Showing papers on "Stochastic programming published in 2009"

Differential Evolution Algorithm With Strategy Adaptation for Global Numerical Optimization

Abstract: Differential evolution (DE) is an efficient and powerful population-based stochastic search technique for solving optimization problems over continuous space, which has been widely applied in many scientific and engineering fields. However, the success of DE in solving a specific problem crucially depends on appropriately choosing trial vector generation strategies and their associated control parameter values. Employing a trial-and-error scheme to search for the most suitable strategy and its associated parameter settings requires high computational costs. Moreover, at different stages of evolution, different strategies coupled with different parameter settings may be required in order to achieve the best performance. In this paper, we propose a self-adaptive DE (SaDE) algorithm, in which both trial vector generation strategies and their associated control parameter values are gradually self-adapted by learning from their previous experiences in generating promising solutions. Consequently, a more suitable generation strategy along with its parameter settings can be determined adaptively to match different phases of the search process/evolution. The performance of the SaDE algorithm is extensively evaluated (using codes available from P. N. Suganthan) on a suite of 26 bound-constrained numerical optimization problems and compares favorably with the conventional DE and several state-of-the-art parameter adaptive DE variants.

Lectures on Stochastic Programming: Modeling and Theory

Abstract: List of notations Preface to the second edition Preface to the first edition 1. Stochastic programming models 2. Two-stage problems 3. Multistage problems 4. Optimization models with probabilistic constraints 5. Statistical inference 6. Risk averse optimization 7. Background material 8. Bibliographical remarks Bibliography Index.

Primal-dual subgradient methods for convex problems

Abstract: In this paper we present a new approach for constructing subgradient schemes for different types of nonsmooth problems with convex structure. Our methods are primal-dual since they are always able to generate a feasible approximation to the optimum of an appropriately formulated dual problem. Besides other advantages, this useful feature provides the methods with a reliable stopping criterion. The proposed schemes differ from the classical approaches (divergent series methods, mirror descent methods) by presence of two control sequences. The first sequence is responsible for aggregating the support functions in the dual space, and the second one establishes a dynamically updated scale between the primal and dual spaces. This additional flexibility allows to guarantee a boundedness of the sequence of primal test points even in the case of unbounded feasible set (however, we always assume the uniform boundedness of subgradients). We present the variants of subgradient schemes for nonsmooth convex minimization, minimax problems, saddle point problems, variational inequalities, and stochastic optimization. In all situations our methods are proved to be optimal from the view point of worst-case black-box lower complexity bounds.

Continuous-time Stochastic Control and Optimization with Financial Applications

Abstract: Stochastic optimization problems arise in decision-making problems under uncertainty, and find various applications in economics and finance. On the other hand, problems in finance have recently led to new developments in the theory of stochastic control. This volume provides a systematic treatment of stochastic optimization problems applied to finance by presenting the different existing methods: dynamic programming, viscosity solutions, backward stochastic differential equations, and martingale duality methods. The theory is discussed in the context of recent developments in this field, with complete and detailed proofs, and is illustrated by means of concrete examples from the world of finance: portfolio allocation, option hedging, real options, optimal investment, etc. This book is directed towards graduate students and researchers in mathematical finance, and will also benefit applied mathematicians interested in financial applications and practitioners wishing to know more about the use of stochastic optimization methods in finance.

Economic Valuation of Reserves in Power Systems With High Penetration of Wind Power

Abstract: This paper proposes a methodology to determine the required level of spinning and nonspinning reserves in a power system with a high penetration of wind power. The computation of the required reserve levels and their costs is achieved through a stochastic programming market-clearing model spanning a daily time horizon. This model considers the network constraints and takes into account the cost of both the load shedding and the wind spillage. The methodology proposed is illustrated using an example and a realistic case study. Some conclusions are finally drawn.

Uncertainty Management in the Unit Commitment Problem

Abstract: Uncertainty in power systems operations has been traditionally managed by multistage decision making and operating reserve requirements. A familiar example of multistage decisions is day-ahead unit commitment and real-time economic dispatch. An alternate approach for managing uncertainty is a stochastic formulation, which allows the explicit modeling of the sources of uncertainty. This paper compares stochastic and reserve methods and evaluates the benefits of a combined approach for the efficient management of uncertainty in the unit commitment problem. Numerical studies show that unit commitment solutions obtained for the combined approach are robust and superior with respect to the traditional approach in terms of both economics and reliability metrics.

Optimal virtual machine placement across multiple cloud providers

Abstract: Cloud computing provides users an efficient way to dynamically allocate computing resources to meet demands. Cloud providers can offer users two payment plans, i.e., reservation and on-demand plans for resource provisioning. Price of resources in reservation plan is generally cheaper than that in on-demand plan. However, since the reservation plan has to be acquired in advance, it may not fully meet future demands in which the on-demand plan can be used to guarantee the availability to the user. In this paper, we propose an optimal virtual machine placement (OVMP) algorithm. This algorithm can minimize the cost spending in each plan for hosting virtual machines in a multiple cloud provider environment under future demand and price uncertainty. OVMP algorithm makes a decision based on the optimal solution of stochastic integer programming (SIP) to rent resources from cloud providers. The performance of OVMP algorithm is evaluated by numerical studies and simulation. The results clearly show that the proposed OVMP algorithm can minimize users' budgets. This algorithm can be applied to provision resources in emerging cloud computing environments.

A Stochastic Model for the Optimal Operation of a Wind-Thermal Power System

Abstract: This paper presents a stochastic cost model and a solution technique for optimal scheduling of the generators in a wind integrated power system considering the demand and wind generation uncertainties. The proposed robust unit commitment solution methodology will help the power system operators in optimal day-ahead planning even with indeterminate information about the wind generation. A particle swarm optimization based scenario generation and reduction algorithm is used for modeling the uncertainties. The stochastic unit commitment problem is solved using a new parameter free self adaptive particle swarm optimization algorithm. The numerical results indicate the low risk involved in day-ahead power system planning when the stochastic model is used instead of the deterministic model.

An integer programming approach for linear programs with probabilistic constraints

Abstract: Linear programs with joint probabilistic constraints (PCLP) are difficult to solve because the feasible region is not convex. We consider a special case of PCLP in which only the right-hand side is random and this random vector has a finite distribution. We give a mixed-integer programming formulation for this special case and study the relaxation corresponding to a single row of the probabilistic constraint. We obtain two strengthened formulations. As a byproduct of this analysis, we obtain new results for the previously studied mixing set, subject to an additional knapsack inequality. We present computational results which indicate that by using our strengthened formulations, instances that are considerably larger than have been considered before can be solved to optimality.

The Concept of Recoverable Robustness, Linear Programming Recovery, and Railway Applications

Abstract: We present a new concept for optimization under uncertainty: recoverable robustness A solution is recovery robust if it can be recovered by limited means in all likely scenarios Specializing the general concept to linear programming we can show that recoverable robustness combines the flexibility of stochastic programming with the tractability and performances guarantee of the classical robust approach We exemplify recoverable robustness in delay resistant, periodic and aperiodic timetabling problems, and train platforming

Scenario tree modeling for multistage stochastic programs

Abstract: An important issue for solving multistage stochastic programs consists in the approximate representation of the (multivariate) stochastic input process in the form of a scenario tree. In this paper, we develop (stability) theory-based heuristics for generating scenario trees out of an initial set of scenarios. They are based on forward or backward algorithms for tree generation consisting of recursive scenario reduction and bundling steps. Conditions are established implying closeness of optimal values of the original process and its tree approximation, respectively, by relying on a recent stability result in Heitsch, Romisch and Strugarek (SIAM J Optim 17:511–525, 2006) for multistage stochastic programs. Numerical experience is reported for constructing multivariate scenario trees in electricity portfolio management.

Scenario Reduction for Futures Market Trading in Electricity Markets

Abstract: To make informed decisions in futures markets of electric energy, stochastic programming models are commonly used. Such models treat stochastic processes via a set of scenarios, which are plausible realizations throughout the decision-making horizon of the stochastic processes. The number of scenarios needed to accurately represent the uncertainty involved is generally large, which may render the associated stochastic programming problem intractable. Hence, scenario reduction techniques are needed to trim down the number of scenarios while keeping most of the stochastic information embedded in such scenarios. This paper proposes a novel scenario reduction procedure that advantageously compares with existing ones for electricity-market problems tackled via two-stage stochastic programming.

Dynamic network design for reverse logistics operations under uncertainty

Abstract: The design of reverse logistics network has attracted growing attention with the stringent pressures from environmental and social requirements In general, decisions about reverse logistics network configurations are made on a long-term basis and factors influencing such reverse logistics network design may also vary over time This paper proposes dynamic location and allocation models to cope with such issues A two-stage stochastic programming model is further developed by which a deterministic model for multiperiod reverse logistics network design can be extended to account for the uncertainties A solution approach integrating a recently proposed sampling method with a heuristic algorithm is also proposed in this research A numerical experiment is presented to demonstrate the significance of the developed stochastic model as well as the efficiency of the proposed solution method

A Bilevel Stochastic Programming Approach for Retailer Futures Market Trading

Abstract: This paper presents a bilevel programming approach to solve the medium-term decision-making problem faced by a power retailer. A retailer decides its level of involvement in the futures market and in the pool as well as the selling price offered to its potential clients with the goal of maximizing the expected profit at a given risk level. Uncertainty on future pool prices, client demands, and rival-retailer prices is accounted for via stochastic programming. Unlike in previous approaches, client response to retail price and competition among rival retailers are both explicitly considered in the proposed bilevel model. The resulting nonlinear bilevel programming formulation is transformed into an equivalent single-level mixed-integer linear programming problem by replacing the lower-level optimization by its Karush-Kuhn-Tucker optimality conditions and converting a number of nonlinearities to linear equivalents using some well-known integer algebra results. A realistic case study is solved to illustrate the efficient performance of the proposed methodology.

Risk Management for a Global Supply Chain Planning Under Uncertainty : Models and Algorithms

Abstract: In this article, we consider the risk management for mid-term planning of a global multi-product chemical supply chain under demand and freight rate uncertainty. A two-stage stochastic linear programming approach is proposed within a multi-period planning model that takes into account the production and inventory levels, transportation modes, times of shipments, and customer service levels. To investigate the potential improvement by using stochastic programming, we describe a simulation framework that relies on a rolling horizon approach. The studies suggest that at least 5% savings in the total real cost can be achieved compared with the deterministic case. In addition, an algorithm based on the multi-cut L-shaped method is proposed to effectively solve the resulting large scale industrial size problems. We also introduce risk management models by incorporating risk measures into the stochastic programming model, and multi-objective optimization schemes are implemented to establish the tradeoffs between cost and risk. To demonstrate the effectiveness of the proposed stochastic models and decomposition algorithms, a case study of a realistic global chemical supply chain problem is presented. 2009 American Institute of Chemical Engineers AIChE J, 55: 931–946, 2009

Light Robustness

Abstract: We consider optimization problems where the exact value of the input data is not known in advance and can be affected by uncertainty. For these problems, one is typically required to determine a robust solution, i.e., a possibly suboptimal solution whose feasibility and cost is not affected heavily by the change of certain input coefficients. Two main classes of methods have been proposed in the literature to handle uncertainty: stochastic programming (offering great flexibility, but often leading to models too large in size to be handled efficiently), and robust optimization (whose models are easier to solve but sometimes lead to very conservative solutions of little practical use). In this paper we investigate a heuristic way to model uncertainty, leading to a modelling framework that we call Light Robustness. Light Robustness couples robust optimization with a simplified two-stage stochastic programming approach, and has a number of important advantages in terms of flexibility and ease to use. In particular, experiments on both random and real word problems show that Light Robustness is sometimes able to produce solutions whose quality is comparable with that obtained through stochastic programming or robust models, though it requires less effort in terms of model formulation and solution time.

Economic valuation of reserves in power systems with high penetration of wind power

Abstract: This paper proposes a methodology to determine the required level of spinning and non-spinning reserves in a power system with a high penetration of wind power. The computation of the required reserve levels and their costs is achieved through a stochastic programming market-clearing model spanning a daily time horizon. This model considers the network constraints and takes into account the cost of both the load shedding and the wind spillage. The methodology proposed is illustrated using an example and a realistic case study. Some conclusions are finally drawn.

Scenario-based model predictive control of stochastic constrained linear systems

Abstract: In this paper we propose a stochastic model predictive control (MPC) formulation based on scenario generation for linear systems affected by discrete multiplicative disturbances. By separating the problems of (1) stochastic performance, and (2) stochastic stabilization and robust constraints fulfillment of the closed-loop system, we aim at obtaining a less conservative control action with respect to classical robust MPC schemes, still enforcing convergence and feasibility properties for the controlled system. Stochastic performance is addressed for very general classes of stochastic disturbance processes, although discretized in the probability space, by adopting ideas from multi-stage stochastic optimization. Stochastic stability and recursive feasibility are enforced through linear matrix inequality (LMI) problems, which are solved off-line; stochastic performance is optimized by an on-line MPC problem which is formulated as a convex quadratically constrained quadratic program (QCQP) and solved in a receding horizon fashion. The performance achieved by the proposed approach is shown in simulation and compared to the one obtained by standard robust and deterministic MPC schemes.

A multistage fuzzy-stochastic programming model for supporting sustainable water-resources allocation and management

Abstract: In this study, a multistage fuzzy-stochastic programming (MFSP) model is developed for tackling uncertainties presented as fuzzy sets and probability distributions. A vertex analysis approach is proposed for solving multiple fuzzy sets in the MFSP model. Solutions under a set of @a-cut levels can be generated by solving a series of deterministic submodels. The developed method is applied to the planning of a case study for water-resources management. Dynamics and uncertainties of water availability (and thus water allocation and shortage) could be taken into account through generation of a set of representative scenarios within a multistage context. Moreover, penalties are exercised with recourse against any infeasibility, which permits in-depth analyses of various policy scenarios that are associated with different levels of economic consequences when the promised water-allocation targets are violated. The modeling results can help to generate a range of alternatives under various system conditions, and thus help decision makers to identify desired water-resources management policies under uncertainty.

A Study of Demand Stochasticity in Service Network Design

Abstract: The objective of this paper is to investigate the importance of introducing stochastic elements into service network design formulations. To offer insights into this issue, we take a basic version of the problem in which periodic schedules are built for a number of vehicles and where only the demand may vary stochastically. We study how solutions based on uncertain demand differ from solutions based on deterministic demand and provide qualitative descriptions of the structural differences. Some of these structural differences provide a hedge against uncertainty by using consolidation. This way we get consolidation as output from the model rather than as an a priori required property. Service networks with such properties are robust, as seen by the customers, by providing operational flexibility.

Fast Approaches to Improve the Robustness of a Railway Timetable

Abstract: The train timetabling problem (TTP) consists of finding a train schedule on a railway network that satisfies some operational constraints and maximizes some profit function that accounts for the efficiency of the infrastructure usage. In practical cases, however, the maximization of the objective function is not enough, and one calls for a robust solution that is capable of absorbing, as much as possible, delays/disturbances on the network. In this paper we propose and computationally analyze four different methods to improve the robustness of a given TTP solution for the aperiodic (noncyclic) case. The approaches combine linear programming (LP) and ad hoc stochastic programming/robust optimization techniques. We computationally compare the effectiveness and practical applicability of the four techniques under investigation on real-world test cases from the Italian railway company Trenitalia. The outcome is that two of the proposed techniques are very fast and provide robust solutions of comparable quality with respect to the standard (but very time consuming) stochastic programming approach.

What You Should Know About Approximate Dynamic Programming

Abstract: Approximate dynamic programming (ADP) is a broad umbrella for a modeling and algorithmic strategy for solving problems that are sometimes large and complex, and are usually (but not always) stochastic. It is most often presented as a method for overcoming the classic curse of dimensionality that is well-known to plague the use of Bellman's equation. For many problems, there are actually up to three curses of dimensionality. But the richer message of approximate dynamic programming is learning what to learn, and how to learn it, to make better decisions over time. This article provides a brief review of approximate dynamic programming, without intending to be a complete tutorial. Instead, our goal is to provide a broader perspective of ADP and how it should be approached from the perspective of different problem classes. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 56: 239-249, 2009