Linear Programming in Infinite-dimensional Spaces: Theory and ApplicationsThis book provides a survey of linear programming in semi-infinite and infinite-dimensional spaces. It includes a treatment of duality theory and of the fundamental theory of simplex-like algorithms for linear programs posed over vector spaces which may be infinite-dimensional. However, more than half the book is devoted to a detailed investigation of various types of infinite-dimensional linear program which occur, for example, in approximation theory, optimal control theory, dynamic networks, mass transfer problems and structural design. The approach is inductive; specific problems and algorithms are discussed in detail and the authors proceed from these to more general concepts and results. |
Contents
Infinitedimensional Linear Programs | 1 |
Algebraic Fundamentals | 16 |
Topology and Duality | 35 |
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asymptotically consistent basic feasible solution basic optimal solution basic solutions capacity Chapter choose compact consider constraint map constraint space continuous functions continuous linear program convex convex cone convex hull convex set defined dual pair dual problem dual solution duality gap duality theory example extreme points finite linear program finite number finite value formulation Hence infinite infinite-dimensional linear programs Lemma linearly independent Mackey topology maximize subject maximum-flow maximum-flow problem MHEP minimize node non-degenerate basic non-negative non-zero objective function p₁ piecewise continuous assignment posed positive cone primal problem Proof r₁ reachable Remez exchange algorithm s₁ SCLP Section semi-infinite program sequence simplex algorithm SIP1 SIP2 solvable strong duality subconsistent SUBUBD subvalue supp Suppose t₁ TCNP Theorem topology v₁ variable space vector spaces w₁ x₁ zero λη λξ