Roberto Andreani

On Augmented Lagrangian Methods with General Lower-Level Constraints

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 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

Augmented Lagrangian methods under the constant positive linear dependence constraint qualification

3 \times 10^6

On sequential optimality conditions for smooth constrained optimization

variables and

A relaxed constant positive linear dependence constraint qualification and applications

14 \times 10^6

On the Resolution of the Generalized Nonlinear Complementarity Problem

constraints are solved in this way, using moderate computer time. All the codes are available at http://www.ime.usp.br/

Convergent algorithms for protein structural alignment

\sim

Two New Weak Constraint Qualifications and Applications

egbirgin/tango/.

A New Sequential Optimality Condition for Constrained Optimization and Algorithmic Consequences

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.

On second-order optimality conditions for nonlinear programming

Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Approximate Karush–Kuhn–Tucker and approximate gradient projection conditions are analysed in this work. These conditions are not necessarily equivalent. Implications between different conditions and counter-examples will be shown. Algorithmic consequences will be discussed.

Second-order negative-curvature methods for box-constrained and general constrained optimization

In this work we introduce a relaxed version of the constant positive linear dependence constraint qualification (CPLD) that we call RCPLD. This development is inspired by a recent generalization of the constant rank constraint qualification by Minchenko and Stakhovski that was called RCRCQ. We show that RCPLD is enough to ensure the convergence of an augmented Lagrangian algorithm and that it asserts the validity of an error bound. We also provide proofs and counter-examples that show the relations of RCRCQ and RCPLD with other known constraint qualifications. In particular, RCPLD is strictly weaker than CPLD and RCRCQ, while still stronger than Abadie’s constraint qualification. We also verify that the second order necessary optimality condition holds under RCRCQ.