5,361 research outputs found
Decomposition-Based Method for Sparse Semidefinite Relaxations of Polynomial Optimization Problems
We consider polynomial optimization problems pervaded by a sparsity pattern. It has been shown in [1, 2] that the optimal solution of a polynomial programming problem with structured sparsity can be computed by solving a series of semidefinite relaxations that possess the same kind of sparsity. We aim at solving the former relaxations with a decompositionbased method, which partitions the relaxations according to their sparsity pattern. The decomposition-based method that we propose is an extension to semidefinite programming of the Benders decomposition for linear programs [3] .Polynomial optimization, Semidefinite programming, Sparse SDP relaxations, Benders decomposition
Accelerating L-shaped Two-stage Stochastic SCUC with Learning Integrated Benders Decomposition
Benders decomposition is widely used to solve large mixed-integer problems.
This paper takes advantage of machine learning and proposes enhanced variants
of Benders decomposition for solving two-stage stochastic security-constrained
unit commitment (SCUC). The problem is decomposed into a master problem and
subproblems corresponding to a load scenario. The goal is to reduce the
computational costs and memory usage of Benders decomposition by creating
tighter cuts and reducing the size of the master problem. Three approaches are
proposed, namely regression Benders, classification Benders, and
regression-classification Benders. A regressor reads load profile scenarios and
predicts subproblem objective function proxy variables to form tighter cuts for
the master problem. A criterion is defined to measure the level of usefulness
of cuts with respect to their contribution to lower bound improvement. Useful
cuts that contain the necessary information to form the feasible region are
identified with and without a classification learner. Useful cuts are
iteratively added to the master problem, and non-useful cuts are discarded to
reduce the computational burden of each Benders iteration. Simulation studies
on multiple test systems show the effectiveness of the proposed learning-aided
Benders decomposition for solving two-stage SCUC as compared to conventional
multi-cut Benders decomposition
Chance Constrained Mixed Integer Program: Bilinear and Linear Formulations, and Benders Decomposition
In this paper, we study chance constrained mixed integer program with
consideration of recourse decisions and their incurred cost, developed on a
finite discrete scenario set. Through studying a non-traditional bilinear mixed
integer formulation, we derive its linear counterparts and show that they could
be stronger than existing linear formulations. We also develop a variant of
Jensen's inequality that extends the one for stochastic program. To solve this
challenging problem, we present a variant of Benders decomposition method in
bilinear form, which actually provides an easy-to-use algorithm framework for
further improvements, along with a few enhancement strategies based on
structural properties or Jensen's inequality. Computational study shows that
the presented Benders decomposition method, jointly with appropriate
enhancement techniques, outperforms a commercial solver by an order of
magnitude on solving chance constrained program or detecting its infeasibility
Stabilized Benders methods for large-scale combinatorial optimization, with appllication to data privacy
The Cell Suppression Problem (CSP) is a challenging Mixed-Integer Linear Problem arising in statistical tabular data protection. Medium sized instances of CSP involve thousands of binary variables and million of continuous variables and constraints. However, CSP has the typical
structure that allows application of the renowned Bendersā decomposition method: once the ācomplicatingā binary variables are fixed, the problem decomposes into a large set of linear subproblems on the āeasyā continuous ones. This allows to project away the easy variables, reducing to a master problem in the complicating ones where the value functions of the subproblems are approximated with the standard cutting-plane approach. Hence, Bendersā decomposition suffers from the same drawbacks of the cutting-plane method, i.e., oscillation and slow convergence, compounded with the fact that the master problem is combinatorial. To overcome this drawback we present a stabilized Benders decomposition whose master is restricted to a neighborhood of successful candidates by local branching constraints, which are dynamically adjusted, and even dropped, during the iterations. Our experiments with randomly generated and real-world CSP instances with up to 3600 binary variables, 90M continuous variables and 15M inequality constraints show that our approach is competitive with both the current state-of-the-art (cutting-plane-based) code for cell suppression, and the Benders implementation in CPLEX 12.7. In some instances, stabilized Benders is able to quickly provide a very good solution in less than one minute, while the other approaches were not able to find any feasible solution in one hour.Peer ReviewedPreprin
A Redesigned Benders Decomposition Approach for Large-Scale In-Transit Freight Consolidation Operations
The growth in online shopping and third party logistics has caused a revival
of interest in finding optimal solutions to the large scale in-transit freight
consolidation problem. Given the shipment date, size, origin, destination, and
due dates of multiple shipments distributed over space and time, the problem
requires determining when to consolidate some of these shipments into one
shipment at an intermediate consolidation point so as to minimize shipping
costs while satisfying the due date constraints. In this paper, we develop a
mixed-integer programming formulation for a multi-period freight consolidation
problem that involves multiple products, suppliers, and potential consolidation
points. Benders decomposition is then used to replace a large number of integer
freight-consolidation variables by a small number of continuous variables that
reduces the size of the problem without impacting optimality. Our results show
that Benders decomposition provides a significant scale-up in the performance
of the solver. We demonstrate our approach using a large-scale case with more
than 27.5 million variables and 9.2 million constraints
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