Two approaches toward constrained vector optimization and identity of the solutions

In this paper we deal with a Fritz John type constrained vector optimization problem. In spite that there are many concepts of solutions for an unconstrained vector optimization problem, we show the possibility “to double” the number of concepts when a constrained problem is considered. In particular we introduce sense I and sense II isolated minimizers, properly efficient points, efficient points and weakly efficient points. As a motivation leading to these concepts we give some results concerning optimality conditions in constrained vector optimization and stability properties of isolated minimizers and properly efficient points. Our main investigation and results concern relations between sense I and sense II concepts. These relations are proved mostly under convexity type conditions. Key words: Constrained vector optimization, Optimality conditions, Stability, Type of solutions and their identity, Vector optimization and convexity type conditions.

Increase-along-rays property for vector functions

In this paper we extend to the vector case the notion of increasing along rays function. The proposed definition is given by means of a nonlinear scalarization through the so-called oriented distance function from a point to a set. We prove that the considered class of functions enjoys properties similar to those holding in the scalar case, with regard to optimization problems, relations with (generalized) convex functions and characterization in terms of Minty type variational inequalities. Key words: generalized convexity, increase-along-rays property, star-shaped set, Minty variational inequality.

Minty variational inequalities, increase-along-rays property and optimization

Let E be a linear space, K E and f : K ? R. We put in terms of the lower Dini directional derivative a problem, referred to as GMV I(f ,K), which can be considered as a generalization of the Minty variational inequality of differential type (for short, MV I(f ,K)). We investigate, in the case of K star-shaped (for short, st-sh), the existence of a solution x of GMV I(f ,K) and the property of f to increase-along-rays starting at x (for short, f IAR(K, x )). We prove that GMV I(f ,K) with radially l.s.c. function f has a solution x ker K if and only if f IAR(K, x ). Further, we prove, that the solution set of GMV I(f ,K) is a convex and radially closed subset of kerK. We show also that, if GMV I(f ,K) has a solution x K, then x is a global minimizer of the problem f(x) ? min, x K. Moreover, we observe that the set of the global minimizers of the related optimization problem, its kernel, and the solution set of the variational inequality can be different. Finally, we prove, that in case of a quasi-convex function f, these sets coincide. Key words: Minty variational inequality, Generalized variational inequality, Existence of solutions, Increase along rays, Quasi-convex functions.

Second-order mollified derivatives and optimization

The class of strongly semicontinuous functions is considered. For these functions the notion of mollified derivatives, introduced by Ermoliev, Norkin and Wets, is extended to the second order. By means of a generalized Taylor's formula, second order necessary and sufficient conditions are proved for both unconstrained and constrained optimizationMollifiers, optimization, smooth approximations, strong semicontinuity

First order optimality conditions in set-valued optimization

A a set-valued optimization problem minC F(x), x 2 X0, is considered, where X0 X, X and Y are Banach spaces, F : X0 Y is a set-valued function and C Y is a closed cone. The solutions of the set-valued problem are defined as pairs (x0, y0), y0 2 F(x0), and are called minimizers. In particular the notions of w-minimizer (weakly efficient points), p-minimizer (properly efficient points) and i-minimizer (isolated minimizers) are introduced and their characterization in terms of the so called oriented distance is given. The relation between p-minimizers and i-minimizers under Lipschitz type conditions is investigated. The main purpose of the paper is to derive first order conditions, that is conditions in terms of suitable first order derivatives of F, for a pair (x0, y0), where x0 2 X0, y0 2 F(x0), to be a solution of this problem. We define and apply for this purpose the directional Dini derivative. Necessary conditions and sufficient conditions a pair (x0, y0) to be a w-minimizer, and similarly to be a i-minimizer are obtained. The role of the i-minimizers, which seems to be a new concept in set-valued optimization, is underlined. For the case of w-minimizers some comparison with existing results is done. Key words: Vector optimization, Set-valued optimization, First-order optimality conditions.

Variational inequalities in vector optimization

In this paper we investigate the links among generalized scalar variational inequalities of differential type, vector variational inequalities and vector optimization problems. The considered scalar variational inequalities are obtained through a nonlinear scalarization by means of the so called ”oriented distance” function [14, 15]. In the case of Stampacchia-type variational inequalities, the solutions of the proposed ones coincide with the solutions of the vector variational inequalities introduced by Giannessi [8]. For Minty-type variational inequalities, analogous coincidence happens under convexity hypotheses. Furthermore, the considered variational inequalities reveal useful in filling a gap between scalar and vector variational inequalities. Namely, in the scalar case Minty variational inequalities of differential type represent a sufficient optimality condition without additional assumptions, while in the vector case the convexity hypothesis was needed. Moreover it is shown that vector functions admitting a solution of the proposed Minty variational inequality enjoy some well-posedness properties, analogously to the scalar case [4].

First order optimality condition for constrained set-valued optimization

A constrained optimization problem with set-valued data is considered. Different kind of solutions are defined for such a problem. We recall weak minimizer, efficient minimizer and proper minimizer. The latter are defined in a way that embrace also the case when the ordering cone is not pointed. Moreover we present the new concept of isolated minimizer for set-valued optimization. These notions are investigated and appear when establishing first-order necessary and sufficient optimality conditions derived in terms of a Dini type derivative for set-valued maps. The case of convex (along rays) data is considered when studying sufficient optimality conditions for weak minimizers. Key words: Vector optimization, Set-valued optimization, First-order optimality conditions.

Adrenomedullin in pancreatic carcinoma. a case-control study of 22 patients

Pancreatic carcinoma is a leading cause of cancer-related death. Reduction of the diagnostic delay is mandatory. Adrenomedullin (AM) is overexpressed in pancreatic cancer. A case-control study including 12 patients with pathological diagnosis of pancreatic carcinoma and 10 healthy controls was conducted at our Institution. Blood samples were obtained at the time of hospitalization and post-operatively for cases. Controls’ samples were obtained from healthy volunteers. AM was measured by using enzyme immunoassay method. AM showed significant increase in pancreatic carcinoma patients vs controls (4.51 ng/ml vs 1.91 ng/ml, p value = 0.04) regardless of tumor stage, differentiation, resecability/unresecability, diabetes. A cut-off of 1.75 ng/ml reaches a sensibility of 83% and a specificity of 70% (p value = 0.0147; CL 95%; AUC 0.767). The increase of AM didn’t correlate with the increase of other common tumor markers (CA 19-9 and CEA), nor direct bilirubin. These data confirm the utility of studying the role of AM in pancreatic cancer, in order to achieve an early diagnosis in high risk populations

Existence of solutions and star-shapedness in Minty variational inequalities

Minty variational inequalities have proven to define a stronger notion of equilibrium than Stampacchia variational inequalities. This conclusion leads to argue that some regularity, e.g. convexity or generalized convexity, has to be implicit for any function that admits a solution of the corresponding integrable Minty variational inequality. Quasi-convexity arises almost naturally when functions of one variable are involved. However some differences appear when considering functions of several variables. In this case we show that existence of a solution does not necessarily imply quasi-convexity of the function and instead we prove that the level sets of the function must be star-shaped at a point which is a solution of the Minty variational inequality.Minty variational inequality, generalized convexity, star-shaped sets, existence of solutions