Lower-order penalty methods for nonlinear optimization and complementarity problems

Abstract

The main purpose of this thesis is to propose efficient numerical methods to solve inequality constrained nonlinear programming problems and complementarity problems by virtue of the ℓ 1/p (p> 1)-penalty function. We propose an interior-point l1/p -penalty method for inequality constrained optimization problems by introducing a technique of the p-order relaxation to the nonconvex and non-Lipschitzian l1/p -penalty function and combining with an interior-point method. We introduce different kinds of constraint qualifications to establish first-order necessary conditions for the relaxed problem. We employ the modified Newton method to solve a sequence of logarithmic barrier subproblems and detail three reliable algorithms by using the Armijo line search. We prove that the iteration sequence generated by the proposed method converges to some KKT (or FJ) point of the original problem under mild conditions. Preliminary numerical experiments on small, medium and large test problems in the literature show that, comparing with some existing interior-point l1 -penalty methods, the proposed method is competitive in terms of the iteration numbers, better when comparing the number of updating the penalty parameters and more reliable when comparing the relative error. We introduce a box-constrained differentiable penalty method for nonlinear complementarity problems, which not only inherits the same convergence rate as the existing ℓ 1/p -penalty method but also overcomes its disadvantage of the non-Lipschitzianness. We introduce a concept of a uniform ξ-P-function with ξε[1, 2), under which we prove that the solution of box-constrained penalized equations converges to a solution of the original problem at an exponential order. Instead of solving the box-constrained penalized equations directly, we solve a corresponding differentiable least squares problem by using a trust-region Gauss-Newton method to design the globally convergent method that allows arbitrary starting points for solving the complementarity problems. Furthermore, we establish the connection between the local solution of the least squares problem and the solution of the original problem under mild conditions. We carry out the numerical experiments on the test problems from MCPLIB, which show that the proposed method is efficient and robust. We investigate an unconstrained differentiable penalty method for general complementarity problems without introducing artificial variables, which shares the exponential convergence rate under the assumption of a uniform ξ-P-function. Instead of solving the unconstrained penalized equations directly, we solve a corresponding differentiable least squares problem by using a trust-region Gauss-Newton method. Preliminary numerical experiments show that the proposed method is more robust than the box-constrained differentiable penalty method.