin et al. in [17]. Alzalg [8] has adopted their work to derive homogeneous selfdual algorithms for stochastic second-order programs with finite event space. In this paper, we generalize these to derive homogeneous selfdual algorithms for stochastic programs with finite event space over the much wider class of all symmetric cones. They include among others, stochastic semidefinite programs and stochastic second-order cone programs.

A STOCHASTIC APPROXIMATION ALGORITHM FOR STOCHASTIC SEMIDEFINITE PROGRAMMING

Motivated by applications to multi-antenna wireless networks, we propose a distributed and asynchronous algorithm for stochastic semidefinite programming. This algorithm is a stochastic approximation of a continuous-time matrix exponential scheme which is further regularized by the addition of an entropy-like term to the problem's objective function. We show that the resulting algorithm converges almost surely to an ɛ-approximation of the optimal solution requiring only an unbiased estimate of the gradient of the problem's stochastic objective. When applied to throughput maximization in wireless systems, the proposed algorithm retains its convergence properties under a wide array of mobility impediments such as user update asynchronicities, random delays and/or ergodically changing channels. Our theoretical analysis is complemented by extensive numerical simulations, which illustrate the robustness and scalability of the proposed method in realistic network conditions.

Volumetric barrier decomposition algorithms for stochastic quadratic second-order cone programming

Ariyawansa and Zhu (2011) have derived volumetric barrier decomposition algorithms for solving two-stage stochastic semidefinite programs and proved polynomial complexity of certain members of the algorithms. In this paper, we utilize their work to derive volumetric barrier decomposition algorithms for solving two-stage stochastic convex quadratic second-order cone programming, and establish polynomial complexity of certain members of the proposed algorithms.

A stochastic semidefinite programming approach for bounds on option pricing under regime switching

We consider bounds for the price of a European-style call option under regime switching. Stochastic semidefinite programming models are developed that incorporate a lattice generated by a finite-state Markov chain regime-switching model as a representation of scenarios (uncertainty) to compute bounds. The optimal first-stage bound value is equivalent to a Value at Risk quantity, and the optimal solution can be obtained via simple sorting. The upper (lower) bounds from the stochastic model are bounded below (above) by the corresponding deterministic bounds and are always less conservative than their robust optimization (min-max) counterparts. In addition, penalty parameters in the model allow controllability in the degree to which the regime switching dynamics are incorporated into the bounds. We demonstrate the value of the stochastic solution (bound) and computational experiments using the S&P 500 index are performed that illustrate the advantages of the stochastic programming approach over the deterministic strategy.

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.

Hybrid Impulsive Control of Stochastic Systems with Multiplicative Noise Under Markovian Switching

A problem of state output feedback stabilization of discrete-time stochastic systems with multiplicative noise under Markovian switching is considered. Under some appropriate assumptions, the stability of this system under pure impulsive control is given. Further under hybrid impulsive control, the output feedback stabilization problem is investigated. The hybrid control action is formulated as a combination of the regular control along with an impulsive control action. The jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the stochastic and the deterministic terms. Sufficient conditions based on stochastic semi-definite programming and linear matrix inequalities (LMIs) for both stochastic stability and stabilization are obtained. Such a nonconvex problem is solved using the existing optimization algorithms and the nonconvex CVX package. The robustness of the stability and stabilization concepts against all admissible uncertainties are also investigated. The parameter uncertainties we consider here are norm bounded. Two examples are given to demonstrate the obtained results.

Logarithmic barrier decomposition-based interior point methods for stochastic symmetric programming

We introduce and study two-stage stochastic symmetric programs with recourse to handle uncertainty in data defining (deterministic) symmetric programs in which a linear function is minimized over the intersection of an affine set and a symmetric cone. We present a Benders’ decomposition-based interior point algorithm for solving these problems and prove its polynomial complexity. Our convergence analysis proved by showing that the log barrier associated with the recourse function of stochastic symmetric programs behaves a strongly self-concordant barrier and forms a self-concordant family on the first stage solutions. Since our analysis applies to all symmetric cones, this algorithm extends Zhao’s results [G. Zhao, A log barrier method with Benders’ decomposition for solving two-stage stochastic linear programs, Math. Program. Ser. A 90 (2001) 507–536] for two-stage stochastic linear programs, and Mehrotra and Özevin’s results [S. Mehrotra, M.G. Özevin, Decomposition-based interior point methods for two-stage stochastic semidefinite programming, SIAM J. Optim. 18 (1) (2007) 206–222] for two-stage stochastic semidefinite programs.

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.

A class of polynomial volumetric barrier decomposition algorithms for stochastic semidefinite programming

A class of polynomial volumetric barrier decomposition algorithms for stochastic semidefinite programming

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 [1,2] 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.

Location-aided routing with uncertainty in mobile ad hoc networks: A stochastic semidefinite programming approach

We study location-aided routing under mobility in wireless ad hoc networks. Due to node mobility, the network topology changes continuously, and clearly there exists an intricate tradeoff between the message passing overhead and the latency in the route discovery process. Aiming to obtain a clear understanding of this tradeoff, we use stochastic semidefinite programming (SSDP), a newly developed optimization model, to deal with the location uncertainty associated with node mobility. In particular, we model both the speed and the direction of node movement by random variables and construct random ellipses accordingly to better capture the location uncertainty and the heterogeneity across different nodes. Based on SSDP, we propose a stochastic location-aided routing (SLAR) strategy to optimize the tradeoff between the message passing overhead and the latency. Our results reveal that in general SLAR can significantly reduce the overall overhead than existing deterministic algorithms, simply because the location uncertainty in the routing problem is better captured by the SSDP model.

Decomposition‐Based Interior Point Methods for Two‐Stage Stochastic Semidefinite Programming

We introduce two-stage stochastic semidefinite programs with recourse and present an interior point algorithm for solving these problems using Bender’s decomposition. This decomposition algorithm and its analysis extend Zhao’s results [Math. Program., 90 (2001), pp. 507-536] for stochastic linear programs. The convergence results are proved by showing that the logarithmic barrier associated with the recourse function of two-stage stochastic semidefinite programs with recourse is a strongly self-concordant barrier on the first stage solutions. The short-step variant of the algorithm requires $O(\sqrt{p+Kr} \ln \mu^0/\epsilon)$ Newton iterations to follow the first stage central path from a starting value of the barrier parameter $\mu^0$ to a terminating value e. The long-step variant requires $O((p+Kr) \ln \mu^0/\epsilon)$ damped Newton iterations. The calculation of the gradient and Hessian of the recourse function and the first stage Newton direction decomposes across the second stage scenarios.

A class of volumetric barrier decomposition algorithms for stochastic quadratic programming

Ariyawansa and Zhu have introduced a class of volumetric barrier decomposition algorithms [K.A. Ariyawansa, Y. Zhu. A class of polynomial volumetric barrier decomposition algorithms for stochastic semidefinite programming, Available as Technical Report 2006-7, Department of Mathematics, Washington State University, Pullman, WA, submitted for publication. Available from: <http://www.math.wsu.edu/math/TRS/2006-7.pdf>] for solving two-stage stochastic semidefinite programs with recourse (SSDPs) [K.A. Ariyawansa, Y. Zhu, Stochastic semidefinite programming: a new paradigm for stochastic optimization, 4OR—The Quarterly Journal of the Belgian, French and Italian OR Societies, (in press). Available as Technical Report 2004-10, Department of Mathematics, Washington State University, Pullman, WA, October 2004. Available from: <http://www.math.wsu.edu/math/TRS/2004-10.pdf>]. In this paper we utilize their work for SSDPs to derive a class of volumetric barrier decomposition algorithms for solving two-stage stochastic quadratic programs with recourse and to establish polynomial complexity of certain members of the class of algorithms.

Stochastic semidefinite programming: a new paradigm for stochastic optimization

Semidefinite programs are a class of optimization problems that have been studied extensively during the past 15 years. Semidefinite programs are naturally related to linear programs, and both are defined using deterministic data. Stochastic programs were introduced in the 1950s as a paradigm for dealing with uncertainty in data defining linear programs. In this paper, we introduce stochastic semidefinite programs as a paradigm for dealing with uncertainty in data defining semidefinite programs.

Stochastic Programs with Recourse II: On the Continuity of the Objective

Previous article Next article Stochastic Programs with Recourse II: On the Continuity of the ObjectiveDavid W. Walkup and Roger J.-B. WetsDavid W. Walkup and Roger J.-B. Wetshttps://doi.org/10.1137/0117010PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] R. Tyrrell Rockafellar, Convex analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997xviii+451 MR1451876 (97m:49001) 0932.90001 Google Scholar[2] David W. Walkup and , Roger J.-B. Wets, Stochastic programs with recourse, SIAM J. Appl. Math., 15 (1967), 1299–1314 10.1137/0115113 MR0233589 (38:1910) 0203.21806 LinkISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Dynamical Behavior of a Stochastic Forward–Backward Algorithm Using Random Monotone Operators25 July 2016 | Journal of Optimization Theory and Applications, Vol. 171, No. 1 Cross Ref Uncertainties in minimax stochastic programsOptimization, Vol. 60, No. 10-11 Cross Ref Stochastic semidefinite programming: a new paradigm for stochastic optimization18 July 2006 | 4OR, Vol. 4, No. 3 Cross Ref Stochastic linear optimisation of an MPMP production planning modelInternational Journal of Production Economics, Vol. 55, No. 1 Cross Ref STOCHASTIC LINEAR PROGRAMMING WITH RECOURSE: A TUTORIALDecision Sciences, Vol. 11, No. 1 Cross Ref Multistage Stochastic Programming with Recourse: The Equivalent Deterministic ProblemPaul Olsen18 July 2006 | SIAM Journal on Control and Optimization, Vol. 14, No. 3AbstractPDF (2106 KB)Multistage Stochastic Programming with Recourse As Mathematical Programming in an $L_p $ SpacePaul Olsen18 July 2006 | SIAM Journal on Control and Optimization, Vol. 14, No. 3AbstractPDF (909 KB)About properties of the mean value functional and of the continuous infimal convolution in stochastic convex analysis21 May 2005 Cross Ref Duality for Stochastic Programming Interpreted as L. P. in $L_p $-SpaceMark J. Eisner and Paul Olsen12 July 2006 | SIAM Journal on Applied Mathematics, Vol. 28, No. 4AbstractPDF (1304 KB)Stochastic Programs with Fixed Recourse: The Equivalent Deterministic ProgramRoger J.-B. Wets18 July 2006 | SIAM Review, Vol. 16, No. 3AbstractPDF (3839 KB)Nota sobre programacion lineal estocastica: Evolucion y estado actual. (I)Trabajos de estadistica y de investigacion operativa, Vol. 23, No. 1-2 Cross Ref REFERENCES Cross Ref Some Practical Regularity Conditions for Nonlinear ProgramsDavid W. Walkup and Roger J.-B. Wets18 July 2006 | SIAM Journal on Control, Vol. 7, No. 3AbstractPDF (742 KB) Volume 17, Issue 1| 1969SIAM Journal on Applied Mathematics History Submitted:17 January 1968Published online:28 July 2006 InformationCopyright © 1969 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0117010Article page range:pp. 98-103ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

Stochastic Programs with Recourse

Abstract : So far the study of stochastic programs with recourse has been limited to the case (called by G. Dantzig programming under uncertainty) when only the right-hand sides or resources of the problem are random. In this paper the authors extend the theory to the general case when essentially all the parameters involved are random. This generalization immediately raises the problem of attributing a precise meaning to the stochastic constraints. They examine a probability formulation (satisfying the constraints almost surely) and a possibility formulation (satisfying the constraints for all values of the random parameters in the support of their joint distribution) and show them equivalent under a rather weak but curious W-condition. Finally, they prove that without restriction the equivalent deterministic form of a stochastic program with recourse is a convex program for which we obtain some additional properties when some of the parameters of the original problem are constant. The applications of the theoretical results of this paper to certain classes of stochastic programs which have arisen from practical problems will be presented in a separate paper: 'Stochastic Programs with Recourse: Special Forms.' (Author)