Two-Stage Stochastic Variational Inequality Arising from Stochastic Programming

We consider a two-stage stochastic variational inequality arising from a general convex two-stage stochastic programming problem, where the random variables have continuous distributions. The equivalence between the two problems is shown under some moderate conditions, and the monotonicity of the two-stage stochastic variational inequality is discussed under additional conditions. We provide a discretization scheme with convergence results and employ the progressive hedging method with double parameterization to solve the discretized stochastic variational inequality. As an application, we show how the water resources management problem under uncertainty can be transformed from a two-stage stochastic programming problem to a two-stage stochastic variational inequality, and how to solve it, using the discretization scheme and the progressive hedging method with double parameterization.

     

Preparation of Papers

We motivate the study of a vector variational inequality by a practical flow equilibrium problem on a network, namely a generalization of the well-known Wardrop equilibrium principle. Both weak and strong forms of the vector variational inequality are discussed and their relationships to a vector optimization problem are established under various convexity assumptions.     

Supply chain networks with global outsourcing and quick-response production under demand and cost uncertainty

This paper develops a modeling and computational framework for supply chain networks with global outsourcing and quick-response production under demand and cost uncertainty. Our model considers multiple off-shore suppliers, multiple manufacturers, and multiple demand markets. Using variational inequality theory, we formulate the governing equilibrium conditions of the competing decision-makers (the manufacturers) who are faced with two-stage stochastic programming problems but who also have to cooperate with the other decision-makers (the off-shore suppliers). Our theoretical and analytical results shed light on the value of outsourcing from novel real option perspectives. Moreover, our simulation studies reveal important managerial insights regarding how demand and cost uncertainty affects the profits, the risks, as well as the global outsourcing and quick-production decisions of supply chain firms under competition.     

Two-stage non-cooperative games with risk-averse players

This paper formally introduces and studies a non-cooperative multi-agent game under uncertainty. The well-known Nash equilibrium is employed as the solution concept of the game. While there are several formulations of a stochastic Nash equilibrium problem, we focus mainly on a two-stage setting of the game wherein each agent is risk-averse and solves a rival-parameterized stochastic program with quadratic recourse. In such a game, each agent takes deterministic actions in the first stage and recourse decisions in the second stage after the uncertainty is realized. Each agent’s overall objective consists of a deterministic first-stage component plus a second-stage mean-risk component defined by a coherent risk measure describing the agent’s risk aversion. We direct our analysis towards a broad class of quantile-based risk measures and linear-quadratic recourse functions. For this class of non-cooperative games under uncertainty, the agents’ objective functions can be shown to be convex in their own decision variables, provided that the deterministic component of these functions have the same convexity property. Nevertheless, due to the non-differentiability of the recourse functions, the agents’ objective functions are at best directionally differentiable. Such non-differentiability creates multiple challenges for the analysis and solution of the game, two principal ones being: (1) a stochastic multi-valued variational inequality is needed to characterize a Nash equilibrium, provided that the players’ optimization problems are convex; (2) one needs to be careful in the design of algorithms that require differentiability of the objectives. Moreover, the resulting (multi-valued) variational formulation cannot be expected to be of the monotone type in general. The main contributions of this paper are as follows: (a) Prior to addressing the main problem of the paper, we summarize several approaches that have existed in the literature to deal with uncertainty in a non-cooperative game. (b) We introduce a unified formulation of the two-stage SNEP with risk-averse players and convex quadratic recourse functions and highlight the technical challenges in dealing with this game. (c) To handle the lack of smoothness, we propose smoothing schemes and regularization that lead to differentiable approximations. (d) To deal with non-monotonicity, we impose a generalized diagonal dominance condition on the players’ smoothed objective functions that facilitates the application and ensures the convergence of an iterative best-response scheme. (e) To handle the expectation operator, we rely on known methods in stochastic programming that include sampling and approximation. (f) We provide convergence results for various versions of the best-response scheme, particularly for the case of private recourse functions. Overall, this paper lays the foundation for future research into the class of SNEPs that provides a constructive paradigm for modeling and solving competitive decision making problems with risk-averse players facing uncertainty; this paradigm is very much at an infancy stage of research and requires extensive treatment in order to meet its broad applications in many engineering and economics domains.     

优化和均衡的等价性

通过向量优化问题, 向量变分不等式问题以及向量变分原理来分析优化问题及均衡问题的一致性.从而显然, 可以用统一的观点来处理数值优化、向量优化以及博弈论等问题.进而为非线性分析提供了一个新的发展空间.     

交通网络下的多厂商两阶段随机非合作博弈问题——基于随机变分不等式

研究集生产、运输和销售为一体的多个制造商在随机市场环境下的两阶段随机非合作博弈问题.首先,建立了该两阶段随机非合作博弈问题的模型,然后将其转化为两阶段随机变分不等式(Stochastic Variational Inequality,简称SVI).在温和的假设条件下,证明了该问题存在均衡解,并通过Progressive Hedging Method(简称PHM)进行求解.最后,通过改变模型中随机变量的分布和成本参数,分析与研究厂商的市场行为.     

On the information-based complexity of stochastic programming

Existing complexity results in stochastic linear programming using the Turing model depend only on problem dimensionality. We apply techniques from the information-based complexity literature to show that the smoothness of the recourse function is just as important. We derive approximation error bounds for the recourse function of two-stage stochastic linear programs and show that their worst case is exponential and depends on the solution tolerance, the dimensionality of the uncertain parameters and the smoothness of the recourse function.     

消费者分布不均匀条件下的具有网络外部性的Hotelling模型

将交通流量分配技术和无穷维变分方法相结合,采用无穷维变分不等式刻画网络产品水平差异的模型,研究了消费者分布不均匀时,具有网络外部性特征的企业进行价格竞争的两阶段完全信息动态博弈问题.得到了厂商价格竞争均衡的存在性条件,分析了产品的网络外部性特征及单位运输成本对两个厂商竞争均衡情况的影响.     

Solving two-stage stochastic programming problems with level decomposition

We propose a new variant of the two-stage recourse model. It can be used e.g., in managing resources in whose supply random interruptions may occur. Oil and natural gas are examples for such resources. Constraints in the resulting stochastic programming problems can be regarded as generalizations of integrated chance constraints. For the solution of such problems, we propose a new decomposition method that integrates a bundle-type convex programming method with the classic distribution approximation schemes. Feasibility and optimality issues are taken into consideration simultaneously, since we use a convex programming method suited for constrained optimization. This approach can also be applied to traditional two-stage problems whose recourse functions can be extended to the whole space in a computationally efficient way. Network recourse problems are an example for such problems. We report encouraging test results with the new method.      

变分不等式与互补问题、双层规划与平衡约束数学规划问题的若干进展

考虑有限维变分不等式与互补问题、双层规划以及均衡约束的数学规划问题. 在简单介绍这些问题之后,重点介绍近年来这些领域中发展迅速的几个研究方向,包括对称锥互补问题的理论与算法、变分不等式的投影收缩算法、随机变分不等式与随机互补问题的模型与方法、双层规划以及均衡约束数学规划问题的新方法. 最后提出几个进一步研究的方向.     

Stochastic utility-efficient programming of organic dairy farms

Opportunities to make sequential decisions and adjust activities as a season progresses and more information becomes available characterise the farm management process. In this paper, we present a discrete stochastic two-stage utility-efficient programming model of organic dairy farms, which includes risk aversion in the decision maker’s objective function as well as both embedded risk (stochastic programming with recourse) and non-embedded risk (stochastic programming without recourse). Historical farm accountancy data and subjective judgements were combined to assess the nature of the uncertainty that affects the possible consequences of the decisions. The programming model was used within a stochastic dominance framework to examine optimal strategies in organic dairy systems in Norway.     

Stochastic variational inequalities: single-stage to multistage

Variational inequality modeling, analysis and computations are important for many applications, but much of the subject has been developed in a deterministic setting with no uncertainty in a problem’s data. In recent years research has proceeded on a track to incorporate stochasticity in one way or another. However, the main focus has been on rather limited ideas of what a stochastic variational inequality might be. Because variational inequalities are especially tuned to capturing conditions for optimality and equilibrium, stochastic variational inequalities ought to provide such service for problems of optimization and equilibrium in a stochastic setting. Therefore they ought to be able to deal with multistage decision processes involving actions that respond to increasing levels of information. Critical for that, as discovered in stochastic programming, is introducing nonanticipativity as an explicit constraint on responses along with an associated “multiplier” element which captures the “price of information” and provides a means of decomposition as a tool in algorithmic developments. That idea is extended here to a framework which supports multistage optimization and equilibrium models while also clarifying the single-stage picture.     

Re-solving stochastic programming models for airline revenue management

We study some mathematical programming formulations for the origin-destination model in airline revenue management. In particular, we focus on the traditional probabilistic model proposed in the literature. The approach we study consists of solving a sequence of two-stage stochastic programs with simple recourse, which can be viewed as an approximation to a multi-stage stochastic programming formulation to the seat allocation problem. Our theoretical results show that the proposed approximation is robust, in the sense that solving more successive two-stage programs can never worsen the expected revenue obtained with the corresponding allocation policy. Although intuitive, such a property is known not to hold for the traditional deterministic linear programming model found in the literature. We also show that this property does not hold for some bid-price policies. In addition, we propose a heuristic method to choose the re-solving points, rather than re-solving at equally-spaced times as customary. Numerical results are presented to illustrate the effectiveness of the proposed approach.     

Stochastic Programming Models for Decreasing Risk Aversion

For a stochastic programming problem with simple recourse, we show how to formulate and analyze a model that encodes the common risk attitude of decreasing risk aversion. We discuss a class of linear fractional utility functions that represent this risk attitude and show that for such a utility function the resulting nonlinear model can be reduced to a linear programming model. The linear model, moreover, has only a slight percentage increase in the number of constraints as compared with the usual linear model representing risk neutrality.     

Dynamical systems and variational inequalities

The variational inequality problem has been utilized to formulate and study a plethora of competitive equilibrium problems in different disciplines, ranging from oligopolistic market equilibrium problems to traffic network equilibrium problems. In this paper we consider for a given variational inequality a naturally related ordinary differential equation. The ordinary differential equations that arise are nonstandard because of discontinuities that appear in the dynamics. These discontinuities are due to the constraints associated with the feasible region of the variational inequality problem. The goals of the paper are two-fold. The first goal is to demonstrate that although non-standard, many of the important quantitative and qualitative properties of ordinary differential equations that hold under the standard conditions, such as Lipschitz continuity type conditions, apply here as well. This is important from the point of view of modeling, since it suggests (at least under some appropriate conditions) that these ordinary differential equations may serve as dynamical models. The second goal is to prove convergence for a class of numerical schemes designed to approximate solutions to a given variational inequality. This is done by exploiting the equivalence between the stationary points of the associated ordinary differential equation and the solutions of the variational inequality problem. It can be expected that the techniques described in this paper will be useful for more elaborate dynamical models, such as stochastic models, and that the connection between such dynamical models and the solutions to the variational inequalities will provide a deeper understanding of equilibrium problems.     

A model of distributionally robust two-stage stochastic convex programming with linear recourse

We consider distributionally robust two-stage stochastic convex programming problems, in which the recourse problem is linear. Other than analyzing these new models case by case for different ambiguity sets, we adopt a unified form of ambiguity sets proposed by Wiesemann, Kuhn and Sim, and extend their analysis from a single stochastic constraint to the two-stage stochastic programming setting. It is shown that under a standard set of regularity conditions, this class of problems can be converted to a conic optimization problem. Numerical results are presented to show the efficiency of the distributionally robust approach.     

Applying the minimax criterion in stochastic recourse programs

We consider an optimization problem in which some uncertain parameters are replaced by random variables. The minimax approach to stochastic programming concerns the problem of minimizing the worst expected value of the objective function with respect to the set of probability measures that are consistent with the available information on the random data. Only very few practicable solution procedures have been proposed for this problem and the existing ones rely on simplifying assumptions. In this paper, we establish a number of stability results for the minimax stochastic program, justifying in particular the approach of restricting attention to probability measures with support in some known finite set. Following this approach, we elaborate solution procedures for the minimax problem in the setting of two-stage stochastic recourse models, considering the linear recourse case as well as the integer recourse case. Since the solution procedures are modifications of well-known algorithms, their efficacy is immediate from the computational testing of these procedures and we do not report results of any computational experiments.     

Logarithmic barrier decomposition-based interior point methods for stochastic symmetric programming

We introduce and study two-stage stochastic symmetric programs with recourse to handle uncertainty in data defining (deterministic) symmetric programs in which a linear function is minimized over the intersection of an affine set and a symmetric cone. We present a Benders’ decomposition-based interior point algorithm for solving these problems and prove its polynomial complexity. Our convergence analysis proved by showing that the log barrier associated with the recourse function of stochastic symmetric programs behaves a strongly self-concordant barrier and forms a self-concordant family on the first stage solutions. Since our analysis applies to all symmetric cones, this algorithm extends Zhao’s results [G. Zhao, A log barrier method with Benders’ decomposition for solving two-stage stochastic linear programs, Math. Program. Ser. A 90 (2001) 507–536] for two-stage stochastic linear programs, and Mehrotra and Özevin’s results [S. Mehrotra, M.G. Özevin, Decomposition-based interior point methods for two-stage stochastic semidefinite programming, SIAM J. Optim. 18 (1) (2007) 206–222] for two-stage stochastic semidefinite programs.     

Decision-dependent probabilities in stochastic programs with recourse

Stochastic programming with recourse usually assumes uncertainty to be exogenous. Our work presents modelling and application of decision-dependent uncertainty in mathematical programming including a taxonomy of stochastic programming recourse models with decision-dependent uncertainty. The work includes several ways of incorporating direct or indirect manipulation of underlying probability distributions through decision variables in two-stage stochastic programming problems. Two-stage models are formulated where prior probabilities are distorted through an affine transformation or combined using a convex combination of several probability distributions. Additionally, we present models where the parameters of the probability distribution are first-stage decision variables. The probability distributions are either incorporated in the model using the exact expression or by using a rational approximation. Test instances for each formulation are solved with a commercial solver, BARON, using selective branching.     

随机交通均衡配流模型及其等价的变分不等式问题

本文讨论了交通网络系统的随机用户均衡原理的数学表述问题.在路段出行成本是流量的单调函数的较弱条件下,对具有固定需求和弹性需求的模式,首次证明了随机均衡配流模型可表示为一个变分不等式问题,同时也说明了该变分不等式问题与相应的互补问题以及一个凸规划问题之间的等价关系.