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

Abstract

Consider a large mixed integer linear problem where structure of the constraint matrix is sparse, with independent blocks, and coupling constraints and variables. There is one of the groups of constraints to make difficult the application of Benders scheme decomposition. 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 resolution of the dual problem, structured starting from the bundle methods. 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.

Handing Optimization Energy Consumption in Heterogeneous Wireless Sensor Networks

“…The use of integer linear programming approach is limited to a certain level of complexity that sometimes is not enough for a real size network (Table 4). It is expected that other hybrid methodologies [15, 34] can be applied in the solution of the problem.…”

Dye Schedule Optimization: A Case Study in a Textile Industry

“…In this aspect, it is necessary to research the possibility of applying large scale decompositions, namely, Benders, Danzig-Wolfe or cross-decomposition decompositions, to the traveling salesman problem to obtain an exact solution. Other options are structured in hybrid methods [29]. Figure 5 presented by the LINGO software [28], shows the fill-in of the matrix model composed of values in zeros and ones, varying only in the number of variables and constraints.…”

“…In this aspect, it is necessary to research the possibility of applying large scale decompositions, namely, Benders, Danzig-Wolfe or cross-decomposition decompositions, to the traveling salesman problem to obtain an exact solution. Other options are structured in hybrid methods [29]. Considering that this is the application of the traveling salesman problem, there is a limitation of the problem considering that it is a large problem that may involve thousands of variables and restrictions.…”

“…In this aspect, it is necessary to research the possibility of applying large scale decompositions, namely, Benders, Danzig-Wolfe or cross-decomposition decompositions, to the traveling salesman problem to obtain an exact solution. Other options are structured in hybrid methods [29].…”

Dealing with Nonregular Shapes Packing

“…A specialization of this problem is the placement of irregular figures with characteristics similar to regular cut, but dealing with irregular figures, the nesting problems [1]. They have been known as NP-hard due to their difficulty where few exact methods have been reported in the literature [2], where it is possible to find promising solutions by applying methodologies addressed in [3,4].…”