This book is a research monograph with specialized mathematical preliminaries. It presents an original range space and conic theory of infinite dimensional polyhedra (closed convex sets) and optimization over polyhedra in separable Hilbert spaces, providing, in infinite dimensions, a continuation of the author's book:
A Conical Approach to Linear Programming, Scalar and Vector Optimization
Problems, Gordon and Breach Science Publishers, Amsterdam, 1997.
It expands and improves author's new approach to the Maximum Priciple for norm oprimal control of PDE, based on theory of convex cones, providing shaper results in various Hilbert space and Banach space settings. It provides a theory for convex hypersurfaces in lts and Hilbert spaces. For these purposes, it introduces new results and concepts, like the generalizations to the non compact case of cone capping and of the Krein Milman Theorem, an extended theory of closure of pointed cones, the notion of beacon points, and a necessary and sufficient condition of support for void interior closed convex set (complementing the Bishop Phelps Theorem), based on a new decomposition of non closed non pointed cones with non closed lineality space.
.- Introduction.
.- Basic Facts of Set Theory.
.- Linear Spaces.
.- Rudiments of General Topology.
.- Filters: the Fifth Equivalence.
.- Hahn Banach andSeparation Theorems.
.- Locally Convex and Barrelled Spaces.
.- Metrics and pseudometrics, Norms and Pseudonorms.
.- Topological Form of Hahn Banach and Separation Theorems.
.- Extreme points, Faces, Support and the KreinMilman Theorem.
.- Function Spaces.
Paolo d'Alessandro is a former professor at the Department of Mathematics of the Third University of Rome (Italy).
His primary research interests lie in System Theory (including foundations, linear time-variant and bilinear systems), Control Theory, Optimization, Linear Programming and foundations of Probability and Stochastic Systems . He has made significant contributions to LP, by introducing various range space and conic methods for linear programming, and has also applied these results to control problems for linear dynamic systems.
His current research focuses on extending methods to infinite-dimensional Optimization and applying range space and conic techniques to norm-optimal control of partial differential equations (PDEs).
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