Resumen de la Tesis:

The main objective of this doctoral thesis is to analyze and characterize the Lagrangian strong duality property for a non-convex optimization scalar problem subject to a single constraint, beyond those existing in the state of art. Our results are applied to the non-convex quadratic case. In particular, we obtained a relaxed version of Dine’s theorem, associated to the quadratic minimization problem with an inequality, also the quadratic type together with several similarities of an affine type. In that sense, we discussed the case when the optimal value is not finite. Next, we established a characterization of the geometric type of strong duality property for the non convex quadratic problem, without the need to assume Slater’s hypothesis, hence obtaining necessary and sufficient conditions of optimality. Finally, in light of the results on non-emptiness of the solution set, obtained by Frank Wolfe, our version sets asymptotically linear sets. In part two of the study, we established a topological and geometric characterization of the property of strong duality, for a general non-convex problem subject to a constraint of the equality type, together with constraints of the geometric type, from which the convexity of the conical envelope associated with the joint image determined by the functions of the original problem. As an application, we checked the validity of KKT conditions without assuming standard regularity condition. In the final part of this thesis, we study in detail the standard quadratic problem, by replacing the usual simplex with a convex cone, pointed, not necessarily polyhedral, that admits a compact base, for which we associate three different duals to this problem, in each case, we characterize the property of strong duality in terms of the Hessian’s copositivity of the objective function, along with some conditions of optimality. In this sense, for the case of two dimensions, we characterized when every local solution is global.

Publicaciones Originadas de la Tesis (ISI)

Gabriel CARCAMO, Fabián FLORES-BAZáN: Strong duality and KKT conditions in nonconvex optimization with a single equality constraint and geometric constraint. Mathematical Programming, vol. 168, 1-2, pp. 369-400, (2018).

Gabriel CARCAMO, Fabián FLORES-BAZáN: A geometric characterization of strong duality in nonconvex quadratic programming with linear and nonconvex quadratic constraints. Mathematical Programming, vol. 145, 1-2, pp. 263-290, (2014).