We treat in this paper Linear Programming (LP) problems with uncertain data. The focus is on uncertainty associated with hard constraints: those which must be satisfied, whatever is the actual realization of the data (within a prescribed uncertainty set). We suggest a modeling methodology whereas an uncertain LP is replaced by its Robust Counterpart (RC). We then develop the analytical and computational optimization tools to obtain robust solutions of an uncertain LP problem via solving the corresponding explicitly stated convex RC program. In particular, it is shown that the RC of an LP with ellipsoidal uncertainty set is computationally tractable, since it leads to a conic quadratic program, which can be solved in polynomial time.

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A large number of problems in production planning and scheduling, location, transportation, finance, and engineering design require that decisions be made in the presence of uncertainty. Uncertainty, for instance, governs the prices of fuels, the availability of electricity, and the demand for chemicals. A key difficulty in optimization under uncertainty is in dealing with an uncertainty space that is huge and frequently leads to very large-scale optimization models. Decision-making under uncertainty is often further complicated by the presence of integer decision variables to model logical and other discrete decisions in a multi-period or multi-stage setting. This paper reviews theory and methodology that have been developed to cope with the complexity of optimization problems under uncertainty. We discuss and contrast the classical recourse-based stochastic programming, robust stochastic programming, probabilistic (chance-constraint) programming, fuzzy programming, and stochastic dynamic programming. The advantages and shortcomings of these models are reviewed and illustrated through examples. Applications and the state-of-the-art in computations are also reviewed. Finally, we discuss several main areas for future development in this field. These include development of polynomial-time approximation schemes for multi-stage stochastic programs and the application of global optimization algorithms to two-stage and chance-constraint formulations.

In this paper we develop and test scenario generation methods for asset liability management models. We propose a multi-stage stochastic programming model for a Dutch pension fund. Both randomly sampled event trees and event trees fitting the mean and the covariance of the return distribution are used for generating the coefficients of the stochastic program. In order to investigate the performance of the model and the scenario generation procedures we conduct rolling horizon simulations. The average cost and the risk of the stochastic programming policy are compared to the results of a simple fixed mix model. We compare the average switching behavior of the optimal investment policies. Our results show that the performance of the multi-stage stochastic program could be improved drastically by choosing an appropriate scenario generation method. Keywords: Stochastic Programming; Finance; Asset Liability Management; Scenarios. Research partially supported through contract "HPC-...

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...

A general decomposition framework for large convex optimization problems based on augmented Lagrangians is described. The approach is then applied to multistage stochastic programming problems in two different ways: by decomposing the problem into scenarios or decomposing it into nodes corresponding to stages. In both cases the method has favorable convergence properties and a structure which makes it convenient for parallel computing environments. Keywords: Stochastic Programming, Decomposition, Augmented Lagrangian, Jacobi Method, Parallel Computation. iii iv On Augmented Lagrangian Decomposition Methods For Multistage Stochastic Programs Andrzej Ruszczy'nski 1 Introduction Multistage stochastic optimization problems belong to the most difficult problems of mathematical programming. Their size grows very quickly with the number of stages and with the number of events (scenarios) incorporated into the model. Although problems of this type occur frequently in applications (like,...

A dynamic (multi-stage) stochastic programming model for the weekly cost-optimal generation of electric power in a hydro-thermal generation system under uncertain load is developed. The model involves a large number of mixed-integer (stochastic) decision variables and constraints linking time periods and operating power units. Astochastic Lagrangian relaxation scheme is designed by assigning (stochastic) multipliers to all constraints coupling power units. It is assumed that the stochastic load process is given (or approximated) by a nite number of realizations (scenarios) in scenario tree form. Solving the dual by a bundle subgradient method leads to a successive decomposition into stochastic single (thermal or hydro) unit subproblems. The stochastic thermal and hydro subproblems are solved by astochastic dynamic programming technique and by a speci c descent algorithm, respectively. A Lagrangian heuristics that provides approximate solutions for the rst stage (primal) decisions starting from the optimal (stochastic) multipliers is developed. Numerical results are presented for realistic data from a German power utility andfornumbers of scenarios ranging from 5 to 100 and a time horizon from 7 to 9 days. The sizes of the corresponding optimization problems go up to 200.000 binary and 350.000 continuous variables, and more than 500.000 constraints.

. Constraints of a mathematical program are distributed among parallel processors together with an appropriately constructed augmented Lagrangian for each processor, which contains Lagrangian information on the constraints handled by the other processors. Lagrange multiplier information is then exchanged between processors. Convergence is established under suitable conditions for strongly convex quadratic programs and for general convex programs. Key words. Parallel Optimization, Augmented Lagrangians, Quadratic Programs, Convex Programs 1. Introduction. We are concerned with the problem minimize f(x) subject to g 1 (x) 0; . . . ; g k (x) 0 (1.1) where f , g 1 ; . . . ; g k are differentiable convex functions from the n--dimensional real space IR n to IR, IR m 1 ; . . . ; IR m k respectively, with f being strongly convex on IR n . Our principal aim is to distribute the k constraint blocks among k parallel processors together with an appropriately modified objective functio...

An algorithm incorporating the logarithmic barrier into the Benders decomposition technique is proposed for solving two-stage stochastic programs. Basic properties concerning the existence and uniqueness of the solution and the underlying path are studied. When applied to problems with a finite number of scenarios, the algorithm is shown to converge globally and to run in polynomial-time. Key Words: Stochastic programming, Large-scale linear programming, Barrier function, Interior point methods, Benders decomposition, Complexity. Abbreviated Title: A log-barrier method with Benders decomposition AMS(MOS) subject classifications: 90C15, 90C05, 90C06, 90C60. 1 1. Introduction In this paper we propose an algorithm for solving two-stage stochastic programs, establish fundamental properties of the algorithm, and analyze the convergence. An example of a two-stage stochastic program is a production planning problem. The production and demand take place in the first and second periods, resp...