Inner and outer estimates for the solution sets and their asymptotic cones in vector optimization (original) (raw)

Weak efficiency in multiobjective quasiconvex optimization on the real-line without derivatives

Optimization, 2009

This paper deals with the problem of existence of weakly efficient solution to quasiconvex vector optimization problems in a finite dimensional setting on the real-line. This consideration is motivated by algorithmic purposes, because it is expected that, like in scalar minimization, one must solve a one-dimensional problem to find the next iterate. We start by recalling a notion of nonconvexity weaker than quasiconvexity for vector functions introduced ealier by one of the author in an previous paper. Afterwards, we characterize the nonemptiness and/or compactness of the weakly efficient solution set. Then, this set is described as much as possible in the multiobjective case, and the bicriteria problem is carefully analized when each component is lower semicontinuous and quasiconvex. Several examples showing the applicability of our results are presented, and various algorithms are stated to compute the overall weakly efficient solution set.

Vector duality for convex vector optimization problems by means of the quasi-interior of the ordering cone

Optimization, 2013

We define the quasi-minimal elements of a set with respect to a convex cone and characterize them via linear scalarization. Then we attach to a general vector optimization problem a dual vector optimization problem with respect to quasi-efficient solutions and establish new duality results. By considering particular cases of the primal vector optimization problem we derive vector dual problems with respect to quasi-efficient solutions for both constrained and unconstrained vector optimization problems and the corresponding weak, strong and converse duality statements.

Modified objective function method in nonsmooth vector optimization over cones

Optimization Letters, 2014

In this paper optimality for a nonsmooth vector optimization problem having generalized cone-invex objective and constraint functions is considered. An equivalent η-approximated vector optimization problem is constructed by a modification of the objective function. The relationships between weakly efficient solutions and saddle points of the two problems are studied.

Vector Critical Points and Cone Efficiency in Nonsmooth Vector Optimization

Taiwanese Journal of Mathematics, 2020

In this paper, a nonsmooth vector optimization problem with cone and equality constraints is considered. We establish some relations between the notions of vector critical points in the sense of Fritz John and in the sense of Karush-Kuhn-Tucker and weakly K-efficient and K-efficient solutions for the constrained vector optimization problem in which every component of the involved functions is locally Lipschitz. These relationships are stated under cone-F J-pseudo-invexity and cone-KT-pseudoinvexity hypotheses defined for the considered vector optimization problem with cone inequality and also equality constraints and via the Clarke generalized gradient for vector-valued functions.

On the set of weakly efficient minimizers for convex multiobjective programming

Operations Research Letters, 2008

In this paper by employing an asymptotic approach we develop an existence and stability theory for convex multiobjective programming. We deal with the set of weakly efficient minimizers. To this end we employ a notion of convergence for vector-valued functions close to that due to Lemaire. (C. Vera). must impose an additional condition (see (b) of Theorem 3.1) for having a nonempty solution set. As is shown below this condition is not only sufficient but also necessary.

Proper or Weak Efficiency via Saddle Point Conditions in Cone-Constrained Nonconvex Vector Optimization Problems

Journal of Optimization Theory and Applications, 2019

Motivated by many applications (for instance, some production models in finance require infinitydimensional commodity spaces, and the preference is defined in terms of an ordering cone having possibly emtpy interior), this paper deals with a unified model which involves preference relations that are not necessarily transitive or reflexive. Our study is carried out by means of saddle point conditions for the generalized Lagrangian associated with a cone constrained nonconvex vector optimization problem. We establish a necessary and sufficient condition for the existence of a saddle point in case the multiplier vector related to the objective function belongs to the quasi interior of the polar of the ordering set. Moreover, exploiting suitable Slater-type constraints qualifications involving the notion of quasi-relative interior, we obtain several results concerning the existence of a saddle point, which serve to get efficiency, weak efficiency and proper efficiency. Such results generalize to the nonconvex vector case existing conditions in the literature. As a byproduct, we propose a notion of proper efficient solution for a vector problem with explicit constraints. Applications to optimality conditions for vector optimization problems are provided with particular attention to bicriteria problems where optimality conditions for efficiency, proper efficiency and weak efficiency, are stated, both in a geometric form and by means of the level sets of the objective functions.

On convex vector optimisation

Bulletin of the Australian Mathematical Society, 2000

In this article we present a simple method to deduce necessary conditions for weak minimisation of a convex vector program in a Banach space. Our main tool here will be the generalised Jabcobian of Ralph.

Semistrictly quasiconvex mappings and non-convex vector optimization

Mathematical Methods of Operations Research, 2004

This paper introduces a new class of non-convex vector functions strictly larger than that of P-quasiconvexity, with P⊆ m being the underlying ordering cone, called semistrictly ( m \ −int P)-quasiconvex functions. This notion allows us to unify various results on existence of weakly efficient (weakly Pareto) optima. By imposing a coercivity condition we establish also the compactness of the set of weakly Pareto solutions. In addition, we provide various characterizations for the non-emptiness, convexity and compactness of the solution set for a subclass of quasiconvex vector optimization problems on the real-line. Finally, it is also introduced the notion of explicit ( m \ −int P)-quasiconvexity (equivalently explicit (int P)-quasiconvexity) which plays the role of explicit quasiconvexity (quasiconvexity and semistrict quasiconvexity) of real-valued functions.

Nonconvex Vector Optimization and Optimality Conditions for Proper Efficiency

International Journal of Analysis and Applications, 2022

In this paper, we consider, a new nonlinear scalarization function in vector spaces which is a generalization of the oriented distance function. Using the algebraic type of closure, which is called vector closure, we introduce the algebraic boundary of a set, without assuming any topology, in our context. Furthermore, some properties of this algebraic boundary set are given and present the concept of the oriented distance function via this set in the concept of vector optimization. We further investigate Q-proper efficiency in a real vector space, where Q is some nonempty (not necessarily convex) set. The necessary and sufficient conditions for Q-proper efficient solutions of nonconvex optimization problems are obtained via the scalarization technique. The scalarization technique relies on the use of two different scalarization functions, the oriented distance function and nonconvex separation function, which allow us to characterize the Q-proper efficiency in vector optimization with and without constraints.