This Special Issue of the Journal ‘Optimization’ is dedicated to the ‘International Seminar on Optimization and Related Areas’ (ISORA), which was held at the Instituto de Matemática y Ciencias Afines (IMCA) in Lima (Peru) on 9–13 October 2017.
This series of seminars began in 1993, as a tribute to the memory of Eugen Blum, a Swiss mathematician who spent the last 20 years of his life in Peru, as a Professor at the Universidad Nacional de Ingeniería (UNI), an institution that promotes this seminar every odd year since 1993 after his death. His area of research was Mathematical Optimization, to which he made fundamental contributions. The ISORA meetings are now among the most important regular conferences on Optimization Theory and applications in South America.
The main focus of this Special Issue addresses the growing research interest in Optimization from the theoretical as well as from the application points of view. Its aim is to present advances in Mathematical Optimization and related areas including their wide range of applications.
The thirteenth ISORA meeting was sponsored by several institutions: Universidad Nacional de Ingeniería (Perú), Instituto de Matemática Pura e Aplicada (Brazil), Ambassade de France au Pérou (France) and CIENCIACTIVA-CONCYTEC (Perú). The financial support of these institutions is gratefully acknowledged.
This special issue consists of eight papers on aspects of variational analysis, optimization, and their applications.
We now briefly describe the main contents of the selected publications.
Maicon Marques Alves and Samara Lima ‘On the convergence rate of the scaled proximal decomposition on the graph of a maximal monotone operator (SPDG) algorithm’. In this paper the authors revisit the scaled PDG algorithm and show that its convergence can be derived directly using the Partial Inverse operator. Furthermore, they improve the convergence rate upper bound in the strongly monotone case.
Alfredo Iusem and Vahid Mohebbi ‘Extragradient method for nonsmooth equilibrium problems in Banach spaces’. Generalizing the extragradient method for variational inequalities, the authors derive the extragradient method for solving nonsmooth equilibrium problems in Banach spaces. The authors prove weak convergence of the generated sequence to a solution of the equilibrium problem under certain standard assumptions. Furthermore, they develop a regularization procedure which ensures strong convergence of the generated sequence to a solution of the problem.
Wilfredo Sosa, Jaime Orrillo and Sinval Braga de Freitas ‘From Arrow-Debreu condition to star shape preferences’. The authors say that an extended real-valued function u satisfies the Arrow-Debreu condition, when every point z in the segment joining two points x,x’ such that u(x) > u(x’) > -∞ satisfies the inequality u(z) > u(x’). This condition is slightly weaker than semistrict quasiconcavity. The authors present the main properties of such functions, analyse the relations existing between the Arrow-Debreu condition and the notions of quasiconcavity and pseudoconcavity, and, among other results, show that, under suitable assumptions, a binary relation induced by a function satisfying the Arrow-Debreu condition is a star shaped preference.
Eladio Ocaña and Luis Flores ‘The relationship of inertias between two representations of linear sub-spaces’. In this paper the authors present complete expressions of inertia of matrices involved in a finite linear subspace when they are presented as image and as well as kernel of linear maps.
Rubén López ‘Global stability of interval optimization problems’. In an interval optimization problem, the objective function f is interval-valued, that is, for every x in its domain, f(x) is a closed interval [fL(x),fU(x)] of real numbers. The author observes that an interval optimization problem can be reformulated as a bicriteria problem with objectives fL and fU. He uses variational convergence tools, as well as other convergence notions, to obtain global stability results for such problems and several other related properties.
Felipe Lara and Alfredo Iusem ‘Quasiconvex Optimization Problems and Asymptotic Analysis in Banach Spaces’. In this paper, the authors studied generalized asymptotic (recession) cones for nonconvex and nonclosed sets and its respective generalized asymptotic functions by asymptotic analysis. The paper established three characterizations results for the nonemptiness and compactness of the solution set for noncoercive quasiconvex minimization problems using different asymptotic functions.
Ellen Fukuda, Luis Mauricio Graña Drummond and Fernanda Raupp ‘A barrier-type method for multiobjective optimization’. In this paper, the authors have applied the barrier method to solve the constrained optimization problem and discussed its monotonic real-valued auxiliary equation. Furthermore, few theoretical results are also furnished with an implementable version for seeking local optima and analyse its convergence
Héctor Ramírez C. and Vera Roschina ‘Refining the partition for multifold conic optimization problems’. This paper deals with conic optimization problems, more precisely the definitions of complementarity partitions of conic optimization problems. The authors claim two main results: (1) Showing that various definitions of complementarity partitions are essentially equivalent; (2) Application of a nonsingular linear map on a cone preserves the complementary partition.