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 where the constraints are only of the lower-level type. Two methods of this class are introduced and analyzed. 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 an exhaustive comparison against Lancelot . All the problems of the Cute collection are used in this comparison. Moreover, 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.
Abstract. Two Augmented Lagrangian algorithms for solving KKT systems are introduced. The algorithms differ in the way in which penalty parameters are updated. Possibly infeasible accumulation points are characterized. It is proved that feasible limit points that satisfy the Constant Positive Linear Dependence constraint qualification are KKT solutions. Boundedness of the penalty parameters is proved under suitable assumptions. Numerical experiments are presented.
We present two new constraint qualifications (CQ) that are weaker than the recently introduced Relaxed Constant Positive Linear Dependence (RCPLD) constraint qualification. RCPLD is based on the assumption that many subsets of the gradients of the active constraints preserve positive linear dependence locally. A major open question was to identify the exact set of gradients whose properties had to be preserved locally and that would still work as a CQ. This is done in the first new constraint qualification, that we call Constant Rank of the Subspace Component (CRSC) CQ. This new CQ also preserves many of the good properties of RCPLD, like local stability and the validity of an error bound. We also introduce an even weaker CQ, called Constant Positive Generator (CPG), that can replace RCPLD in the analysis of the global convergence of algorithms. We close this work extending convergence results of algorithms belonging to all the main classes of nonlinear optimization methods: SQP, augmented Lagrangians, interior point algorithms, and inexact restoration. * This work was supported by PRONEX-Optimization (PRONEX-CNPq/FAPERJ E-26/171.510/2006-APQ1), Fapesp (Grants
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