845 research outputs found
Reformulation and decomposition of integer programs
In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm
Network Migration Problem: A Logic-based Benders Decomposition Approach Driven by Column Generation and Constraint Programming
Telecommunication networks frequently face technological advancements and
need to upgrade their infrastructure. Adapting legacy networks to the latest
technology requires synchronized technicians responsible for migrating the
equipment. The goal of the network migration problem is to find an optimal plan
for this process. This is a defining step in the customer acquisition of
telecommunications service suppliers, and its outcome directly impacts the
network owners' purchasing behaviour. We propose the first exact method for the
network migration problem, a logic-based Benders decomposition approach that
benefits from a hybrid constraint programming-based column generation in its
master problem and a constraint programming model in its subproblem. This
integrated solution technique is applicable to any integer programming problem
with similar structure, most notably the vehicle routing problem with node
synchronization constraints. Comprehensive evaluation of our method over
instances based on six real networks demonstrates the computational efficiency
of the algorithm in obtaining quality solutions. We also show the merit of each
incorporated optimization paradigm in achieving this performance
Pricing High-Dimensional American Options Using Local Consistency Conditions
We investigate a new method for pricing high-dimensional American options. The method is of finite difference type but is also related to Monte Carlo techniques in that it involves a representative sampling of the underlying variables.An approximating Markov chain is built using this sampling and linear programming is used to satisfy local consistency conditions at each point related to the infinitesimal generator or transition density.The algorithm for constructing the matrix can be parallelised easily; moreover once it has been obtained it can be reused to generate quick solutions for a large class of related problems.We provide pricing results for geometric average options in up to ten dimensions, and compare these with accurate benchmarks.option pricing;inequality;markov chains
A Comparison of Optimization Methods for Solving the Depot Matching and Parking Problem
We consider the Train Unit Shunting Problem, an important plan- ning problem for passenger railway operators. This problem entails assigning physical train units to scheduled train services in such a way that the resulting shunting yard operations are feasible. As such, it arises at every shunting yard in the railway network and involves matching train units to arriving and departing train services as well as assigning the selected matchings to appropriate shunting yard tracks. We present a comparison benchmark of multiple solution approaches for this problem. In particular, we have developed a Constraint Pro- gramming formulation, a Column Generation approach, and a random- ized greedy heuristic. We compare and benchmark these approaches against slightly adjusted existing methods based on a a Mixed Inte- ger Linear Program, and a Two-Stage heuristic. The benchmark con- tains multiple real-life instances provided by the Danish State Rail- ways (DSB) and Netherlands Railways (NS). The results highlight the strengths and weaknesses of the considered approaches
Design and implementation of a modular interior-point solver for linear optimization
This paper introduces the algorithmic design and implementation of Tulip, an
open-source interior-point solver for linear optimization. It implements a
regularized homogeneous interior-point algorithm with multiple centrality
corrections, and therefore handles unbounded and infeasible problems. The
solver is written in Julia, thus allowing for a flexible and efficient
implementation: Tulip's algorithmic framework is fully disentangled from linear
algebra implementations and from a model's arithmetic. In particular, this
allows to seamlessly integrate specialized routines for structured problems.
Extensive computational results are reported. We find that Tulip is competitive
with open-source interior-point solvers on the H. Mittelmann's benchmark of
barrier linear programming solvers. Furthermore, we design specialized linear
algebra routines for structured master problems in the context of Dantzig-Wolfe
decomposition. These routines yield a tenfold speedup on large and dense
instances that arise in power systems operation and two-stage stochastic
programming, thereby outperforming state-of-the-art commercial interior point
method solvers. Finally, we illustrate Tulip's ability to use different levels
of arithmetic precision by solving problems in extended precision
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