Stochastic Complementarity Problems Research Articles

Confidence Intervals for Piecewise Normal Distributions and Stochastic Variational Inequalities

In this paper, we first show how to obtain easy-to-compute confidence intervals for the center of a piecewise normal distribution given a sample from this distribution (assuming that the center belongs to a known affine set parallel to the common lineality space of all cones defining the piecewise normal distribution) by using certain skewed projectors on that space. We then extend this method to an asymptotic setting. Next, we apply this method to compute confidence intervals for the true solution of a stochastic variational inequality given a solution to a sample average approximation (SAA) problem for the general situation in which the asymptotic distribution of SAA solutions is piecewise normal. For stochastic complementarity problems, we obtain asymptotic normality of certain estimators of the true solution when the asymptotic distribution of the SAA solutions is piecewise normal. Funding: The research reported in this paper was supported the National Science Foundation [Grant DMS-1814894].

Multi-Stage Modeling With Recourse Decisions for Solving Stochastic Complementarity Problems With an Application in Energy

Multi-Stage Modeling With Recourse Decisions for Solving Stochastic Complementarity Problems With an Application in Energy

Convergence of discrete approximation for differential linear stochastic complementarity systems

In this paper, we investigate a class of differential linear stochastic complementarity system consisting of an ordinary differential equation and a stochastic complementarity problem. The existence of solutions for such system is obtained under two cases of the coefficient matrix of the linear stochastic complementarity problem: P-matrix and positive semi-definite matrix. As for the first case, the sample average approximate method and time-stepping method are adopted to get the numerical solutions. Furthermore, a regularization approximation is introduced to the second case to ensure the uniqueness of solutions. The corresponding convergence analysis is conducted, and numerical examples are presented to illustrate the convergence results we derived. Finally, we provide numerical results which come from applications involving dynamic traffic flow problems to support our theorems.

The subdifferential of measurable composite max integrands and smoothing approximation

The subdifferential calculus for the expectation of nonsmooth random integrands involves many fundamental and challenging problems in stochastic optimization. It is known that for Clarke regular integrands, the Clarke subdifferential of the expectation equals the expectation of their Clarke subdifferential. In particular, this holds for convex integrands. However, little is known about the calculation of Clarke subgradients for the expectation of non-regular integrands. The focus of this contribution is to approximate Clarke subgradients for the expectation of random integrands by smoothing methods applied to the integrand. A framework for how to proceed along this path is developed and then applied to a class of measurable composite max integrands. This class contains non-regular integrands from stochastic complementarity problems as well as stochastic optimization problems arising in statistical learning.

Stochastic structured tensors to stochastic complementarity problems

This paper is concerned with the stochastic structured tensors to stochastic complementarity problems. The definitions and properties of stochastic structured tensors, such as the stochastic strong P-tensors, stochastic P-tensors, stochastic $$P_{0}$$-tensors, stochastic strictly semi-positive tensors and stochastic S-tensors are given. It is shown that the expected residual minimization formulation (ERM) of the stochastic structured tensor complementarity problem has a nonempty and bounded solution set. Interestingly, we partially answer the open questions proposed by Che et al. (Optim Lett 13:261–279, 2019). We also consider the expected value method of stochastic structured tensor complementarity problem with finitely many elements probability space. Finally, based on the expected residual minimization formulation (ERM) of the stochastic structured tensor complementarity problem, a projected gradient method is proposed for solving the stochastic structured tensor complementarity problem and the related numerical results are also given to show the efficiency of the proposed method.

Infeasible interior-point algorithms based on sampling average approximations for a class of stochastic complementarity problems and their applications

This paper considers a class of stochastic complementarity problems (SCP). Different from the classical complementarity problems, the SCP contains a mathematical expectation, which may not be evaluated in an explicit form in general. We combine an interior-point algorithm for deterministic cases with the well-known sample average approximation (SAA) techniques to present an SAA-based infeasible interior-point algorithm for the SCP. We investigate the convergence properties and computational complexity of the proposed algorithm under mild assumptions. Then, we extend these results to a class of mixed SCPs. Furthermore, we apply the proposed algorithms to solve a stochastic natural gas transmission problem and a stochastic oligopoly model. Preliminary numerical experiments indicate that the proposed approach is competitive with some existing methods. • We study a class of stochastic complementarity problems. • We present an infeasible interior-point algorithm for the considered problem. • We investigate convergence properties and computational complexity of the algorithm. • We apply the proposed approach to natural gas transmission and oligopoly problems.

Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging

The concept of a stochastic variational inequality has recently been articulated in a new way that is able to cover, in particular, the optimality conditions for a multistage stochastic programming problem. One of the long-standing methods for solving such an optimization problem under convexity is the progressive hedging algorithm. That approach is demonstrated here to be applicable also to solving multistage stochastic variational inequality problems under monotonicity, thus increasing the range of applications for progressive hedging. Stochastic complementarity problems as a special case are explored numerically in a linear two-stage formulation.

A Kind of Stochastic Eigenvalue Complementarity Problems

With the development of computer science, computational electromagnetics have also been widely used. Electromagnetic phenomena are closely related to eigenvalue problems. On the other hand, in order to solve the uncertainty of input data, the stochastic eigenvalue complementarity problem, which is a general formulation for the eigenvalue complementarity problem, has aroused interest in research. So, in this paper, we propose a new kind of stochastic eigenvalue complementarity problem. We reformulate the given stochastic eigenvalue complementarity problem as a system of nonsmooth equations with nonnegative constraints. Then, a projected smoothing Newton method is presented to solve it. The global and local convergence properties of the given method for solving the proposed stochastic eigenvalue complementarity problem are also given. Finally, the related numerical results show that the proposed method is efficient.

Sensitivity and covariance in stochastic complementarity problems with an application to North American natural gas markets

We provide an efficient method to approximate the covariance between decision variables and uncertain parameters in solutions to a general class of stochastic nonlinear complementarity problems. We also develop a sensitivity metric to quantify uncertainty propagation by determining the change in the variance of the output due to a change in the variance of an input parameter. The covariance matrix of the solution variables quantifies the uncertainty in the output and pairs correlated variables and parameters. The sensitivity metric helps in identifying the parameters that cause maximum fluctuations in the output. The method developed in this paper optimizes the use of gradients and matrix multiplications which makes it particularly useful for large-scale problems. Having developed this method, we extend the deterministic version of the North American Natural Gas Model (NANGAM), to incorporate effects due to uncertainty in the parameters of the demand function, supply function, infrastructure costs, and investment costs. We then use the sensitivity metrics to identify the parameters that impact the equilibrium the most.

On the existence of solutions to stochastic quasi-variational inequality and complementarity problems

Variational inequality problems allow for capturing an expansive class of problems, including convex optimization problems, convex Nash games and economic equilibrium problems, amongst others. Yet in most practical settings, such problems are complicated by uncertainty, motivating the examination of a stochastic generalization of the variational inequality problem and its extensions in which the components of the mapping contain expectations. When the associated sets are unbounded, ascertaining existence requires having access to analytical forms of the expectations. Naturally, in practical settings, such expressions are often difficult to derive, severely limiting the applicability of such an approach. Consequently, our goal lies in developing techniques that obviate the need for integration and our emphasis lies in developing tractable and verifiable sufficiency conditions for claiming existence. We begin by recapping almost-sure sufficiency conditions for stochastic variational inequality problems with single-valued maps provided in our prior work Ravat and Shanbhag (in: Proceedings of the American Control Conference (ACC), 2010), Ravat and Shanbhag (SIAM J Optim 21: 1168–1199, 2011) and provide extensions to multi-valued mappings. Next, we extend these statements to quasi-variational regimes where maps can be either single or set-valued. Finally, we refine the obtained results to accommodate stochastic complementarity problems where the maps are either general or co-coercive. The applicability of our results is demonstrated on practically occuring instances of stochastic quasi-variational inequality problems and stochastic complementarity problems, arising as nonsmooth generalized Nash-Cournot games and power markets, respectively.

The deterministic ERM and CVaR reformulation for the stochastic generalized complementarity problem

In this paper, we consider the stochastic generalized complementarity problem (SGCP), which has a variety of important applications and has the stochastic complementarity problem as a special case. In order to give reasonable solutions of SGCP, we propose a deterministic formulation that minimizes the expected value of the residual function as the objective function and uses CVaR constraints to ensure the probabilistic feasibility of its solutions. The resulting model containsnon-smooth constraints and expected value functions both in the objective function and constraints. We present approximation problems for solving this model using smoothing and Monte-Carlo techniques. We prove two convergence results of the proposed approximation problems. One is to increase the sample size only, the other is to let smoothing parameter tend to zero as sample increase together. The effectiveness of the proposed approach was demonstrated using numerical examples.

Stochastic second-order-cone complementarity problems: expected residual minimization formulation and its applications

This paper considers a class of stochastic second-order-cone complementarity problems (SSOCCP), which are generalizations of the noticeable stochastic complementarity problems and can be regarded as the Karush–Kuhn–Tucker conditions of some stochastic second-order-cone programming problems. Due to the existence of random variables, the SSOCCP may not have a common solution for almost every realization . In this paper, motivated by the works on stochastic complementarity problems, we present a deterministic formulation called the expected residual minimization formulation for SSOCCP. We present an approximation method based on the Monte Carlo approximation techniques and investigate some properties related to existence of solutions of the ERM formulation. Furthermore, we experiment some practical applications, which include a stochastic natural gas transmission problem and a stochastic optimal power flow problem in radial network.

A Note on Stability for Risk-Averse Stochastic Complementarity Problems

In this paper, we propose a new approach to stochastic complementarity problems which allows to take into account various notions of risk aversion. Our model enhances the expected residual minimization formulation of Chen and Fukushima (Math Oper Res 30:1022---1038, 2005) by replacing the expectation with a more general convex, nondecreasing and law-invariant risk functional. Relevant examples of such risk functionals include the Expected Excess and the Conditional Value-at-Risk. We examine qualitative stability of the resulting one-stage stochastic optimization problem with respect to perturbations of the underlying probability measure. Considering the topology of weak convergence, we prove joint continuity of the objective function with respect to both the decision vector and the entering probability measure. By a classical result from parametric optimization, this implies upper semicontinuity of the optimal value function. Throughout the analysis, we assume for building the model that a nonlinear complementarity function fulfills a certain polynomial growth condition. We conclude the paper demonstrating that this assumption holds in the vast majority of all practically relevant cases.

CONSISTENCY ANALYSIS OF AUBIN PROPERTY OF SAA SOLUTION MAPPING FOR A STOCHASTIC COMPLEMENTARITY PROBLEM

In this paper, we investigate properties of sample average approx- imation (SAA) solution mapping for a parametric stochastic complementarity problem, where the underlying function is the expected value of stochastic func- tion. In particular, using the notion of cosmic deviation, which is originated from the concept of cosmic distance in variational analysis, we develop sufficient conditions for the consistency of Aubin property of the solution mapping of the SAA parametric stochastic complementarity problems, namely if the solution map of the true problem has the Aubin property around some point, then so does the SAA problem around reference point with probability one when the sample size is large enough. At last, an example is illustrated to show the application of the analysis.

Minimum mean-squared deviation method for stochastic complementarity problems

In this paper, we propose a new reformulation for stochastic complementarity problems (SCPs). The new formulation is based on the minimum mean-squared deviation rule in statistics. Under mild conditions, we prove the existence of the solution of the new reformulation for SCP. Furthermore, we present a smoothing sample average approximation method for solving the problems. The convergence properties of the optimal solutions of the approximation problems are studied under mild conditions. Finally, some numerical results are listed as well.

A smooth penalty-based sample average approximation method for stochastic complementarity problems

Sample average approximation method is one of the effective methods in the stochastic optimization. A smooth penalty-based sample average approximation method for stochastic nonlinear complementarity problems is presented in this paper. Based on a smooth penalty function, a reformulation is proposed for the equivalent problem of EV formulation for stochastic complementary problems and it is proven that its solutions are existent under some mild assumptions. An implementable sample average approximation method for the reformulation is further established and its convergence is analyzed. The numerical results for some test examples are reported at last to show efficiency of the proposed method.

Decision-Making in Energy Systems With Multiple Technologies and Uncertain Preferences

Energy systems are complex systems consisting of different decision making entities: policy makers (or system operators), generation companies, utility companies, and consumers. In this paper, the focus is on the decisions made by different market competitors, such as the energy generation companies (GENCOs) and their interactions with the policy makers. The goals of the independent system operators (ISOs) in a deregulated market are to minimize the policy cost and to maximize renewable energy penetration. The problem formulation presented in this paper is aimed toward helping the ISO in modeling the lower-level GENCOs' decision problems. GENCOs compete in energy production, while maximizing their net present values and minimizing their capital investments. The market players' decisions result in a lower-level market equilibrium problem, which is formulated as a complementarity problem. The existence and uniqueness of the solution for the lower-level problem are shown. The main aims of the paper are to study the effect of stakeholders' preferences on the solution of the complementarity problem, to analyze the effect of the ISO's design parameters on the generation equilibrium quantities, and to simulate the stochastic complementarity problem using different techniques. The applications of the proposed decision-making framework include computational modeling of policy design scenarios, such as in feed-in-tariff policies.

The Distributionally Robust Optimization Reformulation for Stochastic Complementarity Problems

We investigate the stochastic linear complementarity problem affinely affected by the uncertain parameters. Assuming that we have only limited information about the uncertain parameters, such as the first two moments or the first two moments as well as the support of the distribution, we formulate the stochastic linear complementarity problem as a distributionally robust optimization reformation which minimizes the worst case of an expected complementarity measure with nonnegativity constraints and a distributionally robust joint chance constraint representing that the probability of the linear mapping being nonnegative is not less than a given probability level. Applying the cone dual theory and S-procedure, we show that the distributionally robust counterpart of the uncertain complementarity problem can be conservatively approximated by the optimization with bilinear matrix inequalities. Preliminary numerical results show that a solution of our method is desirable.

Sampling average approximation method for a class of stochastic Nash equilibrium problems

We consider a class of stochastic Nash equilibrium problems (SNEP). Under some mild conditions, we reformulate the SNEP as a stochastic mixed complementarity problem (SMCP). We apply the well-known sample average approximation (SAA) method to solve the SMCP. We further introduce a semismooth Newton method to solve the SAA problems. The comprehensive convergence analysis is given as well. In addition, we demonstrate the proposed approach on a stochastic Nash equilibrium model in the wholesale gas–oil markets.

Properties of Expected Residual Minimization Model for a Class of Stochastic Complementarity Problems

Expected residual minimization (ERM) model which minimizes an expected residual function defined by an NCP function has been studied in the literature for solving stochastic complementarity problems. In this paper, we first give the definitions of stochasticP-function, stochasticP0-function, and stochastic uniformlyP-function. Furthermore, the conditions such that the function is a stochasticPP0-function are considered. We then study the boundedness of solution set and global error bounds of the expected residual functions defined by the “Fischer-Burmeister” (FB) function and “min” function. The conclusion indicates that solutions of the ERM model are robust in the sense that they may have a minimum sensitivity with respect to random parameter variations in stochastic complementarity problems. On the other hand, we employ quasi-Monte Carlo methods and derivative-free methods to solve ERM model.