Graduate Thesis of Felipe Lara
|Program||PhD in Applied Sciences with mention in Mathematical Engineering, Universidad de Concepción|
|Thesis Title||Second Order Asymptotic Analysis In Optimitacion|
Recently, the concepts of second order asymptotic directions and functions have been introduced and applied to global and vector optimization problems. In this work, we establish some new properties for these two concepts. In particular, in case of a convex set, a complete characterization of the second order asymptotic cone is given. Also, formulas that permit the easy computation of the second order asymptotic function of a convex function are established. It is shown that the second order asymptotic function provides a finer description of the behavior of functions at infinity, than the first order asymptotic function. Furthermore, we show the second order asymptotic function of a given convex one can be seen as the first order asymptotic function of another convex function. We use second order asymptotic analysis to develop necessary and sufficient conditions for the scalar minimization problem in the noncoercive convex case. We give a new existence result for a point to be a proper efficient solution in the multiobjective optimization problem and a sufficient condition for the Domination Property. Finer estimates for the efficient and weak efficient solution sets (and for their second order asymptotic cones) to a convex/quasiconvex vector optimization problems are also provided. We use asymptotic analysis to describe in a more systematic way the behaviour at the infinity of functions in the convex and quasiconvex case. Starting from the formulae for the first and second order asymptotic function in the convex case, we introduce similar notions suitable for dealing with quasiconvex functions. We characterize the nonemptiness and boundedness of the set of minimizers of any lsc quasiconvex function; finally, we also characterize boundedness from below, along lines, of any proper and lsc function.
|Thesis Director(s)||Fabián Flores, Nicolas Hadjisavvas|
|Thesis Project Approval Date||2013, August 30|
|Thesis Defense Date||2015, October 14|
|PDF Thesis||Download Thesis PDF|
ISI Publications from the Thesis
Fabián FLORES-BAZáN, Felipe LARA: Inner and outer estimates for solution sets and their asymptotic cones in vector optimization. Optimization Letters, vol. 6, 7, pp. 1233-1249, (2012).