Description:
The paper deals with optimization problems with uncertain constraints and linear perturbations of the objective function, which are associated with given families of perturbation functions whose dual variable depends on the uncertainty parameters. More in detail, the paper provides characterizations of stable strong robust duality and stable robust duality under convexity and closedness assumptions. The paper also reviews the classical Fenchel duality of the sum of two functions by considering a suitable family of perturbation functions.
Description:
The paper deals with optimization problems with uncertain constraints and linear perturbations of the objective function, which are associated with given families of perturbation functions whose dual variable depends on the uncertainty parameters. More in detail, the paper provides characterizations of stable strong robust duality and stable robust duality under convexity and closedness assumptions. The paper also reviews the classical Fenchel duality of the sum of two functions by considering a suitable family of perturbation functions.
Description:
This paper provides new Farkas-type results characterizing the inclusion of a given set, called contained set, into a second given set, called container set, both of them are subsets of some locally convex space, called decision space. The contained and the container sets are described here by means of vector functions from the decision space to other two locally convex spaces which are equipped with the partial ordering associated with given convex cones. These new Farkas lemmas are obtained via the complete characterization of the conic epigraphs of certain conjugate mappings which constitute the core of our approach. In contrast with a previous paper of three of the authors (Dinh et al. in J Optim Theory Appl 173:357-390, 2017), the aimed characterizations of the containment are expressed here in terms of the data.
Description:
This paper provides new Farkas-type results characterizing the inclusion of a given set, called contained set, into a second given set, called container set, both of them are subsets of some locally convex space, called decision space. The contained and the container sets are described here by means of vector functions from the decision space to other two locally convex spaces which are equipped with the partial ordering associated with given convex cones. These new Farkas lemmas are obtained via the complete characterization of the conic epigraphs of certain conjugate mappings which constitute the core of our approach. In contrast with a previous paper of three of the authors (Dinh et al. in J Optim Theory Appl 173:357-390, 2017), the aimed characterizations of the containment are expressed here in terms of the data.
Description:
This paper considers an uncertain convex optimization problem, posed in a locally convex decision space with an arbitrary number of uncertain constraints. To this problem, where the uncertainty only affects the constraints, we associate a robust (pessimistic) counterpart and several dual problems. The paper provides corresponding dual variational principles for the robust counterpart in terms of the closed convexity of different associated cones.
Description:
The main purpose of this paper consists of providing characterizations of the inclusion of the solution set of a given conic system posed in a real locally convex topological space into a variety of subsets of the same space defined by means of vector-valued functions. These Farkas-type results are used to derive characterizations of the weak solutions of vector optimization problems (including multiobjective and scalar ones), vector variational inequalities, and vector equilibrium problems.
Description:
The main purpose of this paper consists of providing characterizations of the inclusion of the solution set of a given conic system posed in a real locally convex topological space into a variety of subsets of the same space defined by means of vector-valued functions. These Farkas-type results are used to derive characterizations of the weak solutions of vector optimization problems (including multiobjective and scalar ones), vector variational inequalities, and vector equilibrium problems.
Description:
This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f(x) ≥ 0 which are consequences of a composite convex inequality (S ° g)(x) ≤ 0 on a closed convex subset of a given locally convex topological vector space X, where f is a proper lower semicontinuous convex function defined on X, S is an extended sublinear function, and g is a vector-valued S-convex function. In parallel, associated versions of a stable Farkas lemma, considering arbitrary linear perturbations of f, are also given. These new versions of the Farkas lemma, and their corresponding stable forms, are established under the weakest constraint qualification conditions (the so-called closedness conditions), and they are actually equivalent to each other, as well as quivalent to an extended version of the so-called Hahn-Banach-Lagrange theorem, and its stable version, correspondingly. It is shown that any of them implies analytic and algebraic versions of the Hahn-Banach theorem and the Mazur-Orlicz theorem for extended sublinear functions.
Description:
This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f(x) ≥ 0 which are consequences of a composite convex inequality (S ° g)(x) ≤ 0 on a closed convex subset of a given locally convex topological vector space X, where f is a proper lower semicontinuous convex function defined on X, S is an extended sublinear function, and g is a vector-valued S-convex function. In parallel, associated versions of a stable Farkas lemma, considering arbitrary linear perturbations of f, are also given. These new versions of the Farkas lemma, and their corresponding stable forms, are established under the weakest constraint qualification conditions (the so-called closedness conditions), and they are actually equivalent to each other, as well as quivalent to an extended version of the so-called Hahn-Banach-Lagrange theorem, and its stable version, correspondingly. It is shown that any of them implies analytic and algebraic versions of the Hahn-Banach theorem and the Mazur-Orlicz theorem for extended sublinear functions.