An Efficient Branch-and-Bound Algorithm for Globally Solving Minimax Linear Fractional Programming Problem

This paper presents an efficient outer space branch-and-bound algorithm for globally solving a minimax linear fractional programming problem (MLFP), which has a wide range of applications in data envelopment analysis, engineering optimization, management optimization, and so on. In this algorithm, by introducing auxiliary variables, we first equivalently transform the problem (MLFP) into the problem (EP). By using a new linear relaxation technique, the problem (EP) is reduced to a sequence of linear relaxation problems over the outer space rectangle, which provides the valid lower bound for the optimal value of the problem (EP). Based on the outer space branch-and-bound search and the linear relaxation problem, an outer space branch-and-bound algorithm is constructed for globally solving the problem (MLFP). In addition, the convergence and complexity of the presented algorithm are given. Finally, numerical experimental results demonstrate the feasibility and efficiency of the proposed algorithm.

An Efficient Algorithm for Solving a Class of Fractional Programming Problems

This paper presents an efficient branch-and-bound algorithm for globally solving a class of fractional programming problems, which are widely used in communication engineering, financial engineering, portfolio optimization, and other fields. Since the kind of fractional programming problems is nonconvex, in which multiple locally optimal solutions generally exist that are not globally optimal, so there are some vital theoretical and computational difficulties. In this paper, first of all, for constructing this algorithm, we propose a novel linearizing method so that the initial fractional programming problem can be converted into a linear relaxation programming problem by utilizing the linearizing method. Secondly, based on the linear relaxation programming problem, a novel branch-and-bound algorithm is designed for the kind of fractional programming problems, the global convergence of the algorithm is proved, and the computational complexity of the algorithm is analysed. Finally, numerical results are reported to indicate the feasibility and effectiveness of the algorithm.

Local Branching Strategy-Based Method for the Knapsack Problem with Setup

In this paper, we propose to solve the knapsack problem with setups by combining mixed linear relaxation and local branching. The problem with setups can be seen as a generalization of 0–1 knapsack problem, where items belong to disjoint classes (or families) and can be selected only if the corresponding class is activated. The selection of a class involves setup costs and resource consumptions thus affecting both the objective function and the capacity constraint. The mixed linear relaxation can be viewed as driving problem, where it is solved by using a special blackbox solver while the local branching tries to enhance the solutions provided by adding a series of invalid / valid constraints. The performance of the proposed method is evaluated on benchmark instances of the literature and new large-scale instances. Its provided results are compared to those reached by the Cplex solver and the best methods available in the literature. New results have been reached.

A Linear Relaxation-Based Heuristic for Iron Ore Stockyard Energy Planning

Planning the use of electrical energy in a bulk stockyard is a strategic issue due to its impact on efficiency and responsiveness of these systems. Empirical planning becomes more complex when the energy cost changes over time. The mathematical models currently studied in the literature consider many actors involved, such as equipment, sources, blends, and flows. Each paper presents different combinations of actors, creating their own transportation flows, thus increasing the complexity of this problem. In this work, we propose a new mixed integer linear programming (MILP) model for stockyard planning solved by a linear relaxation-based heuristic (LRBH) to minimize the plan’s energy cost. The proposed algorithm will allow the planner to find a solution that saves energy costs with an efficient process. The numerical results show a comparison between the exact and heuristic solutions for some different instances sizes. The linear relaxation approach can provide feasible solutions with a 3.99% average distance of the objective function in relation to the optimal solution (GAP) in the tested instances and with an affordable computation time in instances where the MILP was not able to provide a solution. The model is feasible for small and medium-sized instances, and the heuristic proposes a solution to larger problems to aid in management decision making.

Using multiflow formulations to solve the Steiner tree problem in graphs

We present three dfferent mixed integer linear models with a polynomial number of variables and constraints for the Steiner tree problem in graphs. The linear relaxations of these models are compared to show that a good (strong) linear relaxation can be a good approximation for the problem. We present computational results for the the STP OR-Library (J.E. Beasley) instances of type b, c, d and e.