Full-Newton Step Feasible Interior-Point Algorithms For Solving The Fisher Market Equilibrium Problems

As an important extension of the standard complementarity problem(CP),the weighted complementarity problem(WCP)is a class of novel optimization problems.Many equilibrium problems from marketing economics,atmospheric and chemistry can be reformulated as WCP for their numerical solutions.In some cases,when a practical optimization problem can be either transformed into a nonlinear complementarity problem or modeled as a WCP,the latter may usually leads to more highly efficient algorithms for its numerical solution.However,the analysis of the theory and algorithms of WCP become more difficult since the weight vector is nonzero.At present,there are few researches about WCP.In this paper,we propose three full-Newton step feasible interior-point methods(IPMs)for the weighted complementary model,which is the more general optimization of the Fisher market equilibrium problem.The main contents are described as follows:1.Based on a continuous differentiable function,we obtain the new search directions for the linear weighted complementarity problem(LWCP)by extending a technique of algebraic equivalent transformation(AET)for linear optimization(LO)to LWCP.The feasible IPM with full-Newton step is proposed for solving the linear weighted complementarity model of the general Fisher market equilibrium problem.The algorithm does not require any linear search because it uses only full-Newton steps,which saves computation and memory.We show the polynomial complexities of LWCP and the general Fisher market equilibrium problem.The experiment illustrates that the algorithm has good numerical results.2.We design a new full-Newton step feasible IPM to solve the general Fisher market equilibrium problem.For the system defining the smooth central path,we construct its new equivalent transformation,and obtain the new search directions by Newton’s method.The algorithm also uses the full-Newton steps,which improve computational efficiency.We demonstrate that the algorithm is convergent and has the polynomial complexity.It follows from the preliminary numerical results that our algorithm has good performance.3.By a kernel function,we use its negative gradient to the system defining the central path.And then the new search directions are obtained by solving this equivalent system.A new feasible IPM with full-Newton steps is raised to solve the general Fisher market equilibrium problem.With the suitable conditions,the full-Newton steps are locally quadratic convergent and the algorithm has the polynomial complexity.The numerical results illustrate that the algorithm is effective.