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This paper aims to globally solve a generalized affine fractional program problem (GAFPP). Firstly, by introducing some outer space variables and performing equivalent transformations, we can derive the equivalence problem (EP) of the GAFPP. Secondly, by constructing a novel linear relaxation method, we can deduce the affine relaxation problem (ARP) of the EP. Next, by solving the ARP to compute the lower bound, we propose a new outer space branch-and-bound algorithm for tackling the GAFPP. Then, the global convergence of the algorithm is proved, and the computational complexity of the algorithm in the worst case is analyzed. Finally, numerical experimental results are reported to illustrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

This article presents and validates a potential practical algorithm for minimizing the sum of affine fractional functions over a polyhedron. During the branch and bound search, this algorithm computes the lower bounds by solving the affine relaxation problems of the equivalent problem, which are derived by utilizing a two-level affine relaxation technique. By successive refinement and successively solving a series of affine relaxation problems, the algorithm is convergent to the global minimum of the primal problem. Moreover, the gap between the objective function and its affine relaxation function is derived for the first time, and when the partition interval tends to be infinitesimal, the gap is infinitely close to zero. Furthermore, the maximum iterations of the proposed algorithm are derived by estimating its computational complexity. Some test problems are solved to verify the potential practical and computational advantages of the algorithm. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we present a parametric linear relaxation algorithm for globally solving the nonconvex quadratic programming (NQP). In this algorithm, a new parametric linearized technique is proposed for generating parametric linear relaxation programming (PLRP) of the NQP, which can be used to determine the lower bound of global minimum value of the NQP. To improve the convergent speed of the proposed algorithm, a pruning operation is employed to compress the investigated region. By subdividing subsequently the initial domain and solving subsequently a series of parametric linear relaxation programming problems over the subdivided domain, the proposed algorithm is convergent to the global minimum of the NQP. Finally, an engineering problem for the design of heat exchanger network and some test examples are used to verify the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper presents a deterministic global optimization algorithm for solving minimax linear fractional programming (MLFP). In this algorithm, a new linearization technique is proposed, which uses more information of the objective function of the (MLFP) than other techniques. By utilizing this new linearization technique, the initial nonconvex programming problem (MLFP) is reduced to a sequence of linear relaxation programming (LRP) problems, which can provide reliable lower bound of the optimal value. The proposed algorithm is convergent to the global minimum of the (MLFP) through the successive refinement of the feasible region and solutions of a series of the (LRP). Compared with the known algorithms, numerical results show that the proposed algorithm is robust and effective. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, a global optimization algorithm is proposed for solving sum of generalized polynomial ratios problem (P) which arises in various practical problems. Due to its intrinsic difficulty, less work has been devoted to globally solve the problem (P). For such problems, we present a branch and bound algorithm. In this method, by utilizing exponent transformation and new three-level linear relaxation method, a sequence of linear relaxation programming of the initial nonconvex programming problem (P) are derived which are embedded in a branch and bound algorithm. The proposed method need not introduce new variables and constraints and it is convergent to the global minimum of prime problem by means of the subsequent solutions of a series of linear programming problems. Several numerical examples in the literatures are tested to demonstrate that the proposed algorithm can systematically solve these examples to find the approximate ϵ-global optimum. [Copyright &y& Elsevier]

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: Generalized geometric programming (GGP) problem occurs frequently in engineering design and management. In this paper, a branch-and-pruning global optimization algorithm is proposed for GGP. By utilizing some transformations, a linear relaxation of the problem (GGP) is obtained based on the linear lower bound functions of objective and constraint functions inside some hyperrectangle region. Then a new pruning technique is given to accelerate the convergence of the given algorithm, and this pruning technique offers the possibility to cut away a large part of the current investigated region in which there no exist global optimum solution. The proposed algorithm which connects branch-and-bound method with the pruning technique successfully is convergent to the global minimum, according to the successive refinement of the linear relaxation of feasible region of the objective function and the solutions of a series of linear relaxation problems. And finally numerical experiment is given to illustrate the feasibility and efficiency of the proposed algorithm. [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]