scholarly journals Variant of Constants in Subgradient Optimization Method over Planar 3-Index Assignment Problems

2016 ◽  
Vol 21 (1) ◽  
pp. 4
Author(s):  
Sutitar Maneechai
Keyword(s):  
Optimization Method ◽  
Subgradient Optimization ◽  
Assignment Problems ◽  
Index Assignment

scholarly journals An Improved Subgradiend Optimization Technique for Solving IPs with Lagrangean Relaxation

2013 ◽  
Vol 61 (2) ◽  
pp. 135-140
Author(s):  
M Babul Hasan ◽  
Md Toha
Keyword(s):  
Dual Problem ◽  
Optimization Problems ◽  
Optimization Technique ◽  
Concave Function ◽  
Optimization Method ◽  
Duality Gap ◽  
Dual Function ◽  
Subgradient Optimization ◽  
Step Size ◽  
Primal Problem

The objective of this paper is to improve the subgradient optimization method which is used to solve non-differentiable optimization problems in the Lagrangian dual problem. One of the main drawbacks of the subgradient method is the tuning process to determine the sequence of step-lengths to update successive iterates. In this paper, we propose a modified subgradient optimization method with various step size rules to compute a tuning- free subgradient step-length that is geometrically motivated and algebraically deduced. It is well known that the dual function is a concave function over its domain (regardless of the structure of the cost and constraints of the primal problem), but not necessarily differentiable. We solve the dual problem whenever it is easier to solve than the primal problem with no duality gap. However, even if there is a duality gap the solution of the dual problem provides a lower bound to the primal optimum that can be useful in combinatorial optimization. Numerical examples are illustrated to demonstrate the method. DOI: http://dx.doi.org/10.3329/dujs.v61i2.17059 Dhaka Univ. J. Sci. 61(2): 135-140, 2013 (July)

scholarly journals Lagrangian Duality for Robust Problems with Decomposable Functions: The Case of a Robust Inventory Problem

2020 ◽  
Author(s):  
Filipe Rodrigues ◽  
Agostinho Agra ◽  
Cristina Requejo ◽  
Erick Delage
Keyword(s):  
Local Search ◽  
Lot Sizing ◽  
Optimization Method ◽  
Iterated Local Search ◽  
Lagrangian Approach ◽  
Computational Results ◽  
Subgradient Optimization ◽  
Lagrangian Multipliers ◽  
Local Search Heuristic ◽  
Lot Sizing Problem

We consider a class of min-max robust problems in which the functions that need to be “robustified” can be decomposed as the sum of arbitrary functions. This class of problems includes many practical problems, such as the lot-sizing problem under demand uncertainty. By considering a Lagrangian relaxation of the uncertainty set, we derive a tractable approximation, called the dual Lagrangian approach, that we relate with both the classical dualization approximation approach and an exact approach. Moreover, we show that the dual Lagrangian approach coincides with the affine decision rule approximation approach. The dual Lagrangian approach is applied to a lot-sizing problem, in which demands are assumed to be uncertain and to belong to the uncertainty set with a budget constraint for each time period. Using the insights provided by the interpretation of the Lagrangian multipliers as penalties in the proposed approach, two heuristic strategies, a new guided iterated local search heuristic, and a subgradient optimization method are designed to solve more complex lot-sizing problems in which additional practical aspects, such as setup costs, are considered. Computational results show the efficiency of the proposed heuristics that provide a good compromise between the quality of the robust solutions and the running time required in their computation. Summary of Contribution: The paper includes both theoretical and algorithmic contributions for a class of min-max robust optimization problems where the objective function includes the maximum of a sum of affine functions. From the theoretical point of view, a tractable Lagrangian dual model resulting from a relaxation of the well-known adversarial problem is proposed, providing a new perspective of well-known models, such as the affinely adjustable robust counterpart (AARC) and the dualization technique introduced by Bertsimas and Sim. These results are particularized to lot-sizing problems. From the algorithm point of view, efficient heuristic schemes—which exploit the information based on the interpretation of the Lagrangian multipliers to solve large size robust problems—are proposed, and their performance is evaluated through extensive computational results based on the lot-sizing problem. In particular, a guided iterated local search and a subgradient optimization method are proposed and compared against the dualization approach proposed by Bertsimas and Sim and with several heuristics based on the AARC approach, which include an iterated local search heuristic and a Benders decomposition approach. Computational results show the efficiency of the proposed heuristics, which provide a good compromise between the quality of the robust solutions and the running time.

scholarly journals A Lagrangean Heuristic For The Degree Constrained Minimal Spanning Tree Problem

2010 ◽  
Vol 14 (3) ◽  
Author(s):  
Rakesh Kawatra
Keyword(s):  
Lower Bound ◽  
Spanning Tree ◽  
Feasible Solution ◽  
Objective Function Value ◽  
Optimization Method ◽  
Heuristic Procedure ◽  
Minimal Spanning Tree ◽  
Subgradient Optimization ◽  
Lagrangean Heuristic ◽  
Lagrangean Solution

In this paper we present a new heuristic procedure to solve the degree constrained minimal spanning tree problem. This procedure uses Lagrangian relaxation of the integer programming formulation of the problem to get a lower bound for the optimal objective function value. A subgradient optimization method is used to find multipliers that give good lower bounds. A branch exchange procedure is used after each iteration of the subgradient optimization to generate a feasible solution from an infeasible Lagrangean solution. Computational results are given for problems with up to 300 nodes. The heuristic procedure presented here gives optimal solutions in most instances. For problem sets that were not solved optimally, the gap between the lower bound and the feasible solution was less than 10-2 percent.

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