2,361 research outputs found
Absolute semi-deviation risk measure for ordering problem with transportation cost in Supply Chain
We present a decomposition method for stochastic programs with 0-1 variables
in the second-stage with absolute semi-deviation (ASD) risk measure.
Traditional stochastic programming models are risk-neutral where expected costs
are considered for the second-stage. A common approach to address risk is to
include a dispersion statistic in addition with expected costs and weighted
appropriately. Due to the lack of block angular structure, stochastic programs
with ASD risk-measure possess computational challenges. The proposed
decomposition algorithm uses another risk-measure `expected excess', and
provides tighter bounds for ASD stochastic models. We perform computational
study on a supply chain replenishment problem and standard knapsack instances.
The computational results using supply chain instances demonstrate the
usefulness of ASD risk-measure in decision making under uncertainty, and
knapsack instances indicate that the proposed methodology outperforms a direct
solver
On conditional cuts for Stochastic Dual Dynamic Programming
Multi stage stochastic programs arise in many applications from engineering
whenever a set of inventories or stocks has to be valued. Such is the case in
seasonal storage valuation of a set of cascaded reservoir chains in hydro
management. A popular method is Stochastic Dual Dynamic Programming (SDDP),
especially when the dimensionality of the problem is large and Dynamic
programming no longer an option. The usual assumption of SDDP is that
uncertainty is stage-wise independent, which is highly restrictive from a
practical viewpoint. When possible, the usual remedy is to increase the
state-space to account for some degree of dependency. In applications this may
not be possible or it may increase the state space by too much. In this paper
we present an alternative based on keeping a functional dependency in the SDDP
- cuts related to the conditional expectations in the dynamic programming
equations. Our method is based on popular methodology in mathematical finance,
where it has progressively replaced scenario trees due to superior numerical
performance. On a set of numerical examples, we too show the interest of this
way of handling dependency in uncertainty, when combined with SDDP. Our method
is readily available in the open source software package StOpt.Comment: 26 pages, 10 figure
Risk Aversion to Parameter Uncertainty in Markov Decision Processes with an Application to Slow-Onset Disaster Relief
In classical Markov Decision Processes (MDPs), action costs and transition
probabilities are assumed to be known, although an accurate estimation of these
parameters is often not possible in practice. This study addresses MDPs under
cost and transition probability uncertainty and aims to provide a mathematical
framework to obtain policies minimizing the risk of high long-term losses due
to not knowing the true system parameters. To this end, we utilize the risk
measure value-at-risk associated with the expected performance of an MDP model
with respect to parameter uncertainty. We provide mixed-integer linear and
nonlinear programming formulations and heuristic algorithms for such
risk-averse models of MDPs under a finite distribution of the uncertain
parameters. Our proposed models and solution methods are illustrated on an
inventory management problem for humanitarian relief operations during a
slow-onset disaster. The results demonstrate the potential of our risk-averse
modeling approach for reducing the risk of highly undesirable outcomes in
uncertain/risky environments
Multicut decomposition methods with cut selection for multistage stochastic programs
We introduce a variant of Multicut Decomposition Algorithms (MuDA), called
CuSMuDA (Cut Selection for Multicut Decomposition Algorithms), for solving
multistage stochastic linear programs that incorporates strategies to select
the most relevant cuts of the approximate recourse functions. We prove the
convergence of the method in a finite number of iterations and use it to solve
six portfolio problems with direct transaction costs under return uncertainty
and six inventory management problems under demand uncertainty. On all problem
instances CuSMuDA is much quicker than MuDA: between 5.1 and 12.6 times quicker
for the porfolio problems considered and between 6.4 and 15.7 times quicker for
the inventory problems
DASC: a Decomposition Algorithm for multistage stochastic programs with Strongly Convex cost functions
We introduce DASC, a decomposition method akin to Stochastic Dual Dynamic
Programming (SDDP) which solves some multistage stochastic optimization
problems having strongly convex cost functions. Similarly to SDDP, DASC
approximates cost-to-go functions by a maximum of lower bounding functions
called cuts. However, contrary to SDDP, the cuts computed with DASC are
quadratic functions. We also prove the convergence of DASC.Comment: arXiv admin note: text overlap with arXiv:1707.0081
Single cut and multicut SDDP with cut selection for multistage stochastic linear programs: convergence proof and numerical experiments
We introduce a variant of Multicut Decomposition Algorithms (MuDA), called
CuSMuDA (Cut Selection for Multicut Decomposition Algorithms), for solving
multistage stochastic linear programs that incorporates a class of cut
selection strategies to choose the most relevant cuts of the approximate
recourse functions. This class contains Level 1 and Limited Memory Level 1 cut
selection strategies, initially introduced for respectively Stochastic Dual
Dynamic Programming (SDDP) and Dual Dynamic Programming (DDP). We prove the
almost sure convergence of the method in a finite number of iterations and
obtain as a by-product the almost sure convergence in a finite number of
iterations of SDDP combined with our class of cut selection strategies. We
compare the performance of MuDA, SDDP, and their variants with cut selection
(using Level 1 and Limited Memory Level 1) on several instances of a portfolio
problem and of an inventory problem. On these experiments, in general, SDDP is
quicker (i.e., satisfies the stopping criterion quicker) than MuDA and cut
selection allows us to decrease the computational bulk with Limited Memory
Level 1 being more efficient (sometimes much more) than Level 1.Comment: arXiv admin note: substantial text overlap with arXiv:1705.0897
A Uniform-grid Discretization Algorithm for Stochastic Control with Risk Constraints
In this paper, we present a discretization algorithm for finite horizon risk
constrained dynamic programming algorithm in [Chow_Pavone_13]. Although in a
theoretical standpoint, Bellman's recursion provides a systematic way to find
optimal value functions and generate optimal history dependent policies, there
is a serious computational issue. Even if the state space and action space of
this constrained stochastic optimal control problem are finite, the spaces of
risk threshold and the feasible risk update are closed bounded subset of real
numbers. This prohibits any direct applications of unconstrained finite state
iterative methods in dynamic programming found in [Bertsekas_05]. In order to
approximate Bellman's operator derived in [Chow_Pavone_13], we discretize the
continuous action spaces and formulate a finite space approximation for the
exact dynamic programming algorithm. We will also prove that the approximation
error bound of optimal value functions is bound linearly by the step size of
discretization. Finally, details for implementations and possible modifications
are discussed
Risk-Averse Approximate Dynamic Programming with Quantile-Based Risk Measures
In this paper, we consider a finite-horizon Markov decision process (MDP) for
which the objective at each stage is to minimize a quantile-based risk measure
(QBRM) of the sequence of future costs; we call the overall objective a dynamic
quantile-based risk measure (DQBRM). In particular, we consider optimizing
dynamic risk measures where the one-step risk measures are QBRMs, a class of
risk measures that includes the popular value at risk (VaR) and the conditional
value at risk (CVaR). Although there is considerable theoretical development of
risk-averse MDPs in the literature, the computational challenges have not been
explored as thoroughly. We propose data-driven and simulation-based approximate
dynamic programming (ADP) algorithms to solve the risk-averse sequential
decision problem. We address the issue of inefficient sampling for risk
applications in simulated settings and present a procedure, based on importance
sampling, to direct samples toward the "risky region" as the ADP algorithm
progresses. Finally, we show numerical results of our algorithms in the context
of an application involving risk-averse bidding for energy storage.Comment: 39 pages, 7 figure
Convergence analysis of sampling-based decomposition methods for risk-averse multistage stochastic convex programs
We consider a class of sampling-based decomposition methods to solve
risk-averse multistage stochastic convex programs. We prove a formula for the
computation of the cuts necessary to build the outer linearizations of the
recourse functions. This formula can be used to obtain an efficient
implementation of Stochastic Dual Dynamic Programming applied to convex
nonlinear problems. We prove the almost sure convergence of these decomposition
methods when the relatively complete recourse assumption holds. We also prove
the almost sure convergence of these algorithms when applied to risk-averse
multistage stochastic linear programs that do not satisfy the relatively
complete recourse assumption. The analysis is first done assuming the
underlying stochastic process is interstage independent and discrete, with a
finite set of possible realizations at each stage. We then indicate two ways of
extending the methods and convergence analysis to the case when the process is
interstage dependent
Optimal Pump Control for Water Distribution Networks via Data-based Distributional Robustness
In this paper, we propose a data-based methodology to solve a multi-period
stochastic optimal water flow (OWF) problem for water distribution networks
(WDNs). The framework explicitly considers the pump schedule and water network
head level with limited information of demand forecast errors for an extended
period simulation. The objective is to determine the optimal feedback decisions
of network-connected components, such as nominal pump schedules and tank head
levels and reserve policies, which specify device reactions to forecast errors
for accommodation of fluctuating water demand. Instead of assuming the
uncertainties across the water network are generated by a prescribed certain
distribution, we consider ambiguity sets of distributions centered at an
empirical distribution, which is based directly on a finite training data set.
We use a distance-based ambiguity set with the Wasserstein metric to quantify
the distance between the real unknown data-generating distribution and the
empirical distribution. This allows our multi-period OWF framework to trade off
system performance and inherent sampling errors in the training dataset. Case
studies on a three-tank water distribution network systematically illustrate
the tradeoff between pump operational cost, risks of constraint violation, and
out-of-sample performance
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