Showing papers on "Stochastic programming published in 1999"
••
TL;DR: The purpose of this overview is to discuss main theoretical results, some applications, and solution methods for this interesting and important class of programming problems.
Abstract: Mathematical programming problems dealing with functions, each of which can be represented as a difference of two convex functions, are called DC programming problems. The purpose of this overview is to discuss main theoretical results, some applications, and solution methods for this interesting and important class of programming problems. Some modifications and new results on the optimality conditions and development of algorithms are also presented.
657 citations
•
01 Jan 1999TL;DR: 1. Preliminary concepts of one dimensional unconstrained minimization, unconstrained optimization, linear programming, and finite element based optimization are presented.
Abstract: In this revised and enhanced second edition of Optimization Concepts and Applications in Engineering, the already robust pedagogy has been enhanced with more detailed explanations, an increased number of solved examples and end-of-chapter problems. The source codes are now available free on multiple platforms. It is vitally important to meet or exceed previous quality and reliability standards while at the same time reducing resource consumption. This textbook addresses this critical imperative integrating theory, modeling, the development of numerical methods, and problem solving, thus preparing the student to apply optimization to real-world problems. This text covers a broad variety of optimization problems using: unconstrained, constrained, gradient, and non-gradient techniques; duality concepts; multiobjective optimization; linear, integer, geometric, and dynamic programming with applications; and finite element-based optimization. It is ideal for advanced undergraduate or graduate courses and for practising engineers in all engineering disciplines, as well as in applied mathematics.
576 citations
••
TL;DR: An algorithm for solving stochastic integer programming problems with recourse, based on a dual decomposition scheme and Lagrangian relaxation, which can be applied to multi-stage problems with mixed-integer variables in each time stage.
526 citations
••
12 Sep 1999TL;DR: Under the framework of probabilistic optimization, this work proposes to use a most probable point (MPP) based importance sampling method, a method rooted in the field of reliability analysis, for evaluating the feasibility robustness.
Abstract: In robust design, it is important not only to achieve robust design objectives but also to maintain the robustness of design feasibility under the effect of variations (or uncertainties). However, the evaluation of feasibility robustness is often a computationally intensive process. Simplified approaches in existing robust design applications may lead to either over-conservative or infeasible design solutions. In this paper, several feasibility-modeling techniques for robust optimization are examined. These methods are classified into two categories: methods that require probability and statistical analyses and methods that do not. Using illustrative examples, the effectiveness of each method is compared in terms of its efficiency and accuracy. Constructive recommendations are made to employ different techniques under different circumstances. Under the framework of probabilistic optimization, we propose to use a most probable point (MPP) based importance sampling method, a method rooted in the field of reliability analysis, for evaluating the feasibility robustness. The advantages of this approach are discussed. Though our discussions are centered on robust design, the principles presented are also applicable for general probabilistic optimization problems. The practical significance of this work also lies in the development of efficient feasibility evaluation methods that can support quality engineering practice, such as the Six Sigma approach that is being widely used in American industry.
395 citations
•
TL;DR: An empirical application to the problem of stock market predictability is provided, and the conditions under which such predictability could be exploited in the presence of transaction costs are discussed.
Abstract: This paper argues in favour of a closer link between decision and forecast evaluation problems. Although the idea of using decision theory for forecast evaluation appears early in the dynamic stochastic programming literature, and has continued to be used in meteorological forecasts, it is hardly mentioned in standard academic textbooks on economic forecasting. Some of the main issues involved are illustrated in the context of a two-state, two-action decision problem as well as in a more general setting. Relationships between statistical and economic methods of forecast evaluation are discussed and useful links between Kuipers score, used as a measure of forecast accuracy in the meteorology literature, and the market timing tests used in finance, are established. An empirical application to the problem of stock market predictability is also provided, and the conditions under which such predictability could be exploited in the presence of transaction costs are discussed.
350 citations
••
TL;DR: This paper delineates circumstances under which the rollout algorithms are guaranteed to perform better than the heuristics on which they are based, and shows computational results which suggest that the performance of the rollout policies is near-optimal, and is substantially better thanThe performance of their underlying heuristic.
Abstract: Stochastic scheduling problems are difficult stochastic control problems with combinatorial decision spaces. In this paper we focus on a class of stochastic scheduling problems, the quiz problem and its variations. We discuss the use of heuristics for their solution, and we propose rollout algorithms based on these heuristics which approximate the stochastic dynamic programming algorithm. We show how the rollout algorithms can be implemented efficiently, with considerable savings in computation over optimal algorithms. We delineate circumstances under which the rollout algorithms are guaranteed to perform better than the heuristics on which they are based. We also show computational results which suggest that the performance of the rollout policies is near-optimal, and is substantially better than the performance of their underlying heuristics.
298 citations
••
TL;DR: In this paper, the authors focus on robustness of model-predictive control with respect to satisfaction of process output constraints and propose a method of improving such robustness by formulating output constraints as chance constraints.
Abstract: This work focuses on robustness of model-predictive control with respect to satisfaction of process output constraints. A method of improving such robustness is presented. The method relies on formulating output constraints as chance constraints using the uncertainty description of the process model. The resulting on-line optimization problem is convex. The proposed approach is illustrated through a simulation case study on a high-purity distillation column. Suggestions for further improvements are made.
286 citations
••
TL;DR: Structural properties of and algorithms for stochastic integer programming models, mainly considering linear two‐stage models with mixed‐integer recourse (and their multi‐stage extensions) are surveyed.
Abstract: We survey structural properties of and algorithms for stochastic integer programmingmodels, mainly considering linear two‐stage models with mixed‐integer recourse (and theirmulti‐stage extensions).
221 citations
••
TL;DR: This work discusses a variety of LP-based models that can be used for planning under uncertainty, and presents models that range from simple recourse policies to more general two-stage and multistage SLP formulations.
Abstract: Linear programming is a fundamental planning tool. It is often difficult to precisely estimate or forecast certain critical data elements of the linear program. In such cases, it is necessary to address the impact of uncertainty during the planning process. We discuss a variety of LP-based models that can be used for planning under uncertainty. In all cases, we begin with a deterministic LP model and show how it can be adapted to include the impact of uncertainty. We present models that range from simple recourse policies to more general two-stage and multistage SLP formulations. We also include a discussion of probabilistic constraints. We illustrate the various models using examples taken from the literature. The examples involve models developed for airline yield management, telecommunications, flood control, and production planning.
210 citations
••
TL;DR: This paper presents and solves a single-period, multiproduct, downward substitution model that has one raw material as the production input and produces N different products as outputs and compares three different solution methods.
Abstract: in this paper, we present and solve a single-period, multiproduct, downward substitution model. Our model has one raw material as the production input and produces N different products as outputs. The demands and yields for the products are random. We determine the optimal production input and allocation of the N products to satisfy demands. The problem is modeled as a two-stage stochastic program, which we show can be decomposed into a parameterized network flow problem. We present and compare three different solution methods: a stochastic linear program, a decomposition resulting in a series of network flow subproblems, and a decomposition where the same network flow subproblems are solved by a new greedy algorithm.
209 citations
••
01 Dec 1999TL;DR: This tutorial is not meant to be an exhaustive literature search on simulation optimization techniques, but its emphasis is mostly on issues that are specific to simulation optimization.
Abstract: Simulation models can be used as the objective function and/or constraint functions in optimizing stochastic complex systems. This tutorial is not meant to be an exhaustive literature search on simulation optimization techniques. It does not concentrate on explaining well-known general optimization and mathematical programming techniques either. Its emphasis is mostly on issues that are specific to simulation optimization. Even though a lot of effort has been spent to provide a reasonable overview of the field, still there are methods and techniques that have not been covered and valuable works that may not have been mentioned.
••
TL;DR: Stochastic mathematical programs with equilibrium constraints (SMPEC), which generalize MPEC models by explicitly incorporating possible uncertainties in the problem data to obtain robust solutions to hierarchical problems, are introduced.
••
TL;DR: This paper describes a method for estimating the future value function by multivariate adaptive regression splines (MARS) fit over a discretization scheme based on orthogonal array (OA) experimental designs and shows that this method is accurately able to solve higher dimensional SDP problems than previously possible.
Abstract: In stochastic dynamic programming (SDP) with continuous state and decision variables, the future value function is computed at discrete points in the state spac e. Interpolation can be used to approximate the values of the future value function between these discrete points. However, for large dimensional problems the number of discrete points required to obtain a good approximation of the future value function can be prohibitively large. Statistical methods of experimental design and function estimation may be employed to overcome this "curse of dimensionality." In this paper, we describe a method for estimating the future value function by multivariate adaptive regression splines (MARS) fit over a discretization scheme based on orthogonal array (OA) experimental designs. Because orthogonal arrays only grow polynomially in the state-space dimension, our OA/MARS method is accurately able to solve higher dimensional SDP problems than previously possible. To our knowledge, the most efficient method published prior to this work employs tensor-product cubic splines to approximate the future value function (Johnson et al. 1993). The computational advantages of OA/MA RS are demonstrated in comparisons with the method using tensor-product cubic splines for applications of an inventory forecasting SDP with up to nine state variables computed on a small workstation. In particular, the storage of an adequate tensor-product cubic spline for six dimensions exceeds the memory of our workstation, and the run time for an accurate OA/MARS SDP solution would be at least an order of magnitude faster than using tensor-product cubic splines for higher than six dimensions.
••
TL;DR: A class of discrete optimization problems, where the objective function for a given configuration can be expressed as the expectation of a random variable, and an iterative algorithm called the stochastic comparison (SC) algorithm is developed.
Abstract: In this paper we study a class of discrete optimization problems, where the objective function for a given configuration can be expressed as the expectation of a random variable. In such problems, only samples of the random variables are available for the optimization process. An iterative algorithm called the stochastic comparison (SC) algorithm is developed. The convergence of the SC algorithm is established based on an examination of the quasi-stationary probabilities of a time-inhomogeneous Markov chain. We also present some numerical experiments.
••
TL;DR: In this paper, it is shown that linear programming problems with fuzzy coefficients in constraints can be reduced to a linear semi-infinite programming problem, and a cutting plane algorithm is introduced with a convergence proof.
Abstract: This paper presents a new method for solving linear programming problems with fuzzy coefficients in constraints. It is shown that such problems can be reduced to a linear semi-infinite programming problem. The relations between optimal solutions and extreme points of the linear semi-infinite program are established. A cutting plane algorithm is introduced with a convergence proof, and a numerical example is included to illustrate the solution procedure.
••
TL;DR: A new variant of the stochastic comparison method is proved that it is guaranteed to converge almost surely to the set of global optimal solutions and a result is presented that demonstrates that this method is likely to perform well in practice.
Abstract: We discuss the choice of the estimation of the optimal solution when random search methods are applied to solve discrete stochastic optimization problems. At the present time, such optimization methods usually estimate the optimal solution using either the feasible solution the method is currently exploring or the feasible solution visited most often so far by the method. We propose using all the observed objective function values generated as the random search method moves around the feasible region seeking an optimal solution to obtain increasingly more precise estimates of the objective function values at the different points in the feasible region. At any given time, the feasible solution that has the best estimated objective function value (largest one for maximization problems; the smallest one for minimization problems) is used as the estimate of the optimal solution. We discuss the advantages of using this approach for estimating the optimal solution and present numerical results showing that modifying an existing random search method to use tnhis approach for estimating the optimal soluation appears to yield improved performance. We also present sereval rate of convergence results for random search methods using our approach for estimating the optimal solution. One these random search methods is a new variant of the stochastic comparison method; in addition to specifying the rate of convergence of this method, we prove that it is guaranteed to converge almost surely to the set of global optimal solutions and present a result that demonstrates that this method is likely to perform well in practice.
••
TL;DR: In this article, a Minimax Regret formulation suitable for large-scale linear programming models was proposed for Greenhouse Gas abatement in the Province of Quebec, where the minimax regret strategy depends only on the extremal scenarios and not on intermediate ones, thus making the approach computationally efficient.
••
TL;DR: In this article, the authors proposed an algorithm for multistage stochastic linear programs with recourse where random quantities in different stages are independent. And they proved that the algorithm is convergent with probability one.
Abstract: We propose an algorithm for multistage stochastic linear programs with recourse where random quantities in different stages are independent. The algorithm approximates successively expected recourse functions by building up valid cutting planes to support these functions from below. In each iteration, for the expected recourse function in each stage, one cutting plane is generated using the dual extreme points of the next-stage problem that have been found so far. We prove that the algorithm is convergent with probability one.
••
TL;DR: The results show that the ING method finds optimal or close to optimal solutions for the problems presented, and has a wider range of potential application areas than conventional techniques in behavioural modelling.
Abstract: Even though individual-based models (IBMs) have become very popular in ecology during the last decade, there have been few attempts to implement behavioural aspects in IBMs. This is partly due to lack of appropriate techniques. Behavioural and life history aspects can be implemented in IBMs through adaptive models based on genetic algorithms and neural networks (individual-based-neural network-genetic algorithm, ING). To investigate the precision of the adaptation process, we present three cases where solutions can be found by optimisation. These cases include a state-dependent patch selection problem, a simple game between predators and prey, and a more complex vertical migration scenario for a planktivorous fish. In all cases, the optimal solution is calculated and compared with the solution achieved using ING. The results show that the ING method finds optimal or close to optimal solutions for the problems presented. In addition it has a wider range of potential application areas than conventional techniques in behavioural modelling. Especially the method is well suited for complex problems where other methods fail to provide answers.
••
TL;DR: An inexact cut algorithm is described, its convergence under easily verifiable assumptions is proved, and some computational results from applying the algorithm to a class of stochastic programming problems that arise in hydroelectric scheduling.
Abstract: Benders decomposition is a well-known technique for solving large linear programs with a special structure. In particular, it is a popular technique for solving multistage stochastic linear programming problems. Early termination in the subproblems generated during Benders decomposition (assuming dual feasibility) produces valid cuts that are inexact in the sense that they are not as constraining as cuts derived from an exact solution. We describe an inexact cut algorithm, prove its convergence under easily verifiable assumptions, and discuss a corresponding Dantzig--Wolfe decomposition algorithm. The paper is concluded with some computational results from applying the algorithm to a class of stochastic programming problems that arise in hydroelectric scheduling.
••
TL;DR: The concept of comparison interval is introduced and a methodology based on stochastic optimization to achieve global consistency and to accommodate the fuzzy nature of the comparison process is proposed.
••
TL;DR: In this paper, a single-item inventory model under supply uncertainty is presented, where the uncertainty in supply is modeled using a three-point probability mass function, and the objective is to minimize expected holding and backorder costs over a finite planning horizon under the supply constraints.
••
TL;DR: The authors apply stochastic dynamic programming to derive trading strategies that minimize the expected cost of executing a portfolio of securities over a fixed time period.
Abstract: The authors apply stochastic dynamic programming to derive trading strategies that minimize the expected cost of executing a portfolio of securities over a fixed time period. They test their strategies using real-world stock data.
••
01 Apr 1999TL;DR: An effective approach for job-shop scheduling considering uncertain arrival times, processing times, due dates, and part priorities is presented, and a dual cost is proved to be a lower bound to the optimal expected cost for the stochastic formulation considered.
Abstract: This paper presents an effective approach for job-shop scheduling considering uncertain arrival times, processing times, due dates, and part priorities. A separable problem formulation that balances modeling accuracy and solution method complexity is presented with the goal to minimize expected part tardiness and earliness cost. This optimization is subject to arrival time and operation precedence constraints, and machine capacity constraints. A solution methodology based on a combined Lagrangian relaxation and stochastic dynamic programming is developed to obtain dual solutions. A good dual solution is then selected by using "ordinal optimization", and the actual schedule is dynamically constructed based on the dual solution and the realization of random events. The computational complexity of the overall algorithm is only slightly higher than the one without considering uncertainties, and a dual cost is proved to be a lower bound to the optimal expected cost for the stochastic formulation considered.
••
TL;DR: An optimization-based methodology that combines Lagrangian relaxation, stochastic dynamic programming, and “ordinal optimization” is developed and results supported by simulation demonstrate that near optimal solutions are obtained, and uncertainties are effectively managed for problems of practical sizes.
••
TL;DR: Main results obtained so far by using the idea of stochastic orders in financial optimization are described, especially, the emphasis is placed on the demand and shift effect problems in portfolio selection.
Abstract: Stochastic orders and inequalities are very useful tools in various areas of economics and finance. The purpose of this paper is to describe main results obtained so far by using the idea of stochastic orders in financial optimization. Especially, the emphasis is placed on the demand and shift effect problems in portfolio selection. Some other examples, which are not related directly to optimization problems, are also given to demonstrate the wide spectrum of application areas of stochastic orders in finance.
••
TL;DR: In this article, necessary and sufficient conditions for metric regularity of (several joint) probabilistic constraints are derived using recent results from nonsmooth analysis, and a verifiable sufficient condition for quadratic growth of the objective function in a more specific convex stochastic program is indicated and applied to obtain a new result on quantitative stability of solution sets when the underlying probability distribution is subjected to perturbations.
Abstract: Introducing probabilistic constraints leads in general to nonconvex, nonsmooth or even disconti- nuous optimization models. In this paper, necessary and sufficient conditions for metric regularity of (several joint) probabilistic constraints are derived using recent results from nonsmooth analysis. The conditions apply to fairly general constraints and extend earlier work in this direction. Further, a verifiable sufficient condition for quadratic growth of the objective function in a more specific convex stochastic program is indicated and applied in order to obtain a new result on quantitative stability of solution sets when the underlying probability distribution is subjected to perturbations. This is used to derive bounds for the deviation of solution sets when the probability measure is replaced by empirical estimates.
••
TL;DR: In this article, a two-level stochastic optimization approach is proposed to target the performance of chemical reactors with the use of stochastically optimized configurations. But it is not restricted by the dimensionality or the size of the problem.
Abstract: A systematic methodology to target the performance of chemical reactors with the use of stochastic optimization is presented. The approach employs a two-level strategy where targets are followed by the proposition of reactor configurations that match or are near the desired performance. The targets can be used for synthesis and retrofit problems, as they can provide the incentives to modify the operation, and ideas in developing the reactor design. The application of stochastic optimization enables confidence in the optimization results, can afford particularly nonlinear reactor models, and is not restricted by the dimensionality or the size of the problem.
••
TL;DR: A new approximate proximal point method for minimizing the sum of two convex functions is introduced, which replaces the original problem by a sequence of regularized subproblems in which the functions are alternately represented by linear models.
Abstract: A new approximate proximal point method for minimizing the sum of two convex functions is introduced. It replaces the original problem by a sequence of regularized subproblems in which the functions are alternately represented by linear models. The method updates the linear models and the prox center, as well as the prox coefficient. It is monotone in terms of the objective values and converges to a solution of the problem, if any. A dual version of the method is derived and analyzed. Applications of the methods to multistage stochastic programming problems are discussed and preliminary numerical experience is presented.