44 research outputs found
Leveraging Decision Diagrams to Solve Two-stage Stochastic Programs with Binary Recourse and Logical Linking Constraints
Two-stage stochastic programs with binary recourse are challenging to solve
and efficient solution methods for such problems have been limited. In this
work, we generalize an existing binary decision diagram-based (BDD-based)
approach of Lozano and Smith (Math. Program., 2018) to solve a special class of
two-stage stochastic programs with binary recourse. In this setting, the
first-stage decisions impact the second-stage constraints. Our modified problem
extends the second-stage problem to a more general setting where logical
expressions of the first-stage solutions enforce constraints in the second
stage. We also propose a complementary problem and solution method which can be
used for many of the same applications. In the complementary problem we have
second-stage costs impacted by expressions of the first-stage decisions. In
both settings, we convexify the second-stage problems using BDDs and
parametrize either the arc costs or capacities of these BDDs with first-stage
solutions depending on the problem. We further extend this work by
incorporating conditional value-at-risk and we propose, to our knowledge, the
first decomposition method for two-stage stochastic programs with binary
recourse and a risk measure. We apply these methods to a novel stochastic
dominating set problem and present numerical results to demonstrate the
effectiveness of the proposed methods
Lagrangian Dual Decision Rules for Multistage Stochastic Mixed Integer Programming
Multistage stochastic programs can be approximated by restricting policies to
follow decision rules. Directly applying this idea to problems with integer
decisions is difficult because of the need for decision rules that lead to
integral decisions. In this work, we introduce Lagrangian dual decision rules
(LDDRs) for multistage stochastic mixed integer programming (MSMIP) which
overcome this difficulty by applying decision rules in a Lagrangian dual of the
MSMIP. We propose two new bounding techniques based on stagewise (SW) and
nonanticipative (NA) Lagrangian duals where the Lagrangian multiplier policies
are restricted by LDDRs. We demonstrate how the solutions from these duals can
be used to drive primal policies. Our proposal requires fewer assumptions than
most existing MSMIP methods. We compare the theoretical strength of the
restricted duals and show that the restricted NA dual can provide relaxation
bounds at least as good as the ones obtained by the restricted SW dual. In our
numerical study, we observe that the proposed LDDR approaches yield significant
optimality gap reductions compared to existing general-purpose bounding methods
for MSMIP problems
A Branch-and-Price Algorithm Enhanced by Decision Diagrams for the Kidney Exchange Problem
Kidney paired donation programs allow patients registered with an
incompatible donor to receive a suitable kidney from another donor, as long as
the latter's co-registered patient, if any, also receives a kidney from a
different donor. The kidney exchange problem (KEP) aims to find an optimal
collection of kidney exchanges taking the form of cycles and chains. Existing
exact solution methods for KEP either are designed for the case where only
cyclic exchanges are considered, or can handle long chains but are scalable as
long as cycles are short. We develop the first decomposition method that is
able to deal with long cycles and long chains for large realistic instances.
More specifically, we propose a branch-and-price framework, in which the
pricing problems are solved (for the first time in packing problems in a
digraph) through multi-valued decision diagrams. Also, we present a new upper
bound on the optimal value of KEP, stronger than the one proposed in the
literature, which is obtained via our master problem. Computational experiments
show superior performance of our method over the state of the art by optimally
solving almost all instances in the PrefLib library for multiple cycle and
chain lengths