An augmented lagrangian interior point method using diretions of negative curvature

We describe an efficient implementation of an interior-point algorithm for non-convex problems that uses directions of negative curvature. These directions should ensure convergence to second-order KKT points and improve the computational efficiency of the procedure. Some relevant aspects of the implementation are the strategy to combine a direction of negative curvature and a modified Newton direction, and the conditions to ensure feasibility of the iterates with respect to the simple bounds. The use of multivariate barrier and penalty parameters is also discussed, as well as the update rules for these parameters. Finally, numerical results on a set oftest problems are presented

An augmented Lagrangian interior-point method using directions of negative curvature

The original publication is available at www.springerlink.comWe describe an efficient implementation of an interior-point algorithm for non-convex problems that uses directions of negative curvature. These directions should ensure convergence to second-order KKT points and improve the computational efficiency of the procedure. Some relevant aspects of the implementation are the strategy to combine a direction of negative curvature and a modified Newton direction, and the conditions to ensure feasibility of the iterates with respect to the simple bounds. The use of multivariate barrier and penalty parameters is also discussed, as well as the update rules for these parameters.We analyze the convergence of the procedure; both the linesearch and the update rule for the barrier parameter behave appropriately. As the main goal of the paper is the practical usage of negative curvature, a set of numerical results on small test problems is presented. Based on these results, the relevance of using directions of negative curvature is discussed.Research supported by Spanish MEC grant TIC2000-1750-C06-04; Research supported by Spanish MEC grant BEC2000-0167Publicad

Combining search directions using gradient flows

The original publication is available at www.springerlink.comThe efficient combination of directions is a significant problem in line search methods that either use negative curvature, or wish to include additional information such as the gradient or different approximations to the Newton direction. In this paper we describe a new procedure to combine several of these directions within an interior-point primal-dual algorithm. Basically, we combine in an efficient manner a modified Newton direction with the gradient of a merit function and a direction of negative curvature, if it exists.We also show that the procedure is well-defined, and it has reasonable theoretical properties regarding the rate of convergence of the method. We also present numerical results from an implementation of the proposed algorithm on a set of small test problems from the CUTE collection.Research supported by Spanish MEC grants BEC2000-0167 and PB98-0728Publicad

A note on the use of vector barrier parameters for interior-point methods

A key feature to ensure desirable convergence properties in an interior point method is the appropriate choice of an updating rule for the barrier parameter. In this work we analyze and describe updating rules based on the use of a vector of barrier parameters. We show that these updating rules are well defined and satisfy sufficient conditions to ensure con vergence to the correct limit points. We also present some numerical results that illustrate the improved performance of these strategies compared to the use of a scalar barrier parameter.Publicad

Combining search directions using gradient flows

The efficient combination of directions is a significant problem in line search methods that either use negative curvature. or wish to include additional information such as the gradient or different approximations to the Newton direction. In thls paper we describe a new procedure to combine several of these directions within an interior-point primal-dual algorithm. Basically. we combine in an efficient manner a modified Newton direction with the gradient of a merit function and a direction of negative curvature. is it exists. We also show that the procedure is well-defined. and it has reasonable theoretical properties regarding the convergence of the method. We also present numerical results from an implementation of the proposed algorithm on a set of small test problems from the CUTE collection

Nonconvex optimization using negative curvature within a modified linesearch

This paper describes a new algorithm for the solution of nonconvex unconstrained optimization problems, with the property of converging to points satisfying second-order necessary optimality conditions. The algorithm is based on a procedure which, from two descent directions, a Newton-type direction and a direction of negative curvature, selects in each iteration the linesearch model best adapted to the properties of these directions. The paper also presents results of numerical experiments that illustrate its practical efficiency.Publicad

Combining and scaling descent and negative curvature directions

The original publication is available at www.springerlink.comThe aim of this paper is the study of different approaches to combine and scale, in an efficient manner, descent information for the solution of unconstrained optimization problems. We consider the situation in which different directions are available in a given iteration, and we wish to analyze how to combine these directions in order to provide a method more efficient and robust than the standard Newton approach. In particular, we will focus on the scaling process that should be carried out before combining the directions. We derive some theoretical results regarding the conditions necessary to ensure the convergence of combination procedures following schemes similar to our proposals. Finally, we conduct some computational experiments to compare these proposals with a modified Newton’s method and other procedures in the literature for the combination of information.Catarina P. Avelino was partially supported by Portuguese FCT postdoctoral grant SFRH/BPD/20453/2004 and by the Research Unit CM-UTAD of University of Trás-os-Montes e Alto Douro. Javier M. Moguerza and Alberto Olivares were partially supported by Spanish grant MEC MTM2006-14961-C05-05. Francisco J. Prieto was partially supported by grant MTM2007-63140 of the Spanish Ministry of Education.Publicad

Nonconvex optimization using negative curvature within a modified linesearch

This paper describes a new algorithm for the solution of nonconvex unconstrained optimization problems, with the property of converging to points satisfying second-order necessary optimality conditions. The algorithm is based on a procedure which, from two descent directions, a Newton-type direction and a direction of negative curvature, selects in each iteration the linesearch model best adapted to the properties of these directions. The paper also presents results of numerical experiments that illustrate its practical efficiency.