In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worst-case) objective while satisfying the constraints for every possible value of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist as SDPs. When the perturbation is "full," our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique and continuous (Hölder-stable) with respect to the unperturbed problem's data. The approach can thus be used to regularize ill-conditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation, and integer programming.

The stochastic linear programming problem with recourse has a dual block angular structure. It can thus be handled by Benders decomposition or by Kelley's method of cutting planes; equivalently the dual problem has a primal block angular structure and can be handled by Dantzig-Wolfe decomposition--- the two approaches are in fact identical by duality. Here we shall investigate the use of the method of cutting planes from analytic centers applied to similar formulations. The only significant difference form the aforementioned methods is that new cutting planes (or columns, by duality) will be generated not from the optimum of the linear programming relaxation, but from the analytic center of the set of localization. 1 Introduction The study of optimization problems in the presence of uncertainty still taxes the limits of methodology and software. One of the most approachable settings is that of two-staged planning under uncertainty, in which a first stage decision has to be taken bef...

In this paper we consider semidenite programs (SDPs) whose data depends on some unknown-but-bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worst-case) objective while satisfying the constraints for every possible values of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist, as SDPs. When the perturbation is "full", our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique, and continuous (Hölder-stable) with respect to the unperturbed problems' data. The approach can thus be used to regularize ill-conditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation and integer programming.

Multistage stochastic programming # in contrast to stochastic control # has found wide application in the formulation and solution of #nancial problems characterized by a large number of state variables and a generally lownumber of possible decision stages. The literature on the use of multistage recourse modelling to formalize complex portfolio optimization problems dates back to the early seventies, when the technique was #rst adopted to solve a #xed interest security portfolio problem. We present here the CALM model which has been designed to deal with uncertainty a#ecting both assets #in either the portfolio or the market# and liabilities #in the form of scenario dependentpayments or borrowing costs#. We consider as an instance a pension fund problem in which portfolio rebalancing is allowed over a long-term horizon at discrete time points and where liabilities refer to #ve di#erent classes of pension contracts. The portfolio manager, given an initial wealth, seeks the maximization...

Value at risk is an important measure of extent to which a given portfolio is subject to different kinds of risk present in financial markets. Considerable amount of research was dedicated during recent years to development of acceptable methods for evaluation of this risk measure. In this paper we go beyond estimation of value at risk and address the following questions. - Suppose that a boundary for acceptable value at risk is fixed. How to find a portfolio among given set of securities which would provide the maximal yield and at the same time satisfy the constraints on value at risk; - Suppose that the market conditions are changing. How to obtain a portfolio rebalancing strategy which would keep portfolio within given boundary on value at risk and maximize in some sense the yield of resulting sequence of portfolios. In order to solve these problems we adapt and further develop algorithmic tools which have their origin in stochastic optimization and were considered recently in mac...

this paper we formulate a model which aims to combine results on the probabilistic analysis of congestion in the network with optimisation of traffic routing decisions. The optimization model is essentially deterministic multicommodity network flow accounting for the stochasticity of traffic flows in the constraints. In this introduction a brief account of modern technology is given for the purpose of explaining the modelling assumptions. The derivation of a measure of stochastic multi-service traffic is presented in Section 2. Section 3 treats model formulation and implementation. Section 4 discusses model implementation and a simple numerical example. Conclusions and future research directions are contained in Section 5. Emerging from telephony, B-ISDN will carry different traffic such as voice, data and video over the same bearer. A standardised transport, multiplexing and switching technique called Asynchronous Transfer Mode (ATM) is recommended as a common digital format for implementing the B-ISDN concept. The resource model of an ATM network considers demands with very different statistical properties requiring flexible allocation of resources. For example data transmission services are highly bursty (burstiness = peak bit rate / average bit rate), whereas high definition television (HDTV) distribution has low burstiness but will consume a bit rate above 50 Mb/s per HDTV channel.All information coming from different traffic sources is transmitted by the means of short fixed length cells comprising a 5 byte header and a 48 byte information payload. The process of resource allocation in ATM networks exploits statistical multiplexing and is associated with some form of traffic usage contract. The notion of a committed quality or Grade of Service (GoS) per connection...

Year this Degree Granted: 2003

In this paper we provide a general classification of mathematical optimization problems, followed by a matrix of applications that shows the areas in which these problems have been typically applied in process systems engineering. We then provide a review of solution methods of the major types of optimization problems for continuous and discrete variable optimization, particularly nonlinear and mixed-integer nonlinear programming. We also review their extensions to dynamic optimization and optimization under uncertainty. While these areas are still subject to significant research efforts, the emphasis in this paper is on major developments that have taken place over the last twenty five years.

tool for stochastic mixed integer optimization