Narrow Results

The use of evolutionary strategies (ESs) to solve problems with multiple objectives [known as vector optimization problems (VOPs)] has attracted much attention recently. Being population-based approaches, ESs offer a means to find a set of Pareto-optimal solutions in a single run. Differential evolution (DE) is an ES that was developed to handle optimization problems over continuous domains. The objective of this paper is to introduce a novel Pareto-frontier differential evolution (PDE) algorithm to solve VOPs. The solutions provided by the proposed algorithm for two standard test problems, outperform the "strength Pareto evolutionary algorithm", one of the state-of-the-art evolutionary algorithm for solving VOPs.

In this paper cone convex and related functions have been studied. The concept of cone semistrictly convex functions on topological vector spaces is introduced as a generalization of semistrictly convex functions. Certain properties of these functions have been established and their interrelations with cone convex and cone subconvex functions have been explored. Assuming the functions to be cone subconvex, sufficient optimality conditions are proved for a vector valued minimization problem over topological vector spaces, involving Gâteaux derivatives. A Mond-Weir type dual is associated and weak and strong duality results are proved.

In this paper, we establish several proper separation theorems for an element and a convex set and for two convex sets in terms of their quasi-relative interiors. Then, we prove that the separation theorem given by [Cammaroto, F and B Di Bella (2007). On a separation theorem involving the quasi-relative. Proceedings of the Edinburgh Mathematical Society, 50(3), 605–610] in Theorem 2.5, is in fact a proper separation theorem for two convex sets in which the classical interior is replaced by the quasi-relative interior. Besides, we extend some known results in the literature, such as [Adán, M and V Novo (2004). Proper efficiency in vector optimization on real linear spaces. Journal of Optimization Theory and Applications, 121, 515–540] in Theorem 2.1 and [Edwards, R (1965). Functional Analysis: Theory and Applications. New York: Reinhart and Winston] in Corollary 2.2.2, through the quasi-relative interior and the quasi-interior, respectively. As an application, we provide Karush–Kuhn–Tucker multipliers for quasi-relative solutions of vector optimization problems. Several examples are given to illustrate the obtained results.

We study the explicit calculation of the set of superhedging portfolios of contingent claims in a discrete-time market model for d assets with proportional transaction costs. The set of superhedging portfolios can be obtained by a recursive construction involving set operations, going backward in the event tree. We reformulate the problem as a sequence of linear vector optimization problems and solve it by adapting known algorithms. The corresponding superhedging strategy can be obtained going forward in the tree. Examples are given involving multiple correlated assets and basket options. Furthermore, we relate existing algorithms for the calculation of the scalar superhedging price to the set-valued algorithm by a recent duality theory for vector optimization problems. The main contribution of the paper is to establish the connection to linear vector optimization, which allows to solve numerically multi-asset superhedging problems under transaction costs.

We study global stability properties for vector optimization problems of the type:

Vector constrained extremum problems and Vector Variational Inequalities are considered, and a separation scheme is introduced. Starting from such a scheme, several theoretical aspects can be developed as well as methods of solution; for instance, scalarization of Vector Optimization is analysed and a method is described which does not require any assumption on the problem. The analysis is extended to a Vector Variational Inequality; this turns out to be equivalent to a scalar Quasi–Variational Inequality.

This paper puts forward a set of parallel algorithms for calculating FFT, which is suitable to the multiple vector systems. The paper also presents a method for vector optimization that increases the minimum length of the inner loop to N. Whereas, α vector optimal approach given in paper [l] only makes the minimum length of inner loop to be .