Stochastic Complementarity Research Articles

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.

Strategic Bidding for a Large Consumer

The smart grid technology enables an increasing level of responsiveness on the demand side, facilitating demand serving entities—large consumers and retailers—to procure their electricity needs under the best conditions. Such entities generally exhibit a proactive role in the pool, seeking to procure their energy needs at minimum cost. Within this framework, we propose a mathematical model to help large consumers to derive bidding strategies to alter pool prices to their own benefit. Representing the uncertainty involved, we develop a stochastic complementarity model to derive bidding curves, and show the advantages of such bidding scheme with respect to non-strategic ones.

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.

Strategic Wind Power Investment

This paper considers the problem of identifying the optimal investment of a strategic wind power investor that participates in both the day-ahead (DA) and the balancing markets. This investor owns a number of wind power units that jointly with the newly built ones allow it to have a dominant position and to exercise market power in the DA market, behaving as a deviator in the balancing market in which the investor buys/sells its production deviations. The model is formulated as a stochastic complementarity model that can be recast as a mixed-integer linear programming (MILP) model. A static approach is proposed focusing on a future target year, whose uncertainties pertaining to demands, wind power productions, and balancing market prices are precisely described. The proposed model is illustrated using a simple example and two case studies.

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.

A New Smoothing Nonlinear Conjugate Gradient Method for Nonsmooth Equations with Finitely Many Maximum Functions

The nonlinear conjugate gradient method is of particular importance for solving unconstrained optimization. Finitely many maximum functions is a kind of very useful nonsmooth equations, which is very useful in the study of complementarity problems, constrained nonlinear programming problems, and many problems in engineering and mechanics. Smoothing methods for solving nonsmooth equations, complementarity problems, and stochastic complementarity problems have been studied for decades. In this paper, we present a new smoothing nonlinear conjugate gradient method for nonsmooth equations with finitely many maximum functions. The new method also guarantees that any accumulation point of the iterative points sequence, which is generated by the new method, is a Clarke stationary point of the merit function for nonsmooth equations with finitely many maximum functions.

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.

Sample average approximation method for stochastic complementarity problems with applications to supply chain supernetworks

We consider a class of stochastic nonlinear complementarityproblems. We propose a new reformulation of the stochasticcomplementarity problem, that is, a two-stage stochasticmathematical programming model reformulation. Based on thisreformulation, we propose a smoothing-based sample averageapproximation method for stochastic complementarity problem andprove its convergence. As an application, a supply chainsuper-network equilibrium is modeled as a stochastic nonlinearcomplementarity problem and numerical results on the problem arereported.

Pricing American options with uncertain volatility through stochastic linear complementarity models

We consider the problem of pricing American options with uncertain volatility and propose two deterministic formulations based on the expected value method and the expected residual minimization method for a stochastic complementarity problem. We give sufficient conditions that ensure the existence of a solution of those deterministic formulations. Furthermore we show numerical results and discuss the usefulness of the proposed approach.

A Benders Decomposition Method for Solving Stochastic Complementarity Problems with an Application in Energy

In this paper we present a new Benders decomposition method for solving stochastic complementarity problems based on the work by Fuller and Chung (Comput Econ 25:303---326, 2005; Eur J Oper Res 185(1):76---91, 2007). A master and subproblem are proposed both of which are in the form of a complementarity problem or an equivalent variational inequality. These problems are solved iteratively until a certain convergence gap is sufficiently close to zero. The details of the method are presented as well as an extension of the theory from Fuller and Chung (2005, 2007). In addition, extensive numerical results are provided based on an electric power market model of Hobbs (IEEE Trans Power Syst 16(2):194---202, 2001) but for which stochastic elements have been added. These results validate the approach and indicate dramatic improvements in solution times as compared to solving the extensive form of the problem directly.

Penalty-based SAA method of stochastic nonlinear complementarity problems

We consider a class of stochastic nonlinear complementarityproblems. We first formulate the stochastic complementarity problemas a stochastic programming model. Based on this reformulation, wepropose a penalty-based sample average approximation (in short, SAA) method forstochastic complementarity problem and prove its convergence.Finally, we report some numerical test results to show theefficiency of our method.

Stochastic Nonlinear Complementarity Problems: Stochastic Programming Reformulation and Penalty-Based Approximation Method

We consider a class of stochastic nonlinear complementarity problems. We first reformulate the stochastic complementarity problem as a stochastic programming model. Based on the reformulation, we then propose a penalty-based sample average approximation method and prove its convergence. Finally, we report on some numerical test results to show the efficiency of our method.

Solving stochastic complementarity problems in energy market modeling using scenario reduction

In this paper, we analyze market equilibrium models with random aspects that lead to stochastic complementarity problems. While the models presented depict energy markets, the results are believed to be applicable to more general stochastic complementarity problems. The contribution is the development of new heuristic, scenario reduction approaches that iteratively work towards solving the full, extensive form, stochastic market model. The methods are tested on three representative models and supporting numerical results are provided as well as derived mathematical bounds.

The SC1 property of an expected residual function arising from stochastic complementarity problems

The stochastic nonlinear complementarity problem has been recently reformulated as an expected residual minimization problem which minimizes an expected residual function defined by an NCP function. In this work, we show that the expected residual function defined by the Fischer–Burmeister function is an SC 1 function.