Duality in stochastic and dynamic optimization

Ari-Pekka Perkkiö

Research output: ThesisDoctoral ThesisCollection of Articles

Abstract

The main theme in this dissertation is convex duality in stochastic and dynamic optimization. The analysis is based on the conjugate duality framework of Rockafellar and on the theory of convex integral functionals. The dissertation consists of an overview and of three articles. In the first article we study dynamic stochastic optimization problems parameterized by a random variable. Such problems arise in many applications in operations research and mathematical finance. We give sufficient conditions for the existence of solutions and the absence of a duality gap. Our proof uses extended dynamic programming equations, whose validity is established under new relaxed conditions that generalize certain no-arbitrage conditions from mathematical finance. The second article contributes to the theory of integral functionals that is closely connected with set-valued analysis. Given a strictly positive measure, we characterize inner semicontinuous solid convex-valued mappings for which continuous functions which are selections almost everywhere are selections. This class contains continuous mappings as well as fully lower semicontinuous closed-valued mappings that arise in variational analysis and optimization of integral functionals. The characterization allows for extending existing results on convex conjugates of integral functionals on continuous functions. We also give an application to integral functionals on left continuous functions of bounded variation. In the third article we study duality in problems of Bolza over functions of bounded variation. We parameterize the problem by a general Borel measure which has direct economic interpretation in problems of financial economics. Using our results on conjugates of integral functionals, we derive a dual representation for the optimal value function in terms of continuous dual arcs and we give conditions for the existence of solutions. Combined with well-known results on problems of Bolza over absolutely continuous arcs, we obtain optimality conditions in terms of extended Hamiltonian conditions.
Translated title of the contributionDuaalisuus stokastisessa ja dynaamisessa optimoinnissa
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
  • Aalto University
Supervisors/Advisors
  • Kinnunen, Juha, Supervisor
  • Pennanen, Teemu, Advisor, External person
Publisher
Print ISBNs978-952-60-5306-6
Electronic ISBNs978-952-60-5307-3
Publication statusPublished - 2013
MoE publication typeG5 Doctoral dissertation (article)

Keywords

  • convex analysis
  • optimization
  • stochastics

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