Primal-dual nonlinear rescaling methods in constrained optimization.

摘要

The ability to solve large scale Nonlinear Programming problems (NLP) is critical in mechanics, structural optimization, antenna and chips design, optimal control, tomography, image reconstruction, power system optimization just to mention a few. The thesis focuses on developing theoretically well grounded and numerically efficient methods for solving large-scale constrained optimization and discrete minimax problems. Numerical realization of the developed methods, testing the software on real life applications is the second main purpose of our work. The work is based on the Nonlinear Resealing (NR) Principle in Constrained Optimization. The NR principle consists of transforming the objective function and/or the constraints set of a given constrained optimization problem into an equivalent one and using the Classical Lagrangian for the equivalent problem for both theoretical analysis and developing numerical methods. The constraints are scaled by a positive scaling parameter. The NR methods consist of finding the primal minimizer of the Lagrangian for the equivalent problem followed by the Lagrange multipliers update. The scaling parameter can be fixed or can be changed from step to step. Our main focus was on the Primal-Dual NR methods for constrained optimization with inequality constraints and discrete minimax. The Primal-Dual NR method consists of replacing the unconstrained minimization and the Lagrange multipliers update by solving the Primal-Dual (PD) system of equations. The Primal-Dual system consists of the optimality criteria for the primal minimizer and formulas for the Lagrange multipliers update. We solve the PD system by Newton's method. We developed a general Primal-Dual NR method, proved its global convergence and estimated the rate of convergence under standard assumptions on the input data. We also developed a MATLAB code based on the Primal-Dual NR method. The code was tested on a number of NLP and discrete minimax problems including COPS set. The numerical results corroborate the theory and show that the Primal-Dual NR methods are numerically stable and produce results competitive with the best known NLP solvers in terms of accuracy and number of Newton steps.