Studies On Stochastic Variational Inequality Problems And Their Applications In Natural Gas Markets

The variational inequality problem(VIP)is widely used in the field of production distribution,management science,energy market and optimal control,etc.However,many practical problems are affected by some random factors,which may result in the disadvantage of the economic and social benefits if they are ignored.Therefore,there are significant theoretical value and practical meaning in studying the stochastic variational inequality problem(SVIP).At present,the SVIP with mathematical expectation has attracted much attention because of its wide applications.Since the expectation in SVIP is generally too expensive to evaluate in practice,the existing numerical methods for the deterministic VIP may not be applied to solve the SVIP directly.How to solve SVIP efficiently has become one of the most active topics in the area of operational research.On the other hand,natural gas is regarded as one kind of clean,environment-friendly and efficient energy,the rational development and utilization of natural gas has great practical significance for adjusting the consumption structure of the primary energy and promoting energy conservation and emission reduction.However,with the development of natural gas marketization,the randomness during the process of production,transportation and sale of natural gas is becoming increasingly obvious.For instance,the production cost and market price of natural gas are usually affected by some random factors,such as the weather and the market demand.The pressure regulation cost of natural gas will be affected by the regulator temperature which is also the random factor.Those random factors bring great challenges to market participants in making decisions,and then lead to the trade market equilibrium problem as well as the production and transportation problem of natural gas under uncertain environment.Under mild control conditions,those two kinds of problems can be transformed into SVIP,and then efficient algorithms can be designed to solve them.This is a new exploration of the research about the stochastic problem in the natural gas market.This thesis focuses on a class of SVIP with mathematical expectation,several numerical algorithms for SVIP are proposed and,the convergence and complexity of the algorithms are studied.Furthermore,the designed algorithms are used to solve the natural gas production and transportation model as well as the natural gas market equilibrium model under uncertain environment.We study the influences of random factors on strategies of the natural gas market,and then provide the theoretical support for the natural gas market participants.The main contents and innovations of this thesis are summarized as follows:Firstly,the stochastic complementarity problem,which is a special case of SVIP,is investigated.By combining an infeasible interior-point algorithm with the sample average approximation(SAA)methods,an SAA-based infeasible interior-point algorithm for the stochastic complementarity problem is presented,and then the convergence and complexity of the algorithm are analyzed.Furthermore,the algorithm mentioned above is extended to deal with a class of more general stochastic complementarity problems.Preliminary numerical results indicate that the new algorithm is superior to some existing methods in computational efficiency.Secondly,we propose a modified forward-backward algorithm with variance reduction,which employs a stochastic approximation version of Armijo’s line search to determine the stepsize,for solving SVIP.Compared with the extragradient method with variance reduction,a salient feature of the proposed algorithm is to compute only one projection at each iteration.We analyze the proposed algorithm for its convergence,sublinear convergence rate in terms of the mean natural residual function,and complexity under pseudo-monotonicity and other moderate conditions.We also discuss the linear convergence rate with the finite computational budget for the proposed algorithm without strongmonotonicity.Preliminary numerical experiments indicate that the proposed algorithm is competitive with some existing algorithms.Thirdly,inspired by the subgradient extragradient algorithm for the VIP,we present a subgradient extragradient algorithm with variance reduction for SVIP,which employs a stochastic approximation version of Armijo’s line search to determine the stepsize.Simply speaking,the proposed algorithm replaces the orthogonal projection onto the feasible set designed in the variance-based extragradient algorithms with a specific subgradient projection that can be calculated explicitly.In other words,the proposed algorithm computes only one projection at each iteration.We give some properties related to convergence,sublinear convergence rate in terms of the mean natural residual function,and complexity for the proposed algorithm under Minty inequality and other suitable assumptions.Furthermore,we discuss the linear convergence rate with the finite computational budget of the proposed algorithm under the strong-Minty inequality or the bounded proximal error bound condition.Preliminary numerical experiments indicate that the proposed approach is competitive with some existing algorithms.Fourthly,as an extension of SVIP,the existing numerical algorithms for SVIP may not be applied to solve the mixed SVIP directly.Based on the advantage of the variance reduction technique in computational efficiency,we present two proximal iteration algorithms with variance reduction for the mixed SVIP.One is extragradient proximal iteration algorithm and another is modified forward-backward proximal iteration algorithm.Under the monotonicity assumption and other moderate conditions,we derive some convergence properties for the proposed algorithms.Additionally,under the bounded metric subregularity condition,we investigate the convergence rate and complexity for the proposed algorithms.Preliminary numerical experiments show that the proposed algorithms are effective.Finally,we study the application of SVIP in the natural gas market.Considering the influence of random factors such as weather and market demand on the production cost and market price of natural gas,a stochastic production and transportation model of natural gas is established with the aim of the maximum profit.Then this model can be effectively solved by the SAA-based infeasible interior-point algorithm which is used for solving stochastic complementarity problems.Moreover,considering that different natural gas consumer groups have different requirements for the natural gas pressure,the natural gas trading volume of traders is limited by long-distance pipeline transportation pressure drop,and the market price is affected by some random factors,a natural gas market stochastic equilibrium model is established aiming at the maximum profit of traders.The algorithms with variance reduction for solving SVIP designed in this thesis are used to solve the stochastic equilibrium model.Through simulation tests that are carried out on a natural gas transmission network consisting of three compressors,and a natural gas trading market consisting of three traders under uncertain environment,the influence of different random variable ranges on the making decision and optimal profit of natural gas market participants are studied.Combined with the numerical results,the theoretical support is provided for the natural gas market participants.