Infinite programming may be defined as the study of mathematical programming problems in which the number of variables and the number of constraints are both possibly infinite. Many optimization problems in engineering, operations research, and economics have natural formul- ions as infinite programs. For example, the problem of Chebyshev approximation can be posed as a linear program with an infinite number of constraints. Formally, given continuous functions f,gl,g2, ••• ,gn on the interval [a,b], we can find the linear combination of the functions gl,g2, ... ,gn which is the best uniform approximation to f by choosing real numbers a,xl,x2, •.. ,x to n minimize a t€ [a,b]. This is an example of a semi-infinite program; the number of variables is finite and the number of constraints is infinite. An example of an infinite program in which the number of constraints and the number of variables are both infinite, is the well-known continuous linear program which can be formulated as follows. T minimize ~ c(t)Tx(t)dt t b(t) , subject to Bx(t) + fo Kx(s)ds x(t) .. 0, t € [0, T] • If x is regarded as a member of some infinite-dimensional vector space of functions, then this problem is a linear program posed over that space. Observe that if the constraint equations are differentiated, then this problem takes the form of a linear optimal control problem with state IV variable inequality constraints.
|Publisher:||Springer Berlin Heidelberg|
|Series:||Lecture Notes in Economics and Mathematical Systems , #259|
|Edition description:||Softcover reprint of the original 1st ed. 1985|
|Product dimensions:||6.69(w) x 9.61(h) x 0.02(d)|
Table of ContentsOpenness, closedness and duality in Banach spaces with applications to continuous linear programming.- Conditions for the closedness of the characteristic cone associated with an infinite linear system.- Symmetric duality: a prelude.- Algebraic fundamentals of linear programming.- On regular semi-infinite optimization.- Semi-infinite programming and continuum physics.- On the computation of membrane-eigenvalues by semi-infinite programming methods.- Lagrangian methods for semi-infinite programming problems.- A new primal algorithm for semi-infinite linear programming.- Extreme points and purification algorithms in general linear programming.- Network programming in continuous time with node storage.- The theorem of Gale for infinite networks and applications.- Nonlinear optimal control problems as infinite-dimensional linear programming problems.- Continuity and asymptotic behaviour of the marginal function in optimal control.- Alternative theorems for general complementarity problems.- Nonsmooth analysis and optimization for a class of nonconvex mappings.- Minimum norm problems in normed vector lattices.- Stochastic nonsmooth analysis and optimization in Banach spaces.- Titles and authors of other papers presented at the symposium.