This special issue is dedicated to the 80th birthday of Professor Alexander Rubinov (1940–2006) and celebrating his achievements and the impact of his work.
Professor Alexander Rubinov was a leading expert on nonlinear analysis, abstract convexity, optimization, mathematical economics, and their various applications. He made significant contributions to nonsmooth analysis and optimization, monotonic analysis, and non-linear Lagrange-type functions. During his productive career, he has co-authored 14 monographs, 3 textbooks, and more than 250 journal papers, which greatly impacted on research in optimization and its many applications. Professor Rubinov supervised more than 40 doctoral students who now work in many countries around the world.
Alexander Rubinov was born in Leningrad, now S.-Petersburg, Russia, and graduated from Leningrad State University. He completed his PhD at the Institute of Mathematics of the Siberian Branch of the Academy of Sciences of the USSR in Novosibirsk and an advanced doctorate at the Computer Centre of the Academy of Sciences of the USSR in Moscow. He then held positions in Leningrad, Novosibirsk and Kalinin (Russia), Baku (Azerbaijan) and Beersheba (Israel). He spent the last ten years of his career at the University of Ballarat, Australia. These were the most productive years in his career. Thanks to Alexander Rubinov, Ballarat became a centre of attraction for researchers worldwide, his office always hosting visitors, both junior and senior, from all around the world.
Alexander Rubinov was admired not only for his fundamental work in optimization, but also for his great personal character, his readiness to share knowledge, and his avid devotion to promoting research activities in the optimization community.
This special issue contains papers on nonsmooth analysis, abstract convexity, optimization and their applications.
In the paper ‘Two iterative processes generated by regular vector fields in Banach spaces’ by S. Reich and A. Zaslavski, the class of regular vector fields is considered and the behaviour of the values of the objective function for two iterative processes generated by a regular vector field in the presence of computational errors is analysed.
The paper ‘Turnpike properties of discrete dispersive dynamical systems with a Lyapunov function’ by A. Zaslavski studies turnpike properties for discrete disperse dynamical systems generated by set-valued mappings. These properties were introduced by Alexander Rubinov in 1980.
H. Bui, R. Burachik, A. Kruger and D. Yost extend conditions for zero duality gap to the context of non-convex and nonsmooth optimization using tools provided by the theory of abstract convexity in the paper ‘Zero duality gap conditions via abstract convexity’.
In the paper ‘Convex analysis of minimal time and signed minimal time functions’ by N.M. Nam, C. Dang, B. Mordukhovich and M. Wells, the class of minimal time functions in the general setting of locally convex topological vector spaces is considered, and subdifferential formulas for the signed minimal time and distance functions are obtained.
The bi-criteria portfolio optimization model is discussed in the paper: ‘Portfolio optimization under a minimax rule revisited’ by K. Meng, H. Yang, X. Yang and C.K.W. Yu.
In the paper ‘On some efficiency conditions for vector optimization problems with uncertain cone constraints: a robust approach via set-valued inclusions’ by A. Uderzo, several types of efficiency conditions are established for vector optimization problems with cone constraints affected by uncertainty.
Bregman Voronoi cells are considered in the paper: ‘On farthest Bregman Voronoi cells’ by J.E. Martinez-Legaz, J.M. Tamadoni Jahromi and E. Naraghirad.
The paper ‘On duality for nonconvex minimization problems within the framework of abstract convexity’ by M. Syga and E. Bednarczuk applies the perturbation function approach to develop the Lagrangian and conjugate duals for minimization problems of the sum of two, generally nonconvex, functions.
In the paper ‘The smoothing objective penalty function method for two-cardinality sparse constrained optimization problems’ by M. Jiang, Z. Meng, R. Shen and C. Dang, two algorithms are developed to solve two-cardinality sparse constrained optimization problems using the smoothing penalty function and smoothing objective penalty function.
V. Peiris and N. Sukhorukova in the paper ‘The extension of linear inequality method for generalised rational Chebyshev approximation to approximation by general quasilinear functions’ demonstrate that a well-known linear inequality method developed for rational Chebyshev approximation is equivalent to the application of the bisection method used in quasiconvex optimization.
A unifying general framework of quantitative primal and dual sufficient and necessary error bound conditions covering linear and nonlinear, local and global settings is proposed in the paper ‘Error bounds revisited’ by N. Duy Cuong and A.Y. Kruger.
Conditions for Hölder calmness and upper Hölder continuity of optimal solution sets to perturbed optimization problems in finite dimensions are explored in the paper ‘On Hölder calmness of minimizing sets’ by D. Klatte and B. Kummer.
In the paper ‘Efficient projection onto the intersection of a half-space and a box-like set and its generalized Jacobian’ by B. Wang, L. Lin and Y.-J. Liu, projections onto the intersection of a half-space and a box-like set are studied using the Lagrangian duality theory and a semismooth Newton algorithm.
C.Y. Kaya in his paper ‘Observer path planning for maximum information’ studies finding an optimal path for an observer, or sensor, moving at a constant speed, which is to estimate the position of a stationary target, using only bearing angle measurements.
D. Ghilli, D.A. Lorenz and E. Resmerita in the paper ‘Nonconvex flexible sparsity regularization: theory and monotone numerical schemes’ show convergence of the regularization method and establish convergence properties of a couple of majorization approaches for the operator equations.
Infinite horizon optimal control problems with time averaging and time discounting criteria are considered in the paper ‘LP-related representations of Cesáro and Abel limits of optimal value functions’ by V. Gaitsgory and I. Shvartsman.
V. Peiris, N. Sukhorukova, J. Ugon and R. Diaz Millan in the paper: ‘Multivariate approximation by polynomial and generalised rational functions’ consider an optimization-based approach to multivariate Chebyshev approximation on a finite grid. Two models are discussed: multivariate polynomial approximation and multivariate generalized rational approximation.
A robust projection twin support vector machine, where a new truncated L2-norm distance measure is applied to boost the robustness of the classifier when encountering a large number of outliers is proposed in the paper ‘Robust projection twin support vector machine via DC Programming’ by G. Li, L. Yin, L. Yang and Z. Wu.
A. Zaffaroni in his paper ‘Separation, convexity, and polarity, in the space of normlinear functions’ studies separation properties for subsets of the space of normlinear functions on a Banach space, i.e. the sum of a linear function and a multiple of the norm.
The authors and editors of this special issue worked closely with Professor Rubinov as his students and colleagues. We are indebted to Alexander Rubinov for his kindness, his encouragement, for sharing his knowledge and ideas, and for his willingness to help even in his last days. We treasure those moments we were lucky to spend with him, and we greatly miss him. This special issue is our modest tribute to Alexander Rubinov.
Disclosure statement
No potential conflict of interest was reported by the author(s).