In this paper, a class of linear multiplicative problems (LMP) are considered, which cover many applications and are known to be NP-hard. For finding the globally optimal solution to problem (LMP) with a pre-specified ε-tolerance, problem (LMP) is first transformed into an equivalent problem (EP) via introducing the variable transformation. And, a novel linear relaxation technique is presented by exploiting the special structure of problem (EP), for deriving the linear relaxation programming which can be used to acquire the upper bound of the optimal value to problem (EP). A branch and bound algorithm is then located for globally solving problem (LMP). The convergence of the algorithm is established and its computational complexity is estimated. Finally, numerical results are reported to illustrate the feasibility and efficiency of the proposed algorithm.

在本文中，考虑了一类线性乘法问题（LMP），它涵盖了许多应用并且已知是 NP 难的。用于找到具有预先指定ε的问题的全局最优解 (LMP)- 容差问题（LMP）首先通过引入变量变换转化为等价问题（EP）。并且，利用问题（EP）的特殊结构，提出了一种新的线性松弛技术，用于推导线性松弛规划，可用于获取问题（EP）的最优值的上界。然后为全局求解问题 (LMP) 定位分支定界算法。建立算法的收敛性并估计其计算复杂度。最后，报告了数值结果，以说明所提出算法的可行性和效率。