Efficient Management of Multiple Sets to Extract Complex Structures from Mathematical Programs

Most of the applied models written with an algebraic modeling language involve simultaneously several dimensions such as materials, location, time or uncertainty. The information about dimensions available in the algebraic formulation is usually sufficient to retrieve different block structures from mathematical programs. These structured problems can then be solved by adequate solution techniques. To illustrate this idea we focus on stochastic programming problems with recourse. Taking into account both time and uncertainty dimensions of these problems, we are able to retrieve different customized structures in their constraint matrices. We applied the Structure Exploiting Tool to retrieve the structure from models built with the GAMS modeling language. The underlying mathematical programs are solved with the decomposition algorithm that applies interior point methods. The optimization algorithm is run in a sequential and in a parallel computing environment.     

Interior-Point Methods with Decomposition for Solving Large-Scale Linear Programs

This paper deals with an algorithm incorporating the interior-point method into the Dantzig–Wolfe decomposition technique for solving large-scale linear programming problems. The algorithm decomposes a linear program into a main problem and a subproblem. The subproblem is solved approximately. Hence, inexact Newton directions are used in solving the main problem. We show that the algorithm is globally linearly convergent and has polynomial-time complexity.     

Automatic Formulation of Stochastic Programs Via an Algebraic Modeling Language

This paper presents an open source tool that automatically generates the so-called deterministic equivalent in stochastic programming. The tool is based on the algebraic modeling language ampl. The user is only required to provide the deterministic version of the stochastic problem and the information on the stochastic process, either as scenarios or as a transitions-based event tree.     

On robust optimization of two-stage systems

Robust-optimization models belong to a special class of stochastic programs, where the traditional expected cost minimization objective is replaced by one that explicitly addresses cost variability. This paper explores robust optimization in the context of two-stage planning systems. We show that, under arbitrary measures for variability, the robust optimization approach might lead to suboptimal solutions to the second-stage planning problem. As a result, the variability of the second-stage costs may be underestimated, thereby defeating the intended purpose of the model. We propose sufficient conditions on the variability measure to remedy this problem. Under the proposed conditions, a robust optimization model can be efficiently solved using a variant of the L-shaped decomposition algorithm for traditional stochastic linear programs. We apply the proposed framework to standard stochastic-programming test problems and to an application that arises in auctioning excess electric power. Mathematics Subject Classification (1991):90C15, 90C33, 90B50, 90A09, 90A43Supported in part by NSF Grants DMI-0099726 and DMI-0133943     

Stochastic quasigradient methods for optimization of discrete event systems

In this paper, stochastic programming techniques are adapted and further developed for applications to discrete event systems. We consider cases where the sample path of the system depends discontinuously on control parameters (e.g. modeling of failures, several competing processes), which could make the computation of estimates of the gradient difficult. Methods which use only samples of the performance criterion are developed, in particular finite differences with reduced variance and concurrent approximation and optimization algorithms. Optimization of the stationary behavior is also considered. Results of numerical experiments and convergence results are reported.     

Auxiliary problem principle and decomposition of optimization problems

The auxiliary problem principle allows one to find the solution of a problem (minimization problem, saddle-point problem, etc.) by solving a sequence of auxiliary problems. There is a wide range of possible choices for these problems, so that one can give special features to them in order to make them easier to solve. We introduced this principle in Ref. 1 and showed its relevance to decomposing a problem into subproblems and to coordinating the subproblems. Here, we derive several basic or abstract algorithms, already given in Ref. 1, and we study their convergence properties in the framework of i infinite-dimensional convex programming.     

Approximations in proximal bundle methods and decomposition of convex programs

A proximal bundle method is presented for minimizing a nonsmooth convex functionf. At each iteration, it requires only one approximate evaluation off and its -subgradient, and it finds a search direction via quadratic programming. When applied to the Lagrangian decomposition of convex programs, it allows for inexact solutions of decomposed subproblems; yet, increasing their required accuracy automatically, it asymptotically finds both the primal and dual solutions. It is an implementable approximate version of the proximal point algorithm. Some encouraging numerical experience is reported.The author thanks two anonymous referees for their valuable comments.Research supported by the State Committee for Scientific Research under Grant 8550502206.     

基于随机规划的建筑集群冷热电联供系统优化研究

面向建筑集群的冷热电联供系统的设计和优化是实现建筑楼宇能源成本节约的重要途径。随机因素对该联供系统的优化决策,具有显著的影响。考虑建筑楼宇的能源需求为随机变量,构建随机混合整数规划模型,解决以最小化建筑楼宇总费用为目标时建筑集群冷热电联供系统的优化问题;其次,提出采用Benders多割平面方法求解多目标规划问题,从而寻找冷热电联供系统的设备配置和系统运行的Pareto最优决策;最后,通过实验验证了模型和算法的有效性。实验结果表明建筑集群在协作模式下,相比于非协作模式,具有更低的总费用。     

Two methods for large-scale nonlinear optimization and their comparison on a case study of hydropower optimization

This paper presents two methods for the optimization of structured large-scale problems: a decomposition method of dual type for nonlinear problems and a sequential quadratic programming based method. Practical details of application of the methods to the case study problem of the hydropower system of an African river are then given. Comparison of results is presented, indicating that both methods are useful and efficient, having however different features from a practical point of view. General remarks concerning the practical differences between a decomposition-based method and a method exploiting the problem structure within the framework of general purpose optimization routines are finally presented.This work was partly supported by the Water Resource Project, IIASA, Laxenburg, Austria.     

An H2-Type Norm of a Discrete-Time Linear Stochastic System with Periodic Coefficients Simultaneously Affected by an Infinite Markov Chain and Multiplicative White Noise Perturbations

This article investigates the problem of the definition and computation of an H₂-type norm for discrete-time time-varying periodic stochastic linear systems simultaneously affected by multiplicative white noise perturbations and random jumping according to a Markov chain with an infinite countable number of states. Also, we solve an optimization problem that contains, as a special case, the H₂ optimal control problem for the considered class of stochastic systems under the assumption of perfect state measurements.