Multiobjective Programs With Equilibrium Constraints And Their Applications In Biofuel Market

Multiobjective programs with equilibrium constraints(MOPEC)have a wide range of applications in energy,environment,health and transportation.Therefore,it is of great practical significance to conduct in-depth research and design an appropriate algorithm for the MOPEC.At present,the research on the MOPEC mainly focuses on the constraint qualifications and stationary conditions of the MOPEC.The algorithms for solving the MOPEC are rarely involved.In order to solve the MOPEC,most researchers convert the MOPEC into the mathematical programs with equilibrium constrains by multiobjective scalarization approaches,then solve them by applying the methods for the mathematical programs with equilibrium constrains.For multiobjective optimization problems,the scalar method is the most commonly used one.However,because the choice of weights is unknown in advance,it is difficult for the decision makers to choose the weights,and the method only calculates the proper Pareto optimal solutions.In recent years,methods that do not scalarize for multiobjective optimization problems,such as the steepest descent method and the approximal gradient method,have attracted widely attention in the field of optimization.Therefore,it is necessary to study this kind of methods for solving the MOPEC.In order to solve the MOPEC,we propose the MOPEC-type constraint qualifications and proper Pareto stationarity conditions,then construct a proximal gradient method for solving the MOPEC.Combined with the proposed MOPEC-type constraint qualifications given in this thesis,the convergence results of the algorithm are established.At last,we apply the MOPEC to the biofuel market.The multiobjectives are used to weight the different targets in the biofuel market,and the equilibrium constraints are used to show the complex interactions between the components of the biofuel market.We study the influences of the government’s policies include the impact of subsidies and the farmland regulations on the biofuel market.Through a case studied,we provide some suggestions for the formulation of government policies in the biofuel market.The main contents and innovations of this thesis can be summarized as follows:Firstly,motivated by the generalized Guignard constraint qualification of multiobjective optimization problems,by combining the ε approximation problem of the multiobjective problems,we extend the existed constraint qualifications of mathematical programs with equilibrium constraints from single objective to multiobjective problems,and give the definitions of MOPEC-type constraint qualifications and the stationary conditions of MOPEC in the sense of proper Pareto optimality.We study the relationship among the MOPEC-type constraint qualifications and the relationships between the constraint qualifications and the proper Pareto stationary conditions.In particular,when the objective function is locally Lipschitz continuous but not differential everywhere,we extend some weak constraint qualifications of nonlinear programming to the nonsmooth MOPEC and prove that they can be used as the constraint qualifications of nonsmooth strong stationarity in the sense of limit subdifferential.Secondly,we use the partial exact penalty to transform the MOPEC into a approximate multiobjective problem with only simple complementary constraints,and use some smoothing techniques to get rid of the nonsmooth,then by combining the proximal gradient method for the multiobjective problems and the smoothing partial exact penalty algorithm for the MPEC,we propose a smoothing partial exact penalty proximal gradient algorithm to solve the smoothing approximate multiobjective problem with simple complementary constraints.Based on the proposed MOPEC-type constraint qualifications,the solution relationship between the approximate problem and the original problem is discussed,and under mild conditions we establish the convergence results of the proposed algorithm.Preliminary numerical experiments show that the proposed algorithm is effective for the MOPEC.Finally,we apply the MOPEC to the biofuel market,and consider a biofuel supply chain composed by farmers,biofuel producers,biofuel blenders and consumers.We mainly study the government subsidy policies and farmland regulation for the biofuel market.Since the formulation of policies in the biomass energy market is multi-objective and multi-level in nature,we model the problem as a MOPEC,in which the government’s objectives include social surplus,total profits,and greenhouse gas emission reduction,while the equilibrium constraints depict a Nash equilibrium among all followers including farmers,producers and blenders because they essentially play a non-cooperative game.Through a case studied in Iowa,we analyze the impact of government subsidy policies and farmland regulation on social surplus values,total profits and greenhouse gas emission reductions.Combined with the numerical results,the corresponding opinions and suggestions are given for the formulation of government policies in the biofuel market.