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.

A note on the convergence of an augmented Lagrangian algorithm to second-order stationary points

“…Algencan-second algorithm was designed to handle box-constraints. However, this requires a more sophisticated exposure that is unnecessary for our purposes (for more details, see [2]). Hence, we simply assume the following assumption that guarantees the well-definiteness of the algorithm presented here.…”

“…The iterate x k in Step 2 of Algorithm 1 may be computed by the Gencan-second [2] algorithm, a box-constrained solver based on an active-set strategy and in spectral projected gradient steps that is able to deal with directions of negative curvature. …”

“…Many algorithms in the literature were developed in order to guarantee convergence to points that satisfy second-order necessary conditions (see for example [2] and references there in). Unfortunately, checking SSONC is an NP-hard [6] problem.…”

Augmented Lagrangians with possible infeasibility and finite termination for global nonlinear programming

“…The Algencan code, available in http://www.ime.usp.br/∼egbirgin/tango/ and based on the theory presented in [5], has been improved several times in the last few years [7,18,20,24,26,25,29] and, in practice, has been shown to converge to global minimizers more frequently than other Nonlinear Programming solvers. Derivative-free versions of Algencan were introduced in [31] and [49].…”

A note on the convergence of an augmented Lagrangian algorithm to second-order stationary points

“…Thus, one question is whether the same occurs with quadratic penalty-like methods. We are particularly interested in the second-order augmented Lagrangian method developed in [2], named Algencan-second. In this method, each iterate satisfies approximately a second-order condition for the Lagrangian subproblem.…”