An interior-point and decomposition approach to multiple stage stochastic programming

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A parallel computation approach for solving multistage stochastic network problems

The original publication is available at www.springerlink.comThis paper presents a parallel computation approach for the efficient solution of very large multistage linear and nonlinear network problems with random parameters. These problems result from particular instances of models for the robust optimization of network problems with uncertainty in the values of the right-hand side and the objective function coefficients. The methodology considered here models the uncertainty using scenarios to characterize the random parameters. A scenario tree is generated and, through the use of full-recourse techniques, an implementable solution is obtained for each group of scenarios at each stage along the planning horizon. As a consequence of the size of the resulting problems, and the special structure of their constraints, these models are particularly well-suited for the application of decomposition techniques, and the solution of the corresponding subproblems in a parallel computation environment. An augmented Lagrangian decomposition algorithm has been implemented on a distributed computation environment, and a static load balancing approach has been chosen for the parallelization scheme, given the subproblem structure of the model. Large problems – 9000 scenarios and 14 stages with a deterministic equivalent nonlinear model having 166000 constraints and 230000 variables – are solved in 45 minutes on a cluster of four small (11 Mflops) workstations. An extensive set of computational experiments is reported; the numerical results and running times obtained for our test set, composed of large-scale real-life problems, confirm the efficiency of this procedure.Publicad

A parallel computation approach for solving multistage stochastic network problems

This paper presents a parallel computation approach for the efficient solution of very large multistage linear and nonIinear network problems with random parameters. These problems resul t from particular instances of models for the robust optimization of network problems with uncertainty in the values of the right-hand side and the objective function coefficients. The methodology considered here models the uncertainty using scenarios to characterize the random parameters. A. scenario tree is generated and, through the use of full-recourse techniques, an implementable solution is obtained for each group of scenarios at each stage along the planning horizon. As a consequence of the size of the resulting problems, and the special structure of their constraints, these models are particularly well-suited for the application of decomposition techniques, and the solution of the corresponding subproblems in a parallel computation environment. An Augmented Lagrangian decomposition algorithm has been implemented on a distributed computation environment, and a static load balancing approach has been chosen for the parallelization scheme. given the subproblem structure of the model. Large problems -9000 scenarios and 14 stages with a deterministic equivalent nonlinear model having 166000 constraints and 230000 variables- are solved in 15 minutes on a cluster of 4 small (16 Mflops) workstations. An extensive set of computational experiments is reported; the numerical results and running times obtained for our test set, composed of large-scale real-life problems, confirm the efficiency of this procedure

Large-scale unit commitment under uncertainty: an updated literature survey

The Unit Commitment problem in energy management aims at finding the optimal production schedule of a set of generation units, while meeting various system-wide constraints. It has always been a large-scale, non-convex, difficult problem, especially in view of the fact that, due to operational requirements, it has to be solved in an unreasonably small time for its size. Recently, growing renewable energy shares have strongly increased the level of uncertainty in the system, making the (ideal) Unit Commitment model a large-scale, non-convex and uncertain (stochastic, robust, chance-constrained) program. We provide a survey of the literature on methods for the Uncertain Unit Commitment problem, in all its variants. We start with a review of the main contributions on solution methods for the deterministic versions of the problem, focussing on those based on mathematical programming techniques that are more relevant for the uncertain versions of the problem. We then present and categorize the approaches to the latter, while providing entry points to the relevant literature on optimization under uncertainty. This is an updated version of the paper "Large-scale Unit Commitment under uncertainty: a literature survey" that appeared in 4OR 13(2), 115--171 (2015); this version has over 170 more citations, most of which appeared in the last three years, proving how fast the literature on uncertain Unit Commitment evolves, and therefore the interest in this subject

Optimal Power Generation under Uncertainty via Stochastic Programming

A power generation system comprising thermal and pumped-storage hydro plants is considered. Two kinds of models for the cost-optimal generation of electric power under uncertain load are introduced: (i) a dynamic model for the short-term operation and (ii) a power production planning model. In both cases, the presence of stochastic data in the optimization model leads to multi-stage and two-stage stochastic programs, respectively. Both stochastic programming problems involve a large number of mixed-integer (stochastic) decisions, but their constraints are loosely coupled across operating power units. This is used to design Lagrangian relaxation methods for both models, which lead to a decomposition into stochastic single unit subproblems. For the dynamic model a Lagrangian decomposition based algorithm is described in more detail. Special emphasis is put on a discussion of the duality gap, the efficient solution of the multi-stage single unit subproblems and on solving the dual problem by bundle methods for convex nondifferentiable optimization

Stochastic Programming Algorithms

Stochastické programování a optimalizace jsou mocnými nástroji pro řešení široké škály inženýrských problémů zahrnujících neurčitost. Algoritmus progressive hedging je efektivní dekompoziční metoda určená pro řešení scénářových stochastických úloh. Z důvodu vertikální dekompozice je možno tento algoritmus implementovat paralelně, čímž lze významně ušetřit výpočetní čas a ostatní prostředky. Teoretická část této diplomové práce se zabývá matematickým a zejména pak stochastickým programováním a detailně popisuje algoritmus progressive hedging. V praktické části je navržena a diskutována původní paralelní implementace algoritmu progressive hedging, která je pak otestována na jednoduchých úlohách. Dále je uvedená paralelní implementace použita pro řešení inženýrského problému plynulého odlévání ocelové bramy a na závěr jsou získané výsledky zhodnoceny.Stochastic programming and optimization are powerful tools for solving a wide variety of engineering problems including uncertainty. The progressive hedging algorithm is an effective decomposition method for solving scenario-based stochastic programmes. Due to the vertical decomposition, this algorithm can be implemented in parallel thereby the computing time and other resources could be considerably spared. The theoretical part of this master's thesis deals with mathematical and especially with stochastic programming. Further, the progressive hedging algorithm is presented and discussed in detail. In the practical part, the original parallel implementation of the progressive hedging algorithm is suggested, fruitfully discussed and tested to simple problems. Furthermore, the presented parallel implementation is used for solving the continuous casting process of steel slabs and the results are appraised.