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Abstract: In this technical note, we give a short extension application for a deterministic global optimization algorithm proposed in [Y. Ji, K.-C. Zhang, S.-J. Qu, A deterministic global optimization algorithm, Applied Mathematics and Computation 185 (2007) 382–387]. Actually our result is slightly more general, since it does not impose any special sign restrictions on numerators and denominators of the ratios. The only assumption is that the denominators of the ratios are nonzero over the feasible region of the problem. For clarity we use the same notations for the different equivalent problems and the corresponding relaxation linear programming (RLP) as done in Ji et al. (2007). [Copyright &y& Elsevier]

Abstract: In this technical note, we give a short extension application for nonlinear sum of ratios problem (P) considered in [Y.J. Wang, K.C. Zhang, Global optimization of nonlinear sum of ratios problem, Applied Mathematics and Computation 158 (2004) 319–330]. Actually our result is slightly more general, since we do not specify additional positive coefficient for each ratio. In this note, we use different equivalent problem as done in Wang and Zhang (2004). Our method introduce p variables less than other method (Wang and Zhang, 2004), and our approach need not additional special program to obtain the upper and lower bound of numerator and denominator for each ratio in the objective function. [Copyright &y& Elsevier]

Abstract: This paper presents a branch-and-bound algorithm for globally solving a wide class of generalized linear fractional programming problems (GLFP). This class includes such problems as: minimizing a sum, or error for product of a finite number of ratios of linear functions, linear multiplicative programming, polynomial programming, etc. – over nonconvex feasible region. First a problem (Q) is derived which is equivalent to problem (GLFP). In the algorithm, lower bounds are derived by solving a sequence of linear relaxation programming problems, which is based on the construction of the linear lower bounding functions for the objective function and constraint functions of problem (Q) over the feasible region. Convergent property of the presented algorithm is proved and numerical results are given to show the feasibility of the proposed algorithm. [Copyright &y& Elsevier]

Abstract: This paper presents a global optimization algorithm for a class of multiplicative programming with exponent under multiplicative constraints (MPE). By utilizing equivalent problem of MPE in the sense that they have the same optimal solution, tangential hypersurfaces and concave envelope approximations a linear relaxation of equivalent problem is received. Thus the initial nonconvex programming problem (MPE) is reduced to a sequence of linear programming problems through the successive refinement of a linear relaxation of feasible region of the objective function. The proposed algorithm is convergent to the globally optimal solution of MPE by means of the subsequent solutions of a series of linear programming problems. Numerical results indicate that the proposed algorithm is extremely robust and can be used successfully to solve global minimum of MPE on microcomputer. [Copyright &y& Elsevier]