Thesis Summary:

The present mathematical engineer's memory goes into the problematic of finding conditions that allow to characterize, in some sense, the property of Strong Duality, in order to establish optimality conditions in solutions of an optimization problem with a quadratic and a linear constraint. Whenever an optimization problem is being raised, it is worth asking if there is another problem associated with the previous one, which allows, among other things, to solve the first one in a simpler way, taking advantage of the properties that the second one has, such as moving from a problem with Restrictions to one without restrictions, these are known as the primal and dual problems. Strong Duality property is the condition that arises that the dual of a problem has a solution and that it matches the optimal value of the primal problem. We structure the work in the following way: in the theoretical chapters 2 and 3, we introduce definitions and properties to the formulation of the primal and dual problem in a general context of the mathematical point of view. Some references to previous research are given to contextualize the different approaches in which this problem has been addressed. In Chapter 4 we focus on studying the property of Strong Duality according to the approach given in [5], where the result is interesting, with respect to the description of the set cone (F(C)(;0)+R2++) and The Strong Duality characterization for the general case with a constraint. In Chapter 5, we study the Strong Duality property and its relation to the optimality conditions, for this, we introduce the regularized Lagrangian. Chapter 6 presents the novelties for a quadratic problem with a quadratic constraint and a linear one. On the one hand we assure that the set cone (F(C)(;0)+R2++) is always convex. We deliver separately necessary and sufficient conditions for Strong Duality to be met, which, in turn, generates new optimality conditions for the non-convex case. To finalize we give a result that characterizes the existence of a minimum for a quadratic function, on a related subspace.