NDLI: Duality in Countably Infinite Monotropic Programs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Ghate, Archis
Copyright Year	2017
Abstract	Finite-dimensional monotropic programs form a class of convex optimization problems that includes linear programs, convex minimum cost flow problems on networks and hypernetworks, and separable convex programs with linear constraints. Countably infinite monotropic programs arise, for example, in infinite-horizon sequential decision problems and in robust optimization. Their applications encompass (i) countably infinite linear programs such as the shortest path formulations of infinite-horizon nonstationary as well as countable-state Markov decision processes; and (ii) convex minimum cost flow problems on countably infinite networks and hypernetworks. Duality results for finite-dimensional monotropic programs are as powerful as those available for linear programs. On the contrary, applicable duality results for countably infinite monotropic programs are currently nonexistent owing to several mathematical pathologies in infinite-dimensional sequence spaces. This paper overcomes this hurdle by first embedding the dual variables in a sequence space where the Lagrangian function is well-defined and finite. Weak duality and complementary slackness are derived using finite-dimensional proof techniques. Conditions under which zero duality gaps and strong duality between a sequence of finite-dimensional primal-dual projections of the infinite-dimensional problem are preserved in the limit are established. Essentially all known duality results about countably infinite mathematical programs are recovered as special cases.
Sponsorship	National Science Foundation
Starting Page	2010
Ending Page	2033
Page Count	24
File Format	PDF
ISSN	10526234
DOI	10.1137/16M1084328
e-ISSN	10957189
Journal	SIAM Journal on Optimization (SJOPE8)
Issue Number	3
Volume Number	27
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-09-12
Access Restriction	Subscribed
Subject Keyword	Optimality conditions, duality convex optimization Programming in abstract spaces infinite-dimensional optimization duality Convex programming Programming involving graphs or networks
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Theoretical Computer Science Software

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