Monotonic bounds in multistage mixed-integer stochastic programming

Multistage stochastic programs bring computational complexity which may increase exponentially with the size of the scenario tree in real case problems. For this reason approximation techniques which replace the problem by a simpler one and provide lower and upper bounds to the optimal value are very useful. In this paper we provide monotonic lower and upper bounds for the optimal objective value of a multistage stochastic program. These results also apply to stochastic multistage mixed integer linear programs. Chains of inequalities among the new quantities are provided in relation to the optimal objective value, the wait-and-see solution and the expected result of using the expected value solution. The computational complexity of the proposed lower and upper bounds is discussed and an algorithmic procedure to use them is provided. Numerical results on a real case transportation problem are presented.     

An approach to the valuation and decision of ERP investment projects based on real options

The risks and uncertainties inherent in most enterprise resources planning (ERP) investment projects are vast. Decision making in multistage ERP projects investment is also complex, due mainly to the uncertainties involved and the various managerial and/or physical constraints to be enforced. This paper tackles the problem using a real-option analysis framework, and applies multistage stochastic integer programming in formulating an analytical model whose solution will yield optimum or near-optimum investment decisions for ERP projects. Traditionally, such decision problems were tackled using lattice simulation or finite difference methods to compute the value of simple real options. However, these approaches are incapable of dealing with the more complex compound real options, and their use is thus limited to simple real-option analysis. Multistage stochastic integer programming is particularly suitable for sequential decision making under uncertainty, and is used in this paper and to find near-optimal strategies for complex decision problems. Compared with the traditional approaches, multistage stochastic integer programming is a much more powerful tool in evaluating such compound real options. This paper describes the proposed real-option analysis model and uses an example case study to demonstrate the effectiveness of the proposed approach.     

Bounds in Multistage Linear Stochastic Programming

Multistage stochastic programs, which involve sequences of decisions over time, are usually hard to solve in realistically sized problems. Providing bounds for optimal solution may help in evaluating whether it is worth the additional computations for the stochastic program vs. simplified approaches. In this paper we generalize measures from the two-stage case, based on different levels of available information, to the multistage stochastic programming problems. A set of theorems providing chains of inequalities among the new quantities are proved. Numerical results on a case study related to a simple transportation problem illustrate the described relationships.     

Solving Sequences of Refined Multistage Stochastic Linear Programs

Multistage stochastic programs with continuous underlying distributions involve the obstacle of high-dimensional integrals where the integrands' values again are given by solutions of stochastic programs. A common solution technique consists of discretizing the support of the original distributions leading to scenario trees and corresponding LPs which are – up to a certain size – easy to solve. In order to improve the accuracy of approximation, successive refinements of the support result in rapidly expanding scenario trees and associated LPs. Hence, the solvability of the multistage stochastic program is limited by the numerical solvability of sequences of such expanding LPs. This work describes an algorithmic technique for solving the large-scale LP of refinement ν based on the solutions at the previous ν?1 refinements. Numerical results are presented for practical problem statements within financial applications demonstrating significant speedup (depending on the size of the LP instances).     

Stability in multistage stochastic programming

Multistage stochastic programs are regarded as mathematical programs in a Banach spaceX of summable functions. Relying on a result for parametric programs in Banach spaces, the paper presents conditions under which linearly constrained convex multistage problems behave stably when the (input) data process is subjected to (small) perturbations. In particular, we show the persistence of optimal solutions, the local Lipschitz continuity of the optimal value and the upper semicontinuity of optimal sets with respect to the weak topology inX. The linear case with deterministic first-stage decisions is studied in more detail.This research has been supported by the Schwerpunktprogramm Anwendungsbezogene Optimierung und Steuerung of the Deutsche Forschungsgemeinschaft.     

Decision making in multiobjective optimization aided by the multicriteria tournament decision method

This paper proposes a new method for multicriteria analysis, named Multicriteria Tournament Decision (MTD). It provides the ranking of alternatives from best to worst, according to the preferences of a human decision-maker (DM). It has some positive aspects such as: it has a simple algorithm with intuitive appeal; it involves few input parameters (just the importance weight of each criterion).The helpfulness of MTD is demonstrated by using it to select the final solution of multiobjective optimization problems in an a posteriori decision making approach. Having at hand a discrete approximation of the Pareto front (provided by a multiobjective evolutionary search algorithm), the choice of the preferred Pareto-optimal solution is performed using MTD.A simple method, named Gain Analysis method (GAM), for verifying the existence of a better solution (a solution associated to higher marginal rates of return) than the one originally chosen by the DM, is also introduced here. The usefulness of MTD and GAM methods is confirmed by the suitable results shown in this paper.     

Nonlinear Stochastic Difference Equations Driven by Martingales

Abstract

We provide in this paper a systematic development of nonlinear stochastic difference equations driven by martingales (that depend on a spatial parameter); three such equations are considered. We begin with the existence and uniqueness of solutions and continue with the study of stochastic properties, such as the martingale and Markov properties, along with ? irreducibility and recurrence. We discuss in the final section the discrete-time flow and asymptotic flow properties of the solution process.     

A New Decomposition Technique in Solving Multistage Stochastic Linear Programs by Infeasible Interior Point Methods

Multistage stochastic linear programming (MSLP) is a powerful tool for making decisions under uncertainty. A deterministic equivalent problem of MSLP is a large-scale linear program with nonanticipativity constraints. Recently developed infeasible interior point methods are used to solve the resulting linear program. Technical problems arising from this approach include rank reduction and computation of search directions. The sparsity of the nonanticipativity constraints and the special structure of the problem are exploited by the interior point method. Preliminary numerical results are reported. The study shows that, by combining the infeasible interior point methods and specific decomposition techniques, it is possible to greatly improve the computability of multistage stochastic linear programs.     

Decision making in multiobjective optimization aided by the multicriteria tournament decision method

This paper proposes a new method for multicriteria analysis, named Multicriteria Tournament Decision (MTD). It provides the ranking of alternatives from best to worst, according to the preferences of a human decision-maker (DM). It has some positive aspects such as: it has a simple algorithm with intuitive appeal; it involves few input parameters (just the importance weight of each criterion).The helpfulness of MTD is demonstrated by using it to select the final solution of multiobjective optimization problems in an a posteriori decision making approach. Having at hand a discrete approximation of the Pareto front (provided by a multiobjective evolutionary search algorithm), the choice of the preferred Pareto-optimal solution is performed using MTD.A simple method, named Gain Analysis method (GAM), for verifying the existence of a better solution (a solution associated to higher marginal rates of return) than the one originally chosen by the DM, is also introduced here. The usefulness of MTD and GAM methods is confirmed by the suitable results shown in this paper.     

Convergence analysis of sample average approximation for a class of stochastic nonlinear complementarity problems: from two-stage to multistage

In this paper, we consider the sample average approximation (SAA) approach for a class of stochastic nonlinear complementarity problems (SNCPs) and study the corresponding convergence properties. We first investigate the convergence of the SAA counterparts of two-stage SNCPs when the first-stage problem is continuously differentiable and the second-stage problem is locally Lipschitz continuous. After that, we extend the convergence results to a class of multistage SNCPs whose decision variable of each stage is influenced only by the decision variables of adjacent stages. Finally, some preliminary numerical tests are presented to illustrate the convergence results.

     

Subregular recourse in nonlinear multistage stochastic optimization

We consider nonlinear multistage stochastic optimization problems in the spaces of integrable functions. We allow for nonlinear dynamics and general objective functionals, including dynamic risk measures. We study causal operators describing the dynamics of the system and derive the Clarke subdifferential for a penalty function involving such operators. Then we introduce the concept of subregular recourse in nonlinear multistage stochastic optimization and establish subregularity of the resulting systems in two formulations: with built-in nonanticipativity and with explicit nonanticipativity constraints. Finally, we derive optimality conditions for both formulations and study their relations.

     

Quasi-variational problems with non-self map on Banach spaces: Existence and applications

This paper focuses on the analysis of generalized quasi-variational inequality problems with non-self constraint map. To study such problems, in Aussel et al. (2016) the authors introduced the concept of the projected solution and proved its existence in finite-dimensional spaces. The main contribution of this paper is to prove the existence of a projected solution for generalized quasi-variational inequality problems with non-self constraint map on real Banach spaces. Then, following the multistage stochastic variational approach introduced in Rockafellar and Wets (2017), we introduce the concept of the projected solution in a multistage stochastic setting, and we prove the existence of such a solution. We apply this theoretical result in studying an electricity market with renewable power sources.     

应用模糊综合评判和灰色关联度分析评价水平井多级压裂效果

水平井多级压裂是低渗透和非常规油气田开发的有效技术手段,其压后产能评价问题已成为相关研究的热点和难点.水平井多级压裂压后产能的各项影响因素之间关系复杂,通过常规理论分析量化研究困难,压裂效果很难准确预测.本文提出应用模糊综合评判和灰色关联度分析相结合的方法,对多级压裂水平井的压裂效果进行评价.首先运用灰色理论分析影响压裂效果的各因素之间的灰色关联度,计算各影响因素在综合评判中的权重值,再结合模糊综合评判方法,对多级压裂井压后效果进行预测评价.通过目的区块24口多级压裂水平井压后效果评价与实际生产情况对比,表明模型计算准确率达到95.8%,评价结果准确可靠.     

A parallel implementation of the nested decomposition algorithm for multistage stochastic linear programs

Multistage stochastic linear programs can represent a variety of practical decision problems. Solving a multistage stochastic program can be viewed as solving a large tree of linear programs. A common approach for solving these problems is the nested decomposition algorithm, which moves up down the tree by solving nodes and passing information among nodes. The natural independence of subtrees suggests that much of the computational effort of the nested decomposition algorithm can run in parallel across small numbers of fast processors. This paper explores the advantages of such parallel implementations over serial implementations and compares alternative sequencing protocols for parallel processors. Computational experience on a large test set of practical problems with up to 1.5 million constraints and almost 5 million variables suggests that parallel implementations may indeed work well, but they require careful attention to processor load balancing. Supported in part by the National Science Foundation under Grants DDM-9215921 and SES-9211937.     

Aggregation and discretization in multistage stochastic programming

Multistage stochastic programs have applications in many areas and support policy makers in finding rational decisions that hedge against unforeseen negative events. In order to ensure computational tractability, continuous-state stochastic programs are usually discretized; and frequently, the curse of dimensionality dictates that decision stages must be aggregated. In this article we construct two discrete, stage-aggregated stochastic programs which provide upper and lower bounds on the optimal value of the original problem. The approximate problems involve finitely many decisions and constraints, thus principally allowing for numerical solution.      

A fuzzy MCDM approach for stock selection

A fuzzy MCDM approach is applied to the stock selection problem, where the proposed approach can deal with qualitative information in addition to quantitative information. A hierarchy of major–sub criteria is then established to reduce the dependence between criteria. The ratings of alternatives versus qualitative sub-criteria and the weights of major- and sub-criteria are assessed in linguistic terms represented by fuzzy numbers. Each sub-criterion is in a benefit, cost, or balanced nature. New standardization methods for fuzzy numbers in the cost and balanced nature are presented. The algorithms of membership functions of the final aggregation are completely developed instead of approximation. The final aggregations in fuzzy numbers are then defuzzified to crisp values in order to rank the performance of alternatives. Moreover, the ratio of market price to performance (PP) is suggested to filter the over/under-pricing of alternatives. A set of buying/selling strategies are recommended according to the performance and PP. An empirical example then demonstrates the processing of the proposed approach.     

Stability and Controllability in Non-autonomous Time-discrete Dynamical Systems

In this paper stability and attractivity in non-autonomous time-discrete dynamical systems is investigated with the aid of Lyapunov functions. The results are applied to the problem of stabilization of controlled systems by feedback controls. In the final section of the paper we give sufficient conditions for norm-bounded null-controllability of linear systems.     

Multi-attribute decision making: A simulation comparison of select methods

Several methods have been proposed for solving multi-attribute decision making problems (MADM). A major criticism of MADM is that different techniques may yield different results when applied to the same problem. The problem considered in this study consists of a decision matrix input of N criteria weights and ratings of L alternatives on each criterion. The comparative performance of some methods has been investigated in a few, mostly field, studies. In this simulation experiment we investigate the performance of eight methods: ELECTRE, TOPSIS, Multiplicative Exponential Weighting (MEW), Simple Additive Weighting (SAW), and four versions of AHP (original vs. geometric scale and right eigenvector vs. mean transformation solution). Simulation parameters are the number of alternatives, criteria and their distribution. The solutions are analyzed using twelve measures of similarity of performance. Similarities and differences in the behavior of these methods are investigated. Dissimilarities in weights produced by these methods become stronger in problems with few alternatives; however, the corresponding final rankings of the alternatives vary across methods more in problems with many alternatives. Although less significant, the distribution of criterion weights affects the methods differently. In general, all AHP versions behave similarly and closer to SAW than the other methods. ELECTRE is the least similar to SAW (except for closer matching the top-ranked alternative), followed by MEW. TOPSIS behaves closer to AHP and differently from ELECTRE and MEW, except for problems with few criteria. A similar rank-reversal experiment produced the following performance order of methods: SAW and MEW (best), followed by TOPSIS, AHPs and ELECTRE. It should be noted that the ELECTRE version used was adapted to the common MADM problem and therefore it did not take advantage of the method's capabilities in handling problems with ordinal or imprecise information.     

Numerical solution of laminar incompressible generalized Newtonian fluids flow

This paper deals with the numerical solution of laminar viscous incompressible flows for generalized Newtonian fluids in the branching channel. The generalized Newtonian fluids contain Newtonian fluids, shear thickening and shear thinning non-Newtonian fluids. The mathematical model is the generalized system of Navier-Stokes equations. The finite volume method combined with an artificial compressibility method is used for spatial discretization. For time discretization the explicit multistage Runge-Kutta numerical scheme is considered. Steady state solution is achieved for t → ∞ using steady boundary conditions and followed by steady residual behavior. For unsteady solution a dual-time stepping method is considered. Numerical results for flows in two dimensional and three dimensional branching channel are presented.     

An ε-sensitivity analysis in the primal–dual interior point method

This paper presents a method of sensitivity analysis on the cost coefficients and the right-hand sides for most variants of the primal–dual interior point method. We first define an ε-optimal solution to describe the characteristics of the final solution obtained by the primal–dual interior point method. Then an ε-sensitivity analysis is defined to determine the characteristic region where the final solution remains the ε-optimal solution as a cost coefficient or a right-hand side changes. To develop the method of ε-sensitivity analysis, we first derive the expressions for the final solution from data which are commonly maintained in most variants of the primal–dual interior point method. Then we extract the characteristic regions on the cost coefficients and the right-hand sides by manipulating the mathematical expressions for the final solution. Finally, we show that in the nondegenerate case, the characteristic regions obtained by ε-sensitivity analysis are convergent to those obtained by sensitivity analysis in the simplex algorithm.