7,794 research outputs found
Combining Column Generation and Lagrangian Relaxation
Although the possibility to combine column generation and Lagrangian relaxation has been known for quite some time, it has only recently been exploited in algorithms. In this paper, we discuss ways of combining these techniques. We focus on solving the LP relaxation of the Dantzig-Wolfe master problem. In a first approach we apply Lagrangian relaxation directly to this extended formulation, i.e. no simplex method is used. In a second one, we use Lagrangian relaxation to generate new columns, that is Lagrangian relaxation is applied to the compact for-mulation. We will illustrate the ideas behind these algorithms with an application in Lot-sizing. To show the wide applicability of these techniques, we also discuss applications in integrated vehicle and crew scheduling, plant location and cutting stock problems.column generation;Lagrangean relaxation;cutting stock problem;lotsizing;vehicle and crew scheduling
Lagrangian Relaxation and Partial Cover
Lagrangian relaxation has been used extensively in the design of
approximation algorithms. This paper studies its strengths and limitations when
applied to Partial Cover.Comment: 20 pages, extended abstract appeared in STACS 200
Modeling and solving the multi-period inventory routing problem with constant demand rates
The inventory routing problem (IRP) is one of the challenging optimization problems in supply chain logistics. It combines inventory control and vehicle routing optimization. The main purpose of the IRP is to determine optimal delivery times and quantities to be delivered to customers, as well as optimal vehicle routes to distribute these quantities. The IRP is an underlying logistical optimization problem for supply chains implementing vendor-managed inventory (VMI) policies, in which the supplier takes responsibility for the management of the customers' inventory. In this paper, we consider a multi-period inventory routing problem assuming constant demand rates (MP-CIRP). The proposed model is formulated as a linear mixed-integer program and solved with a Lagrangian relaxation method. The solution obtained by the Lagrangian relaxation method is then used to generate a close to optimal feasible solution of the MP-CIRP by solving a series of assignment problems. The numerical experiments carried out so far show that the proposed Lagrangian relaxation approach nds quite good solutions for the MP-CIRP and in reasonable computation times
Solving the p -Median Problem with a Semi-Lagrangian Relaxation
Lagrangian relaxation is commonly used in combinatorial optimization to generate lower bounds for a minimization problem. We study a modified Lagrangian relaxation which generates an optimal integer solution. We call it semi-Lagrangian relaxation and illustrate its practical value by solving large-scale instances of the p-median proble
Lagrangian Relaxation for Mixed-Integer Linear Programming: Importance, Challenges, Recent Advancements, and Opportunities
Operations in areas of importance to society are frequently modeled as
Mixed-Integer Linear Programming (MILP) problems. While MILP problems suffer
from combinatorial complexity, Lagrangian Relaxation has been a beacon of hope
to resolve the associated difficulties through decomposition. Due to the
non-smooth nature of Lagrangian dual functions, the coordination aspect of the
method has posed serious challenges. This paper presents several significant
historical milestones (beginning with Polyak's pioneering work in 1967) toward
improving Lagrangian Relaxation coordination through improved optimization of
non-smooth functionals. Finally, this paper presents the most recent
developments in Lagrangian Relaxation for fast resolution of MILP problems. The
paper also briefly discusses the opportunities that Lagrangian Relaxation can
provide at this point in time
Combining Column Generation and Lagrangian Relaxation
Although the possibility to combine column generation and Lagrangian relaxation has been known for quite some time, it has only recently been exploited in algorithms. In this paper, we discuss ways of combining these techniques. We focus on solving the LP relaxation of the Dantzig-Wolfe master problem. In a first approach we apply Lagrangian relaxation directly to this extended formulation, i.e. no simplex method is used. In a second one, we use Lagrangian relaxation to generate new columns, that is Lagrangian relaxation is applied to the compact for-mulation. We will illustrate the ideas behind these algorithms with an application in Lot-sizing. To show the wide applicability of these techniques, we also discuss applications in integrated vehicle and crew scheduling, plant location and cutting stock problems
Submodular relaxation for inference in Markov random fields
In this paper we address the problem of finding the most probable state of a
discrete Markov random field (MRF), also known as the MRF energy minimization
problem. The task is known to be NP-hard in general and its practical
importance motivates numerous approximate algorithms. We propose a submodular
relaxation approach (SMR) based on a Lagrangian relaxation of the initial
problem. Unlike the dual decomposition approach of Komodakis et al., 2011 SMR
does not decompose the graph structure of the initial problem but constructs a
submodular energy that is minimized within the Lagrangian relaxation. Our
approach is applicable to both pairwise and high-order MRFs and allows to take
into account global potentials of certain types. We study theoretical
properties of the proposed approach and evaluate it experimentally.Comment: This paper is accepted for publication in IEEE Transactions on
Pattern Analysis and Machine Intelligenc
Lagrangian Relaxation for MAP Estimation in Graphical Models
We develop a general framework for MAP estimation in discrete and Gaussian
graphical models using Lagrangian relaxation techniques. The key idea is to
reformulate an intractable estimation problem as one defined on a more
tractable graph, but subject to additional constraints. Relaxing these
constraints gives a tractable dual problem, one defined by a thin graph, which
is then optimized by an iterative procedure. When this iterative optimization
leads to a consistent estimate, one which also satisfies the constraints, then
it corresponds to an optimal MAP estimate of the original model. Otherwise
there is a ``duality gap'', and we obtain a bound on the optimal solution.
Thus, our approach combines convex optimization with dynamic programming
techniques applicable for thin graphs. The popular tree-reweighted max-product
(TRMP) method may be seen as solving a particular class of such relaxations,
where the intractable graph is relaxed to a set of spanning trees. We also
consider relaxations to a set of small induced subgraphs, thin subgraphs (e.g.
loops), and a connected tree obtained by ``unwinding'' cycles. In addition, we
propose a new class of multiscale relaxations that introduce ``summary''
variables. The potential benefits of such generalizations include: reducing or
eliminating the ``duality gap'' in hard problems, reducing the number or
Lagrange multipliers in the dual problem, and accelerating convergence of the
iterative optimization procedure.Comment: 10 pages, presented at 45th Allerton conference on communication,
control and computing, to appear in proceeding
OPERASI PENJADWALAN BEBERAPA PEMBANGKIT TERMAL DENGAN KEKANGAN TRANSMISI MENGGUNAKAN METODE LAGRANGIAN RELAXATION
Operasi Penjadwalan Beberapa Pembangkit Termal dengan Kekangan Transmisi Menggunakan Metode Lagrangian Relaxation membahas tentang perencanaan menjadwalkan pembangkit-pembangkit termal yang akan dioperasikan dengan pengaruh kekangan transmisi yakni rugi-rugi transmisi. Dari penjadwalan tersebut dicari daya pembangkitan yang memenuhi permintaan beban secara optimal dan biaya yang digunakan pun ekonomis. Hasil penjadwalan metode yang dipilih penulis dalam penelitian ini akan dibandingkan dengan hasil realisasi dari Penyaluran dan Pusat Pengatur Beban (P3B) PT PLN (Persero) Jawa Bali. Perbandingan ini bertujuan untuk membuktikan apakah metode yang digunakan penulis lebih baik dari realisasi PLN, sehingga diketahui pula keunggulan dan kelemahan metode Lagrangian Relaxation. Dengan adanya penelitian ini diharapkan dapat mengembangkan metode Lagrangian Relaxation, agar diketahui sejauh mana metode ini mampu memberi kontribusi di lapangan.
Operations scheduling several generating thermal with transmission constrained using Lagrangian Relaxation Method discusses the planning schedule of thermal power plants which will be operated by the confinement effect of the transmission of transmission losses. The scheduling of generation sought the optimally power to satisfy the load demand and costs used were economic. Results scheduling method chosen authors in this study will be compared with actual results of Distribution and Load Control Center (P3B) PT PLN (Persero) Java Bali. This comparison aims to prove whether the method used by the author better than the realization of PLN, so it is also known advantages and disadvantages of the Lagrangian Relaxation method. With the research is expected to develop a Lagrangian Relaxation method, in order to know the extent to which this method is able to contribute in the field
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