Flexible Penalty Methods For Nonlinear Optimization With Equality Constraints

Nonlinear constrained optimization has great important application in many felds.A variety of efective approaches to solve them are studied all the while. The traditionalmethods include various penalty-type methods and penalty-free methods. Penalty-typemethods usually use some penalty function as a merit function and require that thevalue of the merit function is descent sufciently. Specially, exact penalty methods canefectively solve inconsistent constraint linearizations and certain classes of problemsin which standard constraint qualifcations are not satisfed. But the selection of thepenalty parameter is a complex and difcult problem. The second kind of methods arenamed penalty-free methods which do not use any penalty function. Filter methodsare one of the representatives. Now there are so many available penalty-free methods.The numerical results show their efciency. Penalty-type methods and penalty-freemethods have the characteristics respectively. Devising new methods between penalty-type and penalty-free methods are of important theoretical signifcance and applicationvalue.In this paper, We proposed a fexible penalty method for solving nonlinear opti-mization with nonlinear equality constraints. The new method use the penalty param-eter to compute some trial step. To decide whether the trial step is accepted or not, thenew method demands that the measure of constraint violation is improved or the valueof the objective function is improved within the measure of feasibility control. Accept-able criteria of the trial step is dependent on the objective function and the measure ofconstraint violation. The updation of penalty parameter is dependent on the messageof the current iterative point and is diferent from one of the traditional method wherethe penalty parameter is non-decreasing monotonically. The new method combinesthe penalty parameter with penalty-free acceptable criteria, which can solve difcultnonlinear programs and the updating of the penalty parameter is very fexible.Under usual assumption, the well defnition of the algorithm is analyzed and theglobal convergence is given. Finally, several numerical results are reported.