Thesis Summary:

The main goal of this PhD thesis is to investigate weakly efficient vectorial minima under relaxed hypotheses of convexity for the involved functions. The organization of the thesis is as follows. In the first part, our main concern is the existence of solutions for the compact case, without hypotheses of convexity and differentiability. Afterwards, by using notions of generalized convexity and recession analysis we treat the unbounded case for finite dimensional spaces. We study also problems for which the range of the involved vectorial function is contained in a fi- nite or in an infinite dimensional space. For the latter case, we give several conditions for the nonemptiness and the boundedness of the set of weak solutions. In particular, we deal with the case of polyhedral and Lorentz cones. Moreover, we seek for weakly efficient solutions for the case when the domain of the vectorial function is a set of the real numbers. By the way, we obtain relationships for the nonemptiness of the set of weak minima and the set of minima of the component functions. In the second part, we study theorems of alternative for vectorial optimization problems and we obtain optimal conditions for such theorems. With the aid of such results we characterize bi-dimensional spaces, we perform an scalarization by means of the positive polar cone for obtaining weakly efficient points, we characterize the zero duality gap and we obtain optimality conditions of Fritz-John type for vectorial optimization problems. Finally, in connection with the first part, in the third part we seek for weak minima, when the domain of the vectorial function is a subset of the real numbers, the range is bi-dimensional and the component functions are quasiconvex without any hypothesis of differentiability. This case is totally characterized and this allows us to develop a finite time algorithm for calculating weakly efficient solutions and the supremum of the set of weakly efficient minima.

ISI Publications from the Thesis

Fabián FLORES-BAZáN, Cristian VERA: Unifying efficiency and weak efficiency in generalized quasiconvex vector minimization on the real-line. International Journal of Optimization: Theory, Methods and Appl., vol 1, 3, pp. 247-265, (2009)

Fabián FLORES-BAZáN, Nicolás HADJISAVVAS, Cristian VERA: An Optimal Alternative Theorem and Applications to Mathematical Programming. Journal of Global Optimization, vol. 37, 2, pp. 229-243 (2007)

Fabián FLORES-BAZáN, Cristian VERA: Characterization of the nonemptiness and compactnes of solution sets in convex/nonconvex vector optimization. Journal of Optimization Theory and Applications, vol. 130, 2, pp. 185-207, (2006).