An SR۱/BFGS Projected Exact Penalty Method for Constrained Nonlinear Optimization

We propose a combined symmetric rank one (SR۱) and Broyden-Fletcher-Goldfarb-Shanno (BFGS) updating scheme to approximate projected Hessians in an exact penalty algorithm for solving general nonlinear optimization problems. We have three iterations types: “infeasible”, “almost feasible” and “local”. In an infeasible iteration, a descent direction is computed for the penalty function. In an almost feasible iteration, the step direction is a combination of a horizontal direction for reducing the penalty function and a vertical direction for maintaining feasibility. The projected Hessians in infeasible and almost feasible iterations are updated with the SR۱ formula. During local iterations, when an iterate is near stationarity, the Lagrange multipliers are calculated by solving a linear least squares problem. If the computed Lagrange multipliers satisfy the first-order optimality conditions, a Newton step direction is computed to obtain a superlinear rate of convergence to a stationary point of the problem. Otherwise, a dropping step is calculated to decrease the penalty function value. In local iterations, BFGS updating formula is used to update the projected Hessian of the penalty function. Comparative numerical experiments on some test problems from the CUTEst library confirm the efficiency of the proposed algorithm as compared to the four well-known software packages: KNITRO/Active-Set, SNOPT, filterSQP and LANCELOT.

کلیدواژه ها:

Constrained Optimization ، Exact Penalty Function ، SR۱ Update ، BFGS Update ، Local Convergence Rate