Polynomial Approximations for Infinite-Dimensional Optimization Problems
Author(s)
Bampou, Dimitra
Type
Thesis
Abstract
Many real-life decision problems in management science and engineering involve decisions that
are functions of time and/or uncertainty. The resulting optimization models are therefore naturally
formulated on infinite-dimensional function spaces. However, such infinite-dimensional
optimization problems are notoriously difficult, and to solve them one usually has to resort to
approximation methods. The objective of this thesis is to devise polynomial approximations
for solving continuous linear programs and multi-stage stochastic programs, both of which constitute
important classes of infinite-dimensional optimization problems with manifold practical
applications. Approximating the functional decision variables by polynomials allows us to apply
sum-of-squares techniques from algebraic geometry to reformulate the resulting problems as
tractable semidefinite programs, which can be solved efficiently with interior point algorithms.
Continuous linear programs represent deterministic optimization problems whose decision variables
are functions of time subject to pointwise and dynamic linear constraints. They have
attracted considerable interest due to their potential for modelling manufacturing, scheduling
and routing problems. While efficient simplex-type algorithms have been developed for separated
continuous linear programs, crude time discretization remains the method of choice for
solving general (non-separated) problem instances. In this thesis we propose a more generic
approximation scheme for non-separated continuous linear programs, which are believed to be
NP-hard. We approximate the functional decision variables (policies) by polynomial and piecewise
polynomial decision rules. To estimate the approximation error, we also compute a lower
bound by solving a dual continuous linear program in (piecewise) polynomial decision rules.
Multi-stage stochastic programming provides a versatile framework for optimal decision making
under uncertainty, but it gives rise to hard functional optimization problems since the adaptive
recourse decisions must be modelled as functions of some or all uncertain parameters. We
propose to approximate these recourse decisions by polynomial decision rules and show that
the best polynomial decision rule of a fixed degree can be computed efficiently. Again, we
show that the suboptimality of the best polynomial decision rule can be estimated efficiently
by solving a dual version of the stochastic program in polynomial decision rules.
Recent progress in the theory of dynamic risk measures has found a strong echo in stochastic
programming, where the time-consistency of dynamic decision making under uncertainty is currently under scrutiny. We extend the concepts of coherence and time consistency to stochastic
programming models subject to distributional ambiguity, which motivates us to introduce
robust dynamic risk measures. We discuss conditions under which these robust risk measures
inherit coherence and time-consistency from their nominal counterparts. We also propose an approximation
scheme based on polynomial decision rules for solving linear multi-stage stochastic
programs involving robust dynamic risk measures.
are functions of time and/or uncertainty. The resulting optimization models are therefore naturally
formulated on infinite-dimensional function spaces. However, such infinite-dimensional
optimization problems are notoriously difficult, and to solve them one usually has to resort to
approximation methods. The objective of this thesis is to devise polynomial approximations
for solving continuous linear programs and multi-stage stochastic programs, both of which constitute
important classes of infinite-dimensional optimization problems with manifold practical
applications. Approximating the functional decision variables by polynomials allows us to apply
sum-of-squares techniques from algebraic geometry to reformulate the resulting problems as
tractable semidefinite programs, which can be solved efficiently with interior point algorithms.
Continuous linear programs represent deterministic optimization problems whose decision variables
are functions of time subject to pointwise and dynamic linear constraints. They have
attracted considerable interest due to their potential for modelling manufacturing, scheduling
and routing problems. While efficient simplex-type algorithms have been developed for separated
continuous linear programs, crude time discretization remains the method of choice for
solving general (non-separated) problem instances. In this thesis we propose a more generic
approximation scheme for non-separated continuous linear programs, which are believed to be
NP-hard. We approximate the functional decision variables (policies) by polynomial and piecewise
polynomial decision rules. To estimate the approximation error, we also compute a lower
bound by solving a dual continuous linear program in (piecewise) polynomial decision rules.
Multi-stage stochastic programming provides a versatile framework for optimal decision making
under uncertainty, but it gives rise to hard functional optimization problems since the adaptive
recourse decisions must be modelled as functions of some or all uncertain parameters. We
propose to approximate these recourse decisions by polynomial decision rules and show that
the best polynomial decision rule of a fixed degree can be computed efficiently. Again, we
show that the suboptimality of the best polynomial decision rule can be estimated efficiently
by solving a dual version of the stochastic program in polynomial decision rules.
Recent progress in the theory of dynamic risk measures has found a strong echo in stochastic
programming, where the time-consistency of dynamic decision making under uncertainty is currently under scrutiny. We extend the concepts of coherence and time consistency to stochastic
programming models subject to distributional ambiguity, which motivates us to introduce
robust dynamic risk measures. We discuss conditions under which these robust risk measures
inherit coherence and time-consistency from their nominal counterparts. We also propose an approximation
scheme based on polynomial decision rules for solving linear multi-stage stochastic
programs involving robust dynamic risk measures.
Date Issued
2012-08
Date Awarded
2013-01
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Rustem, Berc
Kuhn, damiel
Publisher Department
Computing
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)