On the use of the SYMMBK algorithm for computing negative curvature directions within Newton-Krylov methods

In this paper, we consider the issue of computing negative curvature directions, for nonconvex functions, within Newton-Krylov methods for large scale unconstrained optimization. This issue has been widely investigated in the literature, and different approaches have been proposed. We focus on the well known SYMMBK method proposed for solving large scale symmetric possibly inde finite linear systems [3, 5, 7, 20], and show how to exploit it to yield an effective negative curvature direction. The distinguishing feature of our proposal is that the computation of such negative curvature direction is iteratively carried out, without storing no more than a couple of additional vectors. The results of a preliminary numerical experience are reported showing the reliability of the novel approach we propose

Planar Conjugate Gradient Algorithm for Large-Scale Unconstrained Optimization, Part 1: Theory

Abstract. In this paper, we describe an application of the planar conjugate gradient method introduced in Part 1 (Ref. 1) and aimed at solving indefinite nonsingular sets of linear equations. We prove that it can be used fruitfully within optimization frameworks; in particular, we present a globally convergent truncated Newton scheme, which uses the above planar method for solving the Newton equation. Finally, our approach is tested over several problems from the CUTE collection (Ref. 2). Key Words. Large-scale unconstrained optimization, truncated Newto

Bridging the gap between Trust–Region Methods (TRMs) and Linesearch Based Methods (LBMs) for Nonlinear Programming: quadratic sub–problems

We consider the solution of a recurrent sub–problem within both constrained and unconstrained Nonlinear Programming: namely the minimization of a quadratic function subject to linear constraints. This problem appears in a number of LBM frameworks, and to some extent it reveals a close analogy with the solution of trust–region sub–problems. In particular, we refer to a structured quadratic problem where five linear inequality constraints are included. We show that our proposal retains an appreciable versatility, despite its particular structure, so that a number of different real instances may be reformulated following the pattern in our proposal. Moreover, we detail how to compute an exact global solution of our quadratic sub–problem, exploiting first order KKT conditions

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

Global convergence of the nonmonotone MBFGS method for nonconvex unconstrained minimization

AbstractIn this paper, we propose a new nonmonotone Armijo type line search and prove that the MBFGS method proposed by Li and Fukushima with this new line search converges globally for nonconvex minimization. Some numerical experiments show that this nonmonotone MBFGS method is efficient for the given test problems

Uso delle Direzioni Coniugate negli algoritmi per l'Ottimizzazione Non Vincolata a grande dimensione

Both obesity and dysmenorrhea are prevalent among women. Few population-based longitudinal studies investigate the association between body mass index (BMI) and dysmenorrhea yielding mixed results, especially for obesity. This study aims to investigate the long-term association between BMI and dysmenorrhea.9,688 women from a prospective population-based cohort study were followed for 13 years. Data were collected through self-reported questionnaires. The longitudinal association between dysmenorrhea and BMI or BMI change was investigated by logistic regression analysis using generalized estimating equations to account for the repeated measures.When the women were aged 22 to 27 years, approximately 11% were obese, 7% underweight, and 25% reported dysmenorrhea. Compared to women with a normal weight, significantly higher odds of reporting dysmenorrhea were detected for both women who were underweight (odds ratio (OR) 1.34, 95% confidence interval (CI) 1.15, 1.57) and obese (OR 1.22, 95% CI 1.11, 1.35). Compared to women who remained at normal weight or overweight over time, significant risk was detected for women who: remained underweight or obese (OR 1.33, 95% CI 1.20, 1.48), were underweight despite weight gain (OR 1.33, 95% CI 1.12, 1.58), became underweight (OR 1.28, 95% CI 1.02, 1.61). However the higher risk among obese women disappeared when they lost weight (OR 1.06, 95% CI 0.85, 1.32).A U-shaped association was revealed between dysmenorrhea and BMI, revealing a higher risk of dysmenorrhea for both underweight and obese women. Maintaining a healthy weight over time may be important for women to have pain-free periods