|Stochastic Systems Group|
Prof. Asuman Ozdaglar
In this talk, I will talk about my thesis work, which presents a new development of Lagrange multiplier theory for constrained optimization problems. In the past, the main research in Lagrange multiplier theory has focused on developing general and easily verifiable conditions on the constraint set, called constraint qualifications, that guarantee the existence of Lagrange multipliers for the optimization problem of interest. Our goal in this work is to generalize, unify, and streamline the theory of constraint qualifications. Our development is based on a set of enhanced Fritz John optimality conditions and the notion of constraint pseudonormality. It allows for an abstract set constraint (in addition to equalities and inequalities), and highlights the essential structure of the constraints, based in part upon nonsmooth analysis concepts. This approach also yields identification of different types of Lagrange multipliers, which carry significant sensitivity information regarding the constraints that directly affect the optimal cost change.
In the second part of the talk, I will focus on the extension of this theory to nonsmooth problems under convexity assumptions. Using an approach based on convex analysis, we develop Fritz John-type optimality conditions for these problems. Through an extended notion of constraint pseudonormality, this development provides an alternative pathway to strong duality results of convex programming.
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