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### Lagrangian Smoothing Heuristics for Max-Cut

Hernán Alperin, Ivo Nowak
2005 Journal of Heuristics
This paper presents a smoothing heuristic for an NP-hard combinatorial problem.  ...  Furthermore, to the best of our knowledge, this is the first time that a Lagrangian heuristic is combined with pathfollowing techniques.  ...  Kiwiel for making NOA 3.0 available, and Stefan Vigerske, who helped us with the C++ coding and Linux scripting. We also like to thank two anonymous referees for their critics to our first draft.  ...

### On greedy construction heuristics for the MAX-CUT problem

Sera Kahruman, Elif Kolotoglu, Sergiy Butenko, Illya V. Hicks
2007 International Journal of Computational Science and Engineering (IJCSE)
This paper compares the performance of several greedy construction heuristics for MAX-CUT problem.  ...  Given a graph with non-negative edge weights, the MAX-CUT problem is to partition the set of vertices into two subsets so that the sum of the weights of edges with endpoints in different subsets is maximized  ...  Lagrangian Smoothing Heuristics for Max-Cut. Journal of Heuristics, 11: 447-463, 2005.  ...

### Heuristics for the multi-item capacitated lot-sizing problem with lost sales

Nabil Absi, Boris Detienne, Stéphane Dauzère-Pérès
2013 Computers & Operations Research
To find a good lower bound, we use a Lagrangian relaxation of the capacity constraints, when single-item uncapacitated lot-sizing problems with lost sales have to be solved.  ...  To find feasible solutions, we propose a non-myopic heuristic based on a probing strategy and a refining procedure. We propose also some metaheuristic principles to improve solutions.  ...  Algorithm 2 2 Lagrangian heuristic Let s L be a Lagrangian solution s * ← Cut(s L ) s ← s L ; iter ← 1 while s is not feasible and iter ≤ M axIter do s ← Improved Smooth(s) s ← Cut(s) if cost(s) < cost  ...

### About Lagrangian Methods in Integer Optimization

Antonio Frangioni
2005 Annals of Operations Research
It is less often realized that this equivalence is effective, in that basically all known algorithms for solving the Lagrangian dual either naturally compute an (approximate) optimal solution of the "convexified  ...  It is well-known that the Lagrangian dual of an Integer Linear Program (ILP) provides the same bound as a continuous relaxation involving the convex hull of all the optimal solutions of the Lagrangian  ...  Acknowledgments I'm indebted with Giorgio Gallo, Alberto Borghetti, Fabrizio Lacalandra, Carlo Alberto Nucci, Andrea Lodi, and Giovanni Rinaldi for their fundamental contribution in the various pieces  ...

### Exact makespan minimization of unrelated parallel machines

Edvin Åblad, Ann-Brith Strömberg, Domenico Spensieri
2021 Open Journal of Mathematical Optimization
For this relaxation we derive a criterion for variable fixing and prove the zero duality gap property for the case of two parallel machines.  ...  To improve these bounds and to enable the solution of larger instances, we propose a branch-and-bound method based on a Lagrangian relaxation of the assignment constraints.  ...  Finally, we would like to thank the anonymous reviewers for their valuable feedback.  ...

### An efficient surrogate subgradient method within Lagrangian relaxation for the Payment Cost Minimization problem

M. A. Bragin, P. B. Luh, J. H. Yan, Nanpeng Yu, Xu Han, G. A. Stern
2012 2012 IEEE Power and Energy Society General Meeting
In the presented methodology the problem structure is exploited using Lagrangian relaxation and the relaxation of the integrality constraints is exploited using branch-and-cut.  ...  For large Payment Cost Minimization problems, the method can find significantly better feasible solutions within less CPU time than that obtained by standard branch-and-cut methods implemented in commercial  ...  These directions are smooth from one iteration to the next, thus the zigzagging is avoided.  ...

### A Hybrid Approach of Bundle and Benders Applied Large Mixed Linear Integer Problem

Placido Rogerio Pinheiro, Paulo Roberto Oliveira
2013 Journal of Applied Mathematics
In this work, we propose the following algorithm; a Lagrangian relaxation is made on the mentioned set of constraints; we presented a process heuristic for the calculation of the multiplier through the  ...  According to the methodology proposed, for each iteration of the algorithm, we propose Benders decomposition where quotas are provided for the value function andε-subgradient.  ...  heuristics for selecting cuts, given that the accumulations of all inequalities explode the subproblem.  ...

### A dual ascent heuristic for obtaining a lower bound of the generalized set partitioning problem with convexity constraints

Stefania Pan, Roberto Wolfler Calvo, Mahuna Akplogan, Lucas Létocart, Nora Touati
2019 Discrete Optimization
The proposed dual ascent heuristic is based on a reformulation and it uses Lagrangian relaxation and subgradient method.  ...  In this paper we propose a dual ascent heuristic for solving the linear relaxation of the generalized set partitioning problem with convexity constraints, which often models the master problem of a column  ...  Acknowledgement The authors would like to thank the anonymous reviewers and associate editor for their helpful suggestions.  ...

### A Review on the Performance of Linear and Mixed Integer Two-Stage Stochastic Programming Software

Juan J. Torres, Can Li, Robert M. Apap, Ignacio E. Grossmann
2022 Algorithms
This paper presents a tutorial on the state-of-the-art software for the solution of two-stage (mixed-integer) linear stochastic programs and provides a list of software designed for this purpose.  ...  The best lower bound of (PNAC) is computed by solving the following maximization problem, referred to as the Lagrangian dual problem: Z LD = max λ D(λ) (18) The Lagrangian dual is a concave non-smooth  ...  Algorithm 2: Cutting-plane dual search 1 Set k ←− 0, z LB ←− −∞ and λ 0 ←− 0 2 repeat 3 SOLVE (17b) to obtain (x k ω , y k ω ) and D ω (λ k ω ) for given λ k ω for each ω ∈ Ω 4 set z LB ←− max{z LB , D  ...

### A Bundle Approach for SDPs with Exact Subgraph Constraints [article]

Elisabeth Gaar, Franz Rendl
2019 arXiv   pre-print
Computational experiments on the Max-Cut, stable set and coloring problem show the efficiency of this approach.  ...  We introduce a computational framework for these relaxations designed to cope with these difficulties.  ...  We first look at the exact subgraph relaxations for Max-Cut.  ...

### Multicommodity Network Flows: A Survey, Part II: Solution Methods

I-Lin Wang
2018 International Journal of Operations Research
Finally the computational performance of different solution methods in literature is compared and directions for future research are suggested.  ...  price-directive, resourcedirective, and basis partitioning methods, then summarizes recent progress in applying approximation algorithms, interior-point algorithms, quadratic programming algorithms, and heuristics  ...  Most of them are related to augmented Lagrangian methods which find a saddle point of a smooth min-max function obtained by augmenting the Lagrangian with quadratic terms of both primal and dual variables  ...

### Improving problem reduction for 0–1 Multidimensional Knapsack Problems with valid inequalities

Hanyu Gu
2016 Computers & Operations Research
Recently a novel core based heuristic  is proposed which is based on the Lagrangian Relaxation (LR) method. The Lagrangian relaxation of MKP can  ...  Although cuts are commonly used in the branch and cut algorithms for general ILP solvers, our 40 application allows for classes of cuts with more expensive computational costs.  ...  LR(λ) provides an upper bound for the MKP problem, and can be further strengthened by solving the Lagrangian dual problem LD = min λ LR(λ) (3) which is a non-smooth convex optimisation problem, and can  ...

### Lagrangian relaxation based decomposition for well scheduling in shale-gas systems

Brage Rugstad Knudsen, Ignacio E. Grossmann, Bjarne Foss, Andrew R. Conn
2014 Computers and Chemical Engineering
In this paper, we present a Lagrangian relaxation based scheme for shut-in scheduling of distributed shale multiwell systems.  ...  The scheme optimizes shut-in times and a reference rate for each multi-well pad, such that the total produced rate tracks a given short-term gas demand for the field.  ...  Still, any effi- cient implementation of a Lagrangian relaxation requires a stable and reliable method for finding optimal Lagrangian multipliers, as well as a heuristic for finding primal feasible solutions  ...

### Fast Trust Region for Segmentation

Lena Gorelick, Frank R. Schmidt, Yuri Boykov
2013 2013 IEEE Conference on Computer Vision and Pattern Recognition
In this paper we propose a Fast Trust Region (FTR) approach for optimization of segmentation energies with nonlinear regional terms, which are known to be challenging for existing algorithms.  ...  In general, this approach can be used only when some efficient constrained optimization algorithm is available for the selected non-linear (more accurate) approximation model.  ...  We use the floating point precision in the standard code for graph-cuts  .  ...

### Optimization of an Integrated Lot Sizing and Cutting Stock Problem in the Paper Industry

Sônia Cristina Poltroniere, Silvio Alexandre Araujo, Kelly Cristina Poldi
2016 TEMA
Two mathematical models for the integrated problem are considered, and these models are solved both heuristically and using an optimization package.  ...  of items for each period.  ...  The authors proposed two heuristic methods for the solution based on the Lagrangian relaxation of the integration constraints, which proved to be appropriated to address the integrated problem.  ...
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