Showing papers by "José Mario Martínez published in 2010"

Global minimization using an Augmented Lagrangian method with variable lower-level constraints

Abstract: A novel global optimization method based on an Augmented Lagrangian framework is introduced for continuous constrained nonlinear optimization problems. At each outer iteration k the method requires the $${\varepsilon_{k}}$$ -global minimization of the Augmented Lagrangian with simple constraints, where $${\varepsilon_k \to \varepsilon}$$ . Global convergence to an $${\varepsilon}$$ -global minimizer of the original problem is proved. The subproblems are solved using the źBB method. Numerical experiments are presented.

A New Sequential Optimality Condition for Constrained Optimization and Algorithmic Consequences

Abstract: Necessary first-order sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Sequential optimality conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constraint qualifications. A new condition of this type is introduced in the present paper. It is proved that a well-established augmented Lagrangian algorithm produces sequences whose limits satisfy the new condition. Practical consequences are discussed.

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

Abstract: A Nonlinear Programming algorithm that converges to second-order stationary points is introduced in this paper. The main tool is a second-order negative-curvature method for box-constrained minimization of a certain class of functions that do not possess continuous second derivatives. This method is used to define an Augmented Lagrangian algorithm of PHR (Powell-Hestenes-Rockafellar) type. Convergence proofs under weak constraint qualifications are given. Numerical examples showing that the new method converges to second-order stationary points in situations in which first-order methods fail are exhibited.

Partial spectral projected gradient method with active-set strategy for linearly constrained optimization

Abstract: A method for linearly constrained optimization which modifies and generalizes recent box-constraint optimization algorithms is introduced. The new algorithm is based on a relaxed form of Spectral Projected Gradient iterations. Intercalated with these projected steps, internal iterations restricted to faces of the polytope are performed, which enhance the efficiency of the algorithm. Convergence proofs are given and numerical experiments are included and commented. Software supporting this paper is available through the Tango Project web page: http://www.ime.usp.br/∼egbirgin/tango/.

Addressing the greediness phenomenon in Nonlinear Programming by means of Proximal Augmented Lagrangians

Abstract: When one solves Nonlinear Programming problems by means of algorithms that use merit criteria combining the objective function and penalty feasibility terms, a phenomenon called greediness may occur. Unconstrained minimizers attract the iterates at early stages of the calculations and, so, the penalty parameter needs to grow excessively, in such a way that ill-conditioning harms the overall convergence. In this paper a regularization approach is suggested to overcome this difficulty. An Augmented Lagrangian method is defined with the addition of a regularization term that inhibits the possibility that the iterates go far from a reference point. Convergence proofs and numerical examples are given.