Extensions of incomplete oblique projections method for solving rank-deficient least-squares problems

The aim of this paper is to extend the applicability of an algorithm for solving inconsistent linear systems to the rank-deficient case, by employing incomplete projections onto the set of solutions of the augmented system Ax-r = b. The extended algorithm converges to the unique minimal norm solution of the least squares solutions. For that purpose, incomplete oblique projections are used, defined by means of matrices that penalize the norm of the residuals. The theoretical properties of the new algorithm are analyzed, and numerical experiences are presented comparing its performance with some well-known projection methods.Facultad de Ciencias ExactasFacultad de Ingenierí

Extensions of incomplete oblique projections method for solving rank-deficient least-squares problems

The aim of this paper is to extend the applicability of an algorithm for solving inconsistent linear systems to the rank-deficient case, by employing incomplete projections onto the set of solutions of the augmented system Ax-r = b. The extended algorithm converges to the unique minimal norm solution of the least squares solutions. For that purpose, incomplete oblique projections are used, defined by means of matrices that penalize the norm of the residuals. The theoretical properties of the new algorithm are analyzed, and numerical experiences are presented comparing its performance with some well-known projection methods.Facultad de Ciencias ExactasFacultad de Ingenierí

Extensions of incomplete oblique projections method for solving rank-deficient least-squares problems

The aim of this paper is to extend the applicability of an algorithm for solving inconsistent linear systems to the rank-deficient case, by employing incomplete projections onto the set of solutions of the augmented system Ax-r = b. The extended algorithm converges to the unique minimal norm solution of the least squares solutions. For that purpose, incomplete oblique projections are used, defined by means of matrices that penalize the norm of the residuals. The theoretical properties of the new algorithm are analyzed, and numerical experiences are presented comparing its performance with some well-known projection methods.Facultad de Ciencias ExactasFacultad de Ingenierí

Combining filter method and dynamically dimensioned search for constrained global optimization

In this work we present an algorithm that combines the filter technique and the dynamically dimensioned search (DDS) for solving nonlinear and nonconvex constrained global optimization problems. The DDS is a stochastic global algorithm for solving bound constrained problems that in each iteration generates a randomly trial point perturbing some coordinates of the current best point. The filter technique controls the progress related to optimality and feasibility defining a forbidden region of points refused by the algorithm. This region can be given by the flat or slanting filter rule. The proposed algorithm does not compute or approximate any derivatives of the objective and constraint functions. Preliminary experiments show that the proposed algorithm gives competitive results when compared with other methods.The first author thanks a scholarship supported by the International Cooperation Program CAPES/ COFECUB at the University of Minho. The second and third authors thanks the support given by FCT (Funda¸c˜ao para Ciˆencia e Tecnologia, Portugal) in the scope of the projects: UID/MAT/00013/2013 and UID/CEC/00319/2013. The fourth author was partially supported by CNPq-Brazil grants 308957/2014-8 and 401288/2014-5.info:eu-repo/semantics/publishedVersio

Filter-based stochastic algorithm for global optimization

We propose the general Filter-based Stochastic Algorithm (FbSA) for the global optimization of nonconvex and nonsmooth constrained problems. Under certain conditions on the probability distributions that generate the sample points, almost sure convergence is proved. In order to optimize problems with computationally expensive black-box objective functions, we develop the FbSA-RBF algorithm based on the general FbSA and assisted by Radial Basis Function (RBF) surrogate models to approximate the objective function. At each iteration, the resulting algorithm constructs/updates a surrogate model of the objective function and generates trial points using a dynamic coordinate search strategy similar to the one used in the Dynamically Dimensioned Search method. To identify a promising best trial point, a non-dominance concept based on the values of the surrogate model and the constraint violation at the trial points is used. Theoretical results concerning the sufficient conditions for the almost surely convergence of the algorithm are presented. Preliminary numerical experiments show that the FbSA-RBF is competitive when compared with other known methods in the literature.The authors are grateful to the anonymous referees for their fruitful comments and suggestions.The first and second authors were partially supported by Brazilian Funds through CAPES andCNPq by Grants PDSE 99999.009400/2014-01 and 309303/2017-6. The research of the thirdand fourth authors were partially financed by Portuguese Funds through FCT (Fundação para Ciência e Tecnologia) within the Projects UIDB/00013/2020 and UIDP/00013/2020 of CMAT-UM and UIDB/00319/2020

Cutting Planes and a Biased Newton Direction for Minimizing Quasiconvex Functions

H. Scolnik 2 A biased Newton direction is introduced for minimizing quasiconvex functions with bounded level sets. It is a generalization of the usual Newton's direction for strictly convex quadratic functions. This new direction can be derived from the intersection of approximating hyperplanes to the epigraph at points on the boundary of the same level set. Based on that direction, an unconstrained minimization algorithm is presented. It is proved to have global and local-quadratic convergence under standard hypotheses. These theoretical results may lead to di erent methods based oncomputing search directions using only rst order information at points on the level sets. Most of all if the computational cost can be reduced byrelaxing some of the conditions according for instance totheresults presented intheAppendix. Some tests are presented to show the qualitative behavior of the new direction and with the purpose to stimulate further research on these kind of algorithms