Research on Numerical Methods for Large-Scale Nonlinear Optimization Problems and their Applications to Software Codes

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We have studied numerical methods for solving unconstrained and constrained optimization problems. Specifically, we have done the following.(1) We have proposed new nonlinear conjugate gradient methods based on the modified secant and the multi-step secant conditions for solving large-scale unconstrained optimization problems. We have proved their global convergence properties. Our numerical experiments show that our proposed methods perform well.(2) We have proposed a new memory gradient method for solving large-scale unconstrained optimization problems. We have proved their global convergence properties. Our numerical experiments show that our proposed method performs well.(3) We have combined the limited memory quasi-Newton method, which was proposed by us, and the primal-dual interior point method to solve nonlinearly constrained optimization problems.(4) We have analyzed local behavior of the primal-dual interior point method for degenerate nonlinear optimization problems.(5) We have proposed primal-dual interior point methods for solving nonlinear second-order cone programming and nonlinear semidefinite programming problems. We have proved their global convergence properties by using primal-dual merit function within the framework of the line search strategy.(6) We have proved the local and q-superlinear convergence of the quasi-Newton method with the Broyden family based on the modified secant condition for solving unconstrained optimization problems.(7) We have dealt with the Barzilai-Borwein method to improve numerical performance of the steepest descent method, and we have proposed the extended Barzilai-Borwein method.