A QP-Free Algorithm Without A Penalty Function Or A Filter For Optimization Problems With Nonlinear Complementarity Constraints

The optimization problem with nonlinear complementarity constraints is investigated in this dissertation,which is a special type of mathematical programs with equilibrium constraints.These problems have strong practical background and wide applications in many fields,for instance,mechanics,economics,engineering design,traffic and transportation.Therefore,researching on stable and efficient algorithms for optimization problems with nonlinear complementarity constraints has important theoretical significance and application value.In this dissertation,we propose a QP-free algorithm without a penalty function or a filter for optimization problems with nonlinear complementarity constraints.Firstly,based on Fischer-Burmeister function,the investigated problem is equivalently transformed into a nonsmooth constrained optimization problem.By smoothing techniques,the nonsmooth optimization problem is approximated by a smooth constrained optimization problem.The working set is generated by pivoting operation,which is used to construct systems of linear equations.The use of the working set can reduce the dimension of the coefficient matrix of the systems of linear equations.Therefore,the amount of calculation is reduced and the numerical effect is improved.Secondly,line search technology ensures that the value of objective function or the value of the constraint violation function decreases sufficiently.Under some mild conditions,there exists an accumulation point of the iterative point sequence which is a KKT point for the discussed problem.Finally,in order to overcome the Maratos effect,the algorithm is improved by using the second-order correction technique.Under further assumptions,the modified algorithm has superlinear convergence.At the same time,a preliminary numerical test is carried out and numerical results show that the proposed algorithm is feasible and effective.