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Proximal normal analysis in dynamic optimization Loewen, Philip Daniel


Proximal normal analysis is a relatively new technique whose power and breadth of applicability are only now being realized. Given an optimization problem, there are many ways to define a "value function" which describes the changes in the problem's minimum value as certain parameters are varied. The epigraph of this function, namely the set of points lying on or above its graph, is a set whose geometry is intimately connected both with necessary conditions for optimality in the original problem and with the problem's sensitivity to perturbations. Proximal normal analysis is the geometrical technique which allows such information to be derived from a study of this fundamental set. In the first chapter we illustrate the technique in the simple model framework of a finite-dimensional mathematical programming problem, and describe its consequences for parameter sensitivity in optimal control. Chapter II presents a detailed proof of the fundamental geometric result, called the "proximal normal formula", in Hilbert space. The proof is distilled from the more general work of Borwein and Strojwas (1985), who were the first to make this basic ingredient of the method available in infinite dimensions. This extension is of considerable practical interest: in Chapter III it makes possible a proximal normal analysis of state constraints in optimal control, which gives rise to a new form of the maximum principle for state constrained problems. Limiting techniques and existence theorems are key ingredients in proximal normal analysis. Chapter IV gives a new existence theorem for open-loop stochastic optimal control problems in which compactness of the control set is not required, but instead a growth condition is imposed on the problem's running cost. In addition to their independent interest, the methods and results of Chapter IV enable us to use proximal normal analysis to investigate parameter sensitivity in stochastic optimal control in Chapter V. A byproduct of this analysis is a new proof of the Stochastic Maximum Principle which is more direct (if slightly more technical) than the proofs current in the literature, and which provides a rigorous interpretation of the multipliers.

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