A study on global optimization algorithms for multiplicative programming problems

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In this research, we studied practical algorithms for solving multiplicative programming problems, a class of optimization problems involving products of some convex functions. Although this class is known as a typical multi-extremal global optimization problem, we showed that it is possible to design efficient algorithms both in theoretical and practical senses, by exploiting its special structures. A few of the results are listed below :1 We studied a problem maximizing a single linear function over an efficient set. This problem is associated with multi-criteria decision making and belongs to multi-extremal global optimization. When the number of criteria is up to three, we showed that the problem can be solved efficiently in the same way as the low-rank linear multiplicative programming problem.2 We developed a finite branch-and-bound algorithm for minimizing a product of several affine functions over a polyhedral set. Since the logarithm of the objective function is separable into … More a sum of concave functions, we use this special structure and propose a rectangular branch-and-bound algorithm. We carried out bounding operations in two stages to strengthen the lower bound. The computational result indicated that the algorithm is remarkably efficient.3 The sum-of-linear-ratio problem is an important subclass of multiplicative programming problems. We developed a rectangular branch-and-bound algorithm for solving this problem. Since the number of ratios is less than ten in most applications, we carried out branching operations in the vector space of ratios. As a result, we could obtain globally optimal solutions much efficiently than using the existing algorithms.4 When using the branch-and-bound algorithm to solve multiplicative programming problems, we need to solve linear and/or quadratic programming problems iteratively. Therefore, the procedure for linear and/or quadratic programming problems seriously affects on the efficiency of the algorithm. We then studied some iterative algorithms for the linear complementarity problem, the class of these problems, and showed their worst-case computational complexity. Less