Algebraic duality theorems for infinite LP problems

In this paper we consider a primal-dual infinite linear programming problem-pair, i.e. LPs on infinite dimensional spaces with infinitely many constraints. We present two duality theorems for the problem-pair: a weak and a strong duality theorem. We do not assume any topology on the vector spaces, therefore our results are algebraic duality theorems. As an application, we consider transferable utility cooperative games with arbitrarily many players

Duality in infinite dimensional linear programming

We consider the class of linear programs with infinitely many variables and constraints having the property that every constraint contains at most finitely many variables while every variable appears in at most finitely many constraints. Examples include production planning and equipment replacement over an infinite horizon. We form the natural dual linear programming problem and prove strong duality under a transversality condition that dual prices are asymptotically zero. That is, we show, under this transversality condition, that optimal solutions are attained in both primal and dual problems and their optimal values are equal. The transversality condition, and hence strong duality, is established for an infinite horizon production planning problem.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47922/1/10107_2005_Article_BF01585695.pd

Polubeskonačno programiranje

Polubeskonačno programiranje je optimizacijski problem u kojem minimiziramo realnu funkciju na konačnodimenzionalnom prostoru, ali uz beskonačno mnogo uvjeta tipa nejednakosti. Ta klasa problema je u ovom radu predstavljena kroz četiri poglavlja u kojima su navedeni važni rezultati iz dualnosti, diskretizacije, uvjeta optimalnosti prvog reda te konvergencije. Posebno su istaknuti slučajevi konveksnog problema i glatkog problema polubeskonačnog programiranja radi njihovih matematički pogodnih svojstava. U posljednjem su poglavlju navedene poznate primjene, odnosno načini modeliranja suvremenih problema koji se pojavljuju u robotici, minimizaciji troškova i mjerenju efikasnosti procesa koristeći teoriju polubeskonačnog programiranja.Semi-infinite programming is an optimization problem in which a real function is being minimized on a finite-dimensional space, but with infinite number of constrains that must be satisfied, which are represented by inequalities. In this Master thesis, the class of problems is presented through four chapters in which we gave an overview of important results used in solving semi-infinite programming problems, which include results from duality theory, discretization, first order optimality conditions and convergence. We emphasized the cases of convex problems and linear problems because of their mathematically nicer properties. In the last chapter, we gave examples of well known problems from areas of robotics, minimizing costs and measuring efficiency of processes that can be solved by modeling them like semi-infinite programming problems

From Infinite to Finite Programs: Explicit Error Bounds with Applications to Approximate Dynamic Programming

We consider linear programming (LP) problems in infinite dimensional spaces that are in general computationally intractable. Under suitable assumptions, we develop an approximation bridge from the infinite-dimensional LP to tractable finite convex programs in which the performance of the approximation is quantified explicitly. To this end, we adopt the recent developments in two areas of randomized optimization and first order methods, leading to a priori as well as a posterior performance guarantees. We illustrate the generality and implications of our theoretical results in the special case of the long-run average cost and discounted cost optimal control problems for Markov decision processes on Borel spaces. The applicability of the theoretical results is demonstrated through a constrained linear quadratic optimal control problem and a fisheries management problem.Comment: 30 pages, 5 figure