Nonstandard duality concepts in conic and quadratic optimization

This thesis is centred around the topic of duality. It presents the classical duality theories in optimization and identifies their key ingredients as convexity and constraint qualification. The thesis answers questions on what we can salvage from the theories if these conditions fail to hold.;Then we look at relaxing the other key factor in duality theories; the constraint qualification or regularity condition. We present a strong duality theory for optimization over symmetric cones without assuming any constraint qualification. We show that the dual problem is defined over a homogeneous cone and we analyze its complexity. Specializing the dual to semidefinite optimization the result improves the best currently known complexity bound.;Finally, we show how the classical theory changes if the computations are carried out in finite precision arithmetic. This is an important issue in practice since duality theorems provide the theoretical foundation for stopping criteria and optimality conditions in optimization software. Along with a brief overview of the existing results we present a new set of stopping criteria for convex conic optimization. We prove that the criteria do not change the complexity of the algorithm they are applied with.;A long list of open problems and topics for future research conclude the thesis.;First we present a special duality theory for systems of quadratic functions without assuming convexity. The results stern from the S-lemma, a fundamental result in control theory. We show several proofs for this basic results, then investigate the theory behind each of them. New theorems about nonconvex quadratic systems are also proved.