带补偿的随机二次规划的近似Lagrange-Newton方法

In this paper, two-stage stochastic quadratic programming problems with equality constraints are considered. By Monte Carlo simulation-based approximations of the objective function and its first (second)derivative,an inexact Lagrange-Newton type method is proposed.It is showed that this method is globally convergent with probability one. In particular, the convergence is local superlinear under an integral approximation error bound condition.Moreover, this method can be easily extended to solve stochastic quadratic programming problems with inequality constraints.     

An asynchronous parallel newton method

A parallel Newton method is described for the minimization of a twice continuously differentiable uniformly convex functionF(x). The algorithm generates a sequence {x _j } which converges superlinearly to the global minimizer ofF(x).     

Statistical approximations for recourse constrained stochastic programs

The two stage stochastic program with recourse is known to have numerous applications in financial planning, energy modeling, telecommunications systems etc. Notwithstanding its applicability, the two stage stochastic program is limited in its ability to incorporate a decision maker's attitudes towards risk. In this paper we present an extension via the inclusion of a recourse constraint. This results in a convex integrated chance constraint (ICC), which inherits the convexity properties of two stage programs. However, it also inherits some of the difficulties associated with the evaluation of recourse functions. This motivates our study of conditions that may be applicable to algorithms using statistical approximations of such ICC. We present a set of sufficient conditions that these approximations may satisfy in order to assure convergence. Our conditions are satisfied by a wide range of statistical approximations, and we demonstrate that these approximations can be generated within standard algorithmic procedures.This work was supported in part by Grant No. NSF-DDM-9114352 from the National Science Foundation.     

一类带有MaxEMin评判的补偿型随机规划算法

补偿型随机规划一般假定随机变量的概率分布具有完备信息, 但实际情况往往只能获得部分信息. 针对离散概率具有一类线性部分信息条件而建立了带有MaxEMin评判的两阶段随机规划模型, 借助二次规划和对偶分解方法得到了可行性切割和最优切割, 给出了基于L-型的求解算法, 并证明了算法的收敛性. 通过数值实验表明了算法的有效性.     

Adaptive multicut aggregation for two-stage stochastic linear programs with recourse

Outer linearization methods for two-stage stochastic linear programs with recourse, such as the L-shaped algorithm, generally apply a single optimality cut on the nonlinear objective at each major iteration, while the multicut version of the algorithm allows for several cuts to be placed at once. In general, the L-shaped algorithm tends to have more major iterations than the multicut algorithm. However, the trade-offs in terms of computational time are problem dependent. This paper investigates the computational trade-offs of adjusting the level of optimality cut aggregation from single cut to pure multicut. Specifically, an adaptive multicut algorithm that dynamically adjusts the aggregation level of the optimality cuts in the master program, is presented and tested on standard large-scale instances from the literature. Computational results reveal that a cut aggregation level that is between the single cut and the multicut can result in substantial computational savings over the single cut method.     

Statistical verification of optimality conditions for stochastic programs with recourse

Statistically motivated algorithms for the solution of stochastic programming problems typically suffer from their inability to recognize optimality of a given solution algorithmically. Thus, the quality of solutions provided by such methods is difficult to ascertain. In this paper, we develop methods for verification of optimality conditions within the framework of Stochastic Decomposition (SD) algorithms for two stage linear programs with recourse. Consistent with the stochastic nature of an SD algorithm, we provide termination criteria that are based on statistical verification of traditional (deterministic) optimality conditions. We propose the use of bootstrap methods to confirm the satisfaction of generalized Kuhn-Tucker conditions and conditions based on Lagrange duality. These methods are illustrated in the context of a power generation planning model, and the results are encouraging.This work was supported in part by Grant No. AFOSR-88-0076 from the Air Force Office of Scientific Research and Grant No. DDM-89-10046 from the National Science Foundation.     

L-shaped decomposition of two-stage stochastic programs with integer recourse

We consider two-stage stochastic programming problems with integer recourse. The L-shaped method of stochastic linear programming is generalized to these problems by using generalized Benders decomposition. Nonlinear feasibility and optimality cuts are determined via general duality theory and can be generated when the second stage problem is solved by standard techniques. Finite convergence of the method is established when Gomory’s fractional cutting plane algorithm or a branch-and-bound algorithm is applied.     

Sublinear upper bounds for stochastic programs with recourse

Separable sublinear functions are used to provide upper bounds on the recourse function of a stochastic program. The resulting problem's objective involves the inf-convolution of convex functions. A dual of this problem is formulated to obtain an implementable procedure to calculate the bound. Function evaluations for the resulting convex program only require a small number of single integrations in contrast with previous upper bounds that require a number of function evaluations that grows exponentially in the number of random variables. The sublinear bound can often be used when other suggested upper bounds are intractable. Computational results indicate that the sublinear approximation provides good, efficient bounds on the stochastic program objective value.This research has been partially supported by the National Science Foundation. The first author's work was also supported in part by Office of Naval Research Grant N00014-86-K-0628 and by the National Research Council under a Research Associateship at the Naval Postgraduate School, Monterey, California.     

An algorithm for stochastic programs with first-order dominance constraints induced by linear recourse

We derive a cutting plane decomposition method for stochastic programs with first-order dominance constraints induced by linear recourse models with continuous variables in the second stage.     

一种求解合作博弈最公平核心的非精确平行分裂算法

针对合作博弈核心和Shapley值的特点, 将最公平核心问题转化为带有两个变 量的可分离凸优化问题, 引入结构变分不等式的算子分裂方法框架, 提出了求解最公平核心的一种非精确平行分裂算法. 而且, 该算法充分利用了所求解问题的可行域的简单闭凸性, 子问题的非精确求解是容易的. 最后, 简单算例的数值实验表明了算法的收敛性和有效性.     

A descent method with inexact linear search for nonsmooth equilibrium problems

A descent method with respect to the gap function is formulated and justified for the nonsmooth equilibrium problem. It uses the procedure of inexact linear search of the Armijo type. The proposed method converges under the same assumptions as the methods with exact linear search.     

Distribution sensitivity analysis for stochastic programs with complete recourse

In this paper we study the stability of solutions to stochastic programming problems with complete recourse and show the Lipschitz continuity of optimal solutions as well as the associated Lagrange multipliers with respect to the parameters of the distribution function.     

An inexact Newton method for nonlinear two-point boundary-value problems

The method of quasilinearization for nonlinear two-point boundary-value problems is an application of Newton's method to a nonlinear differential operator equation. Since the linear boundary-value problem to be solved at each iteration must be discretized, it is natural to consider quasilinearization in the framework of an inexact Newton method. More importantly, each linear problem is only a local model of the nonlinear problem, and so it is inefficient to try to solve the linear problems to full accuracy. Conditions on size of the relative residual of the linear differential equation can then be specified to guarantee rapid local convergence to the solution of the nonlinear continuous problem. If initial-value techniques are used to solve the linear boundary-value problems, then an integration step selection scheme is proposed so that the residual criteria are satisfied by the approximate solutions. Numerical results are presented that demonstrate substantial computational savings by this type of economizing on the intermediate problems.This work was supported in part by DOE Contract DE-AS05-82-ER13016 and NSF Grant RII-89-17691 and was part of the author's doctoral thesis at Rice University. It is a pleasure to thank the author's thesis advisors, Professor R. A. Tapia and Professor J. E. Dennis, Jr.     

On augmented Lagrangian decomposition methods for multistage stochastic programs

A general decomposition framework for large convex optimization problems based on augmented Lagrangians is described. The approach is then applied to multistage stochastic programming problems in two different ways: by decomposing the problem into scenarios and by decomposing it into nodes corresponding to stages. Theoretical convergence properties of the two approaches are derived and a computational illustration is presented.     

Solving stochastic programs with integer recourse by enumeration: A framework using Gröbner basis

In this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right-hand side vector, using Gröbner basis methods from computational algebra to solve the numerous second-stage integer programs. Using structural properties of the expected integer recourse function, we prove that under mild conditions an optimal solution is contained in a finite set. Furthermore, we present a basic scheme to enumerate this set and suggest improvements to reduce the number of function evaluations needed.     

A tighter variant of Jensen’s lower bound for stochastic programs and separable approximations to recourse functions

In this paper, we propose a new method to compute lower bounds on the optimal objective value of a stochastic program and show how this method can be used to construct separable approximations to the recourse functions. We show that our method yields tighter lower bounds than Jensen’s lower bound and it requires a reasonable amount of computational effort even for large problems. The fundamental idea behind our method is to relax certain constraints by associating dual multipliers with them. This yields a smaller stochastic program that is easier to solve. We particularly focus on the special case where we relax all but one of the constraints. In this case, the recourse functions of the smaller stochastic program are one dimensional functions. We use these one dimensional recourse functions to construct separable approximations to the original recourse functions. Computational experiments indicate that our lower bounds can significantly improve Jensen’s lower bound and our recourse function approximations can provide good solutions.     

可分离凸优化问题的非精确平行分裂算法

针对一类可分离凸优化问题提出了一种非精确平行分裂算法.该算法充分利用了所求解问题的可分离结构,并对子问题进行非精确求解.在适当的条件下,证明了所提出的非精确平行分裂算法的全局收敛性,初步的数值实验说明了算法有效性.     

Continuity of parametric mixed-integer quadratic programs and its application to stability analysis of two-stage quadratic stochastic programs with mixed-integer recourse

For mixed-integer quadratic program where all coefficients in the objective function and the right-hand sides of constraints vary simultaneously, we show locally Lipschitz continuity of its optimal value function, and derive the corresponding global estimation; furthermore, we also obtain quantitative estimation about the change of its optimal solutions. Applying these results to two-stage quadratic stochastic program with mixed-integer recourse, we establish quantitative stability of the optimal value function and the optimal solution set with respect to the Fortet-Mourier probability metric, when the underlying probability distribution is perturbed. The obtained results generalize available results on continuity properties of mixed-integer quadratic programs and extend current results on quantitative stability of two-stage quadratic stochastic programs with mixed-integer recourse.     

On structure and stability in stochastic programs with random technology matrix and complete integer recourse

For two-stage stochastic programs with integrality constraints in the second stage, we study continuity properties of the expected recourse as a function both of the first-stage policy and the integrating probability measure.Sufficient conditions for lower semicontinuity, continuity and Lipschitz continuity with respect to the first-stage policy are presented. Furthermore, joint continuity in the policy and the probability measure is established. This leads to conclusions on the stability of optimal values and optimal solutions to the two-stage stochastic program when subjecting the underlying probability measure to perturbations.This research is supported by the Schwerpunktprogramm Anwendungsbezogene Optimierung und Steuerung of the Deutsche Forschungsgemeinschaft.The main part of the paper was written while the author was an assistant at the Department of Mathematics at Humboldt University Berlin.     

Approximate nonlinear programming algorithms for solving stochastic programs with recourse

This paper summarizes the main results on approximate nonlinear programming algorithms investigated by the author. These algorithms are obtained by combining approximation and nonlinear programming algorithms. They are designed for programs in which the evaluation of the objective functions is very difficult so that only their approximate values can be obtained. Therefore, these algorithms are particularly suitable for stochastic programming problems with recourse.Project supported by the National Natural Science Foundation of China.