Abstract. Augmented Lagrangian methods with general lower-level constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems in which the constraints are only of the lower-level type. Inexact resolution of the lower-level constrained subproblems is considered. Global convergence is proved using the Constant Positive Linear Dependence constraint qualification. Conditions for boundedness of the penalty parameters are discussed. The reliability of the approach is tested by means of a comparison against Ipopt and Lancelot B. The resolution of location problems in which many constraints of the lower-level set are nonlinear is addressed, employing the Spectral Projected Gradient method for solving the subproblems. Problems of this type with more than 3 × 10 6 variables and 14 × 10 6 constraints are solved in this way, using moderate computer time. The codes are free for download in www.ime.usp.br/∼egbirgin/tango/

We propose two line search primal-dual interior-point methods that have a generic barrier-SQP outer structure and approximately solve a sequence of equality constrained barrier subproblems. To enforce convergence for each subproblem, these methods use an # 2 -exact penalty function eliminating the need to drive the corresponding penalty parameter to infinity when finite multipliers exist. Instead of directly decreasing an equality constraint infeasibility measure, these methods attain feasibility by forcing this measure to zero whenever the steps generated by the methods tend to zero. Our analysis shows that under standard assumptions, our methods have strong global convergence properties. Specifically, we show that if the penalty parameter remains bounded, any limit point of the iterate sequence is either a KKT point of the barrier subproblem, or a FritzJohn (FJ) point of the original problem that fails to satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ); if the penalty parameter tends to infinity, there is a limit point that is either an infeasible FJ point of the inequality constrained feasibility problem (an infeasible stationary point of the infeasibility measure if slack variables are added) or a FJ point of the original problem at which the MFCQ fails to hold. Numerical results are given that illustrate these outcomes.

Abstract. We study a smoothing Newton method for solving a nonsmooth matrix equation that includes semidefinite programming and the semidefinite complementarity problem as special cases. This method, if specialized for solving semidefinite programs, needs to solve only one linear system per iteration and achieves quadratic convergence under strict complementarity and nondegeneracy. We also establish quadratic convergence of this method applied to the semidefinite complementarity problem under the assumption that the Jacobian of the problem is positive definite on the affine hull of the critical cone at the solution. These results are based on the strong semismoothness and complete characterization of the B-subdifferential of a corresponding squared smoothing matrix function, which are of general theoretical interest. Key words. Matrix equations, Newton’s method, nonsmooth optimization, semidefinite complementarity problem, semidefinite programming. AMS subject classifications. 65K05, 90C25, 90C33. 1. Introduction. 1.1. Motivation. Let S(n1,..., nm) be the linear space of symmetric blockdiagonal matrices with m blocks of sizes nk × nk, k = 1,..., m, respectively, and let

convergence analysis of line search interior point methods for nonlinear programming without regularity assumptions ∗