In this paper, we propose a new global optimization algorithm, which can better solve a class of linear fractional programming problems on a large scale. First, the original problem is equivalent to a nonlinear programming problem: It introduces p auxiliary variables. At the same time, p new nonlinear equality constraints are added to the original problem. By classifying the coefficient symbols of all linear functions in the objective function of the original problem, four sets are obtained, which are I i + , I i − , J i + and J i − . Combined with the multiplication rule of real number operation, the objective function and constraint conditions of the equivalent problem are linearized into a lower bound linear relaxation programming problem. Our lower bound determination method only needs e i T x + f i ≠ 0 , and there is no need to convert molecules to non-negative forms in advance for some special problems. A output-space branch and bound algorithm based on solving the linear programming problem is proposed and the convergence of the algorithm is proved. Finally, in order to illustrate the feasibility and effectiveness of the algorithm, we have done a series of numerical experiments, and show the advantages and disadvantages of our algorithm by the numerical results.

Quadratically constrained quadratic programs (QCQP), which often appear in engineering practice and management science, and other fields, are investigated in this paper. By introducing appropriate auxiliary variables, QCQP can be transformed into its equivalent problem (EP) with non-linear equality constraints. After these equality constraints are relaxed, a series of linear relaxation subproblems with auxiliary variables and bound constraints are generated, which can determine the effective lower bound of the global optimal value of QCQP. To enhance the compactness of sub-rectangles and improve the ability to remove sub-rectangles, two rectangle-reduction strategies are employed. Besides, two ϵ-subproblem deletion rules are introduced to improve the convergence speed of the algorithm. Therefore, a relaxation and bound algorithm based on auxiliary variables are proposed to solve QCQP. Numerical experiments show that this algorithm is effective and feasible.