854 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
A Two-Level Approach to Large Mixed-Integer Programs with Application to Cogeneration in Energy-Efficient Buildings
We study a two-stage mixed-integer linear program (MILP) with more than 1
million binary variables in the second stage. We develop a two-level approach
by constructing a semi-coarse model (coarsened with respect to variables) and a
coarse model (coarsened with respect to both variables and constraints). We
coarsen binary variables by selecting a small number of pre-specified daily
on/off profiles. We aggregate constraints by partitioning them into groups and
summing over each group. With an appropriate choice of coarsened profiles, the
semi-coarse model is guaranteed to find a feasible solution of the original
problem and hence provides an upper bound on the optimal solution. We show that
solving a sequence of coarse models converges to the same upper bound with
proven finite steps. This is achieved by adding violated constraints to coarse
models until all constraints in the semi-coarse model are satisfied. We
demonstrate the effectiveness of our approach in cogeneration for buildings.
The coarsened models allow us to obtain good approximate solutions at a
fraction of the time required by solving the original problem. Extensive
numerical experiments show that the two-level approach scales to large problems
that are beyond the capacity of state-of-the-art commercial MILP solvers
A polynomial-time algorithm for optimizing over N-fold 4-block decomposable integer programs
In this paper we generalize N-fold integer programs and two-stage integer
programs with N scenarios to N-fold 4-block decomposable integer programs. We
show that for fixed blocks but variable N, these integer programs are
polynomial-time solvable for any linear objective. Moreover, we present a
polynomial-time computable optimality certificate for the case of fixed blocks,
variable N and any convex separable objective function. We conclude with two
sample applications, stochastic integer programs with second-order dominance
constraints and stochastic integer multi-commodity flows, which (for fixed
blocks) can be solved in polynomial time in the number of scenarios and
commodities and in the binary encoding length of the input data. In the proof
of our main theorem we combine several non-trivial constructions from the
theory of Graver bases. We are confident that our approach paves the way for
further extensions
Data-driven linear decision rule approach for distributionally robust optimization of on-line signal control
We propose a two-stage, on-line signal control strategy for dynamic networks using a linear decision rule (LDR) approach and a distributionally robust optimization (DRO) technique. The first (off-line) stage formulates a LDR that maps real-time traffic data to optimal signal control policies. A DRO problem is solved to optimize the on-line performance of the LDR in the presence of uncertainties associated with the observed traffic states and ambiguity in their underlying distribution functions. We employ a data-driven calibration of the uncertainty set, which takes into account historical traffic data. The second (on-line) stage implements a very efficient linear decision rule whose performance is guaranteed by the off-line computation. We test the proposed signal control procedure in a simulation environment that is informed by actual traffic data obtained in Glasgow, and demonstrate its full potential in on-line operation and deployability on realistic networks, as well as its effectiveness in improving traffic
An approximation framework for two-stage ambiguous stochastic integer programs under mean-MAD information
We consider two-stage recourse models in which only limited information is available on the probability distributions of the random parameters in the model. If all decision variables are continuous, then we are able to derive the worst-case and best-case probability distributions under the assumption that only the means and mean absolute deviations of the random parameters are known. Contrary to most existing results in the literature, these probability distributions are the same for every first-stage decision. The ambiguity set that we use in this paper also turns out to be particularly suitable for ambiguous recourse models involving integer decisions variables. For such problems, we develop a general approximation framework and derive error bounds for using these approximatons. We apply this approximation framework to mixed-ambiguous mixed-integer recourse models in which some of the probability distributions of the random parameters are known and others are ambiguous. To illustrate these results we carry out numerical experiments on a surgery block allocation problem. (C) 2018 Elsevier B.V. All rights reserved
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Simulation and Evaluation of Zonal Electricity Market Designs
Zonal pricing with countertrading (a market-based redispatch) gives arbitrage opportunities to the power producers located in the export-constrained nodes. They can increase their profit by increasing the output in the day-ahead market and decrease it in the real-time market (the inc-dec game). We show that this leads to large inefficiencies in a standard zonal market. We also show how the inefficiencies can be significantly mitigated by changing the design of the real-time market. We consider a two-stage game with oligopoly producers, wind-power shocks and real-time shocks. The game is formulated as a two-stage stochastic equilibrium problem with equilibrium constraints (EPEC), which we recast into a two-stage stochastic Mixed-Integer Bilinear Program (MIBLP). We present numerical results for a six-node and the IEEE 24-node system
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