Functional optimization by variable-basis approximation schemes

This is a summary of the author’s PhD thesis, supervised by Marcello Sanguineti and defended on April 2, 2009 at Università degli Studi di Genova. The thesis is written in English and a copy is available from the author upon request. Functional optimization problems arising in Operations Research are investigated. In such problems, a cost functional Φ has to be minimized over an admissible set S of d-variable functions. As, in general, closed-form solutions cannot be derived, suboptimal solutions are searched for, having the form of variable-basis functions, i.e., elements of the set spann  G of linear combinations of at most n elements from a set G of computational units. Upper bounds on $${\inf_{f \in S \cap {\rm span}_n\, G}\Phi(f)-\inf_{f \in S}\Phi(f)}$$ are obtained. Conditions are derived, under which the estimates do not exhibit the so-called “curse of dimensionality” in the number n of computational units, when the number d of variables grows. The problems considered include dynamic optimization, team optimization, and supervised learning from data.