Stochastic programming

About: Stochastic programming is a research topic. Over the lifetime, 12343 publications have been published within this topic receiving 421049 citations.

Price of Correlations in Stochastic Optimization

Abstract: When decisions are made in the presence of high-dimensional stochastic data, handling joint distribution of correlated random variables can present a formidable task, both in terms of sampling and estimation as well as algorithmic complexity. A common heuristic is to estimate only marginal distributions and substitute joint distribution by independent (product) distribution. In this paper, we study possible loss incurred on ignoring correlations through a distributionally robust stochastic programming model, and we quantify that loss as price of correlations (POC). Using techniques of cost sharing from game theory, we identify a wide class of problems for which POC has a small upper bound. To our interest, this class will include many stochastic convex programs, uncapacitated facility location, Steiner tree, and submodular functions, suggesting that the intuitive approach of assuming independent distribution acceptably approximates the robust model for these stochastic optimization problems. Additionally, we demonstrate hardness of bounding POC via examples of subadditive and supermodular functions that have large POC. We find that our results are also useful for solving many deterministic optimization problems like welfare maximization, k-dimensional matching, and transportation problems, under certain conditions.

Cost Minimization in an Energy-Intensive Plant Using Mathematical Programming Approaches

Abstract: This work addresses the problem of determining the optimal operating schedule that minimizes the operating cost in an energy-intensive air separation plant. The difficulty arises from the fact that the rate at which the utility company supplies electricity to the plant is subject to high fluctuations. This creates a potential opportunity to reduce average operating costs by changing the operating mode and production rates depending on the power costs. However, constraints occur due to product distribution requirements and plant capabilities. The scheduling optimization problem is made more challenging because the power prices are only known for a portion of the desired optimization horizon. These challenges were addressed by developing an efficient two-stage stochastic programming approach. Extensive analysis was done which resulted in a MILP problem formulation that uses an ARIMA model to generate the necessary scenarios for future power prices. The proposed problem was solved by utilizing commercial sof...

Optimal asset allocation for DC pension plans under inflation

Abstract: In this paper, the stochastic dynamic programming approach is used to investigate the optimal asset allocation for a defined-contribution pension plan with downside protection under stochastic inflation. The plan participant invests the fund wealth and the stochastic interim contribution flows into the financial market. The nominal interest rate model is described by the Cox–Ingersoll–Ross ( Cox et al., 1985 ) dynamics. To cope with the inflation risk, the inflation indexed bond is included in the asset menu. The retired individuals receive an annuity that is indexed by inflation and a downside protection on the amount of this annuity is considered. The closed-form solution is derived under the CRRA utility function. Finally, a numerical application is presented to characterize the dynamic behavior of the optimal investment strategy.

Optimal Power Generation under Uncertainty via Stochastic Programming

Abstract: A power generation system comprising thermal and pumpedstorage 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.