A Penalty-free Line Search Method With Infeasibility Detection

This paper considers the general nonlinear optimization problems with nonlinear inequality constraints,which arises naturally in many areas such as optimal control,resource allocation,solution of equilibrium models and structural engineering.Popular methods for solving the problem are the sequential quadratic programming(SQP)methods which need some penalty function as a merit function for determining when one point is“better”than the other.However,the value of a merit function is very sensitive to the value of the penalty parameter.At the present,the choice of the penalty parameter is still a problematic.Therefore,some researches present the various methods without any penalty function.Most of those methods need a feasibility restoration phase to tackle the inconsistence of the linearized constraints,which may spend a large amount of computation and is also cumbersome in the algorithmic implementation.As a result,studying a penalty-free method with the detection of infeasibility and avoiding feasibility restoration phase are of great value in theory and practice.In this paper,we propose a penalty-free line search method with infeasibility detection for the general nonlinear optimization.At first,a linear programming subproblem is solved whose solution can justify whether the original problem is locally feasible or not.If it is feasible,we call the solution as a feasible direction.Next,we solve a quadratic programming subproblem which is always compatible and whose solution will improve the measure of the optimality.We call it as optimal direction.And then we choose some convex combination between the feasible direction and the optimal direction as a line search direction.If the objective function or the measure of the constraint violation decreases sufficiently along the search direction,the current iteration is successful.The method proposed does not need the feasibility restoration phase.If the objective function and constraint functions all are smooth,the method is well defined and there exists a limit point which is an infeasible stationary,or MF constraint qualification does not hold at the point,or which is a stationary point of the original problem.To avoid Maratos effect,a second order correction(SOC)step is adopted.Under the usual assumptions,the method with SOC is locally superlinear convergent.Finally,some preliminary numerical results are reported.