1,720,984 research outputs found
SOME REMARKS ON THE ENVELOPE OF A FAMILY OF CURVES
No full textBy means of Clarke’s differentiability theory, a generalization of the notion of envelope for a nondifferentiable family of curves is considered. In particular, we propose an extension of the celebrated conditions fulfilled by an envelope curve in the continuously differentiable case. An application in the theory of long-run cost curves in Economics, is also provided
Convex Analysis in Groups and Semigroups and Applications to Separation and Optimization
No full textThe aim of this paper is to investigate some fundamental properties of convex sets, cones, and functions in groups and semigroups. In particular, we introduce the concept of quasi core of a convex set, which allows us to provide separation theorems between convex sets in groups where at least one of them has nonempty quasi core. We also analyze optimization problems defined on groups and semigroups and provide saddle point necessary or sufficient optimality conditions by considering suitable Lagrangian functions associated with the given problem
On the Image Space Analysis for Vector Variational Inequalities
No full textThe theory of Vector Variational Inequalities can be based on the image space analysis and theorems of the alternative or separation theorems. Exploiting the separation approach for suitable approximations of the image associated to a Vector Variational Inequality, Lagrangian-type necessary optimality conditions are obtained. Applications to vector optimization problems and to vector traffic equilibria are briefly outlined
Conic separation for vector optimization problems
No full textIn this paper, we analyse the relationships between conic and vector separation of two sets.
Applying such results in the Image Space associated with a vector optimization problem, we show that the existence of a particular instance of vector separation is equivalent to the existence of a vector saddle point for the Lagrangian function associated with the vector optimization problem
Optimality conditions for vector variational inequalities via image space analysis
No full textIn this paper, we outline the main features of the image space analysis for vector variational inequalities with cone constraints. Exploiting a suitable separation scheme in the associated image space, we derive saddle point and Karush-Kuhn-Tucker type optimality conditions for the given vector variational inequality
Near equality and almost convexity of functions with applications to optimization and error bounds
No full textIn this paper, we investigate near equality and almost convexity of extended real valued functions defined on finite-dimensional Euclidean spaces. The main result states that an almost convex function (respectively, its domain, lower level set) is nearly equal to (respectively, the domain, lower level set of) its closure, convex hull and closed convex hull. It is proved that almost convexity of an extended real valued function is equivalent to near equality of itself and another almost convex function. Moreover, it is shown that the operations given by sum, scalar multiple, pointwise supremum, epi-sum and epi-multiple of almost convex functions preserve almost convexity, the formulation of the subdifferential of sum and scalar multiple of almost convex functions on the relative interior of their domain is analogous to that of convex functions under suitable additional assumptions and the proximal average of almost convex functions enjoys analogous properties of lower semi-continuous and convex functions. The episum of almost convex functions is proved to be convex under additional assumptions and the Moreau envelope of an almost convex function is shown to be convex and continuously differentiable with the gradient given by the one related to its closure. Applications to almost convex optimization problems are provided, in particular, under suitable assumptions, the solution set is proved to be nearly convex and the solution sets of two almost convex optimization problems are shown to be nearly equal if the related objective functions are nearly equal. Another application shows that the classical Hoffman' error bound holds for almost convex inequalities under a generalized Slater condition. Several examples are given to illustrate these results
Convex analysis in Zn and applications to integer linear programming
No full textIn this paper, we compare the definitions of convex sets and convex functions in finite dimensional integer spaces introduced by Adivar and Fang, Borwein, and Giladi, respectively. We show that their definitions of convex sets and convex functions are equivalent. We also provide exact formulations for convex sets, convex cones, affine sets, and convex functions and we analyze the separation between convex sets in finite dimensional integer spaces. As an application, we consider an integer linear programming problem with linear inequality constraints and obtain some necessary or sufficient optimality conditions by employing the image space analysis. We finally provide some computational results based on the above-mentioned optimality conditions
Linear separation for G-semidifferentiable problems
No full textThis paper aims to study, in the image space, a certain approximation of a constrained extremum problem: the approximation is obtained by substituting the functions involved with their G-derivatives. It is shown that, in the hypothesis of G-differentiability, the linear separation between a given convex set and the image of the approximated problem is equivalent to the semistationarity of the Lagrangian function associated to the original problem. Similar results are obtained considering approximations involving the Dini-Hadamard derivatives
Optimization Problems with Cone Constraints in Groups and Semigroups: An Approach Based on Image Space Analysis
No full textIn this paper, a class of optimization problems with cone constraints in groups and semigroups is investigated by exploiting the image space analysis. Optimality is proved by means of separation arguments in the image space associated with the given problem, which turns out to be equivalent to the existence of saddle points of generalized Lagrangian functions under suitable assumptions. In particular, Lagrangian-type sufficient or necessary optimality conditions are obtained by introducing convex-like functions and using separation theorems between convex sets in groups and semigroups obtained by Li and Mastroeni
First- and Second-Order Optimality Conditions for Quadratically Constrained Quadratic Programming Problems
No full textWe consider a quadratic programming problem with quadratic cone constraints and an additional geometric constraint. Under suitable assumptions, we establish necessary and sufficient conditions for optimality of a KKT point and, in particular, we characterize optimality by using strong duality as a regularity condition. We consider in details the case where the feasible set is defined by two quadratic equality constraints and, finally, we analyse simultaneous diagonalizable quadratic problems, where the Hessian matrices of the involved quadratic functions are all diagonalizable by means of the same orthonormal matrix
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