28 research outputs found
Generalized Decision Rule Approximations for Stochastic Programming via Liftings
Stochastic programming provides a versatile framework for decision-making under uncertainty, but the resulting optimization problems can be computationally demanding. It has recently been shown that, primal and dual linear decision rule approximations can yield tractable upper and lower bounds on the optimal value of a stochastic program. Unfortunately, linear decision rules often provide crude approximations that result in loose bounds. To address this problem, we propose a lifting technique that maps a given stochastic program to an equivalent problem on a higherdimensional probability space. We prove that solving the lifted problem in primal and dual linear decision rules provides tighter bounds than those obtained from applying linear decision rules to the original problem. We also show that there is a one-to-one correspondence between linear decision rules in the lifted problem and families of non-linear decision rules in the original problem. Finally, we identify structured liftings that give rise to highly flexible piecewise linear decision rules and assess their performance in the context of a stylized investment planning problem.
Design of Near Optimal Decision Rules in Multistage Adaptive Mixed-Integer Optimization
In recent years, decision rules have been established as the preferred solution method for addressing computationally demanding, multistage adaptive optimization problems. Despite their success, existing decision rules (a) are typically constrained by their a priori design and (b) do not incorporate in their modeling adaptive binary decisions. To address these problems, we first derive the structure for optimal decision rules involving continuous and binary variables as piecewise linear and piecewise constant functions, respectively. We then propose a methodology for the optimal design of such decision rules that have a finite number of pieces and solve the problem robustly using mixed-integer optimization. We demonstrate the effectiveness of the proposed methods in the context of two multistage inventory control problems. We provide global lower bounds and show that our approach is (i) practically tractable and (ii) provides high quality solutions that outperform alternative methods
The Decision Rule Approach to Optimisation under Uncertainty: Theory and Applications
Optimisation under uncertainty has a long and distinguished history in operations research.
Decision-makers realised early on that the failure to account for uncertainty in optimisation
problems can lead to substantial unexpected losses or even infeasible solutions. Therefore,
approximating the uncertain parameters by their average or nominal values may result in
decisions that perform poorly in scenarios that deviate from the average.
For the last sixty years, scenario tree-based stochastic programming has been the method
of choice for solving optimisation problems affected by parameter uncertainty. This method
approximates the random problem parameters by finite scenarios that can be arranged as a
tree. Unfortunately, this approximation suffers from a curse of dimensionality: the tree needs
to branch whenever new uncertainties are revealed, and thus its size grows exponentially with
the number of decision stages.
It has recently been argued that stochastic programs can quite generally be made tractable
by restricting the space of recourse decisions to those that exhibit a linear data dependence.
An attractive feature of this linear decision rule approximation is that it typically leads to
polynomial-time solution schemes. Unfortunately, the simple structure of linear decision rules
sacrifices optimality in return for scalability. The worst-case performance of linear decision
rules is in fact rather disappointing. When applied to two-stage robust optimisation problems
with m linear constraints, the underlying worst-case approximation ratio has been shown to
be of the order O(√m). Therefore, in this thesis we endeavour to construct efficiently computable
instance-wise bounds on the loss of optimality incurred by the linear decision rule
approximation.
The contributions of this thesis are as follows. (i)We develop an efficient algorithm for assessing
the loss of optimality incurred by the linear decision rule approximation. The key idea is to
apply the linear decision rule restriction not only to the primal but also to a dual version of
the stochastic program. Since both problems share a similar structure, both problems can
be solved in polynomial-time. The gap between their optimal values estimates the loss of
optimality incurred by the linear decision rule approximation. (ii) We design an improved
approximation based on non-linear decision rules, which can be useful if the optimality gap of
the linear decision rules is deemed unacceptably high. The idea takes advantage of the fact that one can always map a linearly parameterised non-linear function into a higher dimensional
space, where it can be represented as a linear function. This allows us to utilise the machinery
developed for linear decision rules to produce superior quality approximations that can be
obtained in polynomial time. (iii) We assess the performance of the approximations developed
in two operations management problems: a production planning problem and a supply chain
design problem. We show that near-optimal solutions can be found in problem instances with
many stages and random parameters. We additionally compare the quality of the decision rule
approximation with classical approximation techniques. (iv) We develop a systematic approach
to reformulate multi-stage stochastic programs with a large (possibly infinite) number of stages
as static robust optimisation problem that can be solved with a constraint sampling technique.
The method is motivated via an investment planning problem in the electricity industry
Robust Data-driven Prescriptiveness Optimization
The abundance of data has led to the emergence of a variety of optimization
techniques that attempt to leverage available side information to provide more
anticipative decisions. The wide range of methods and contexts of application
have motivated the design of a universal unitless measure of performance known
as the coefficient of prescriptiveness. This coefficient was designed to
quantify both the quality of contextual decisions compared to a reference one
and the prescriptive power of side information. To identify policies that
maximize the former in a data-driven context, this paper introduces a
distributionally robust contextual optimization model where the coefficient of
prescriptiveness substitutes for the classical empirical risk minimization
objective. We present a bisection algorithm to solve this model, which relies
on solving a series of linear programs when the distributional ambiguity set
has an appropriate nested form and polyhedral structure. Studying a contextual
shortest path problem, we evaluate the robustness of the resulting policies
against alternative methods when the out-of-sample dataset is subject to
varying amounts of distribution shift
Exact and Approximate Schemes for Robust Optimization Problems with Decision Dependent Information Discovery
Uncertain optimization problems with decision dependent information discovery allow the decision maker to control the timing of information discovery, in contrast to the classic multistage setting where uncertain parameters are revealed sequentially based on a prescribed filtration. This problem class is useful in a wide range of applications, however, its assimilation is partly limited by the lack of efficient solution schemes. In this paper we study two-stage robust optimization problems with decision dependent information discovery where uncertainty appears in the objective function. The contributions of the paper are twofold: (i) we develop an exact solution scheme based on a nested decomposition algorithm, and (ii) we improve upon the existing K-adaptability approximate by strengthening its formulation using techniques from the integer programming literature. Throughout the paper we use the orienteering problem as our working example, a challenging problem from the logistics literature which naturally fits within this framework. The complex structure of the routing recourse problem forms a challenging test bed for the proposed solution schemes, in which we show that exact solution method outperforms at times the K-adaptability approximation, however, the strengthened K-adaptability formulation can provide good quality solutions in larger instances while significantly outperforming existing approximation schemes even in the decision independent information discovery setting. We leverage the effectiveness of the proposed solution schemes and the orienteering problem in a case study from Alrijne hospital in the Netherlands, where we try to improve the collection process of empty medicine delivery crates by co-optimizing sensor placement and routing decisions
The decision rule approach to optimization under uncertainty: methodology and applications
Dynamic decision-making under uncertainty has a long and distinguished history in operations research. Due to the curse of dimensionality, solution schemes that naïvely partition or discretize the support of the random problem parameters are limited to small and medium-sized problems, or they require restrictive modeling assumptions (e.g., absence of recourse actions). In the last few decades, several solution techniques have been proposed that aim to alleviate the curse of dimensionality. Amongst these is the decision rule approach, which faithfully models the random process and instead approximates the feasible region of the decision problem. In this paper, we survey the major theoretical findings relating to this approach, and we investigate its potential in two applications areas