Abstract
By reformulating the linear multiplicative programming problem (LMP) as an equivalent nonconvex programming problem (EP), we present a new accelerating outcome space branch-and-bound algorithm for globally solving the problem (LMP). Firstly, a linear relaxed programming problem is constructed, which can be used to compute the lower bound of the global optimal value of the problem (EP). Then, by subsequently subdividing the initial outcome space rectangle, and by solving a series of linear relaxed programming problems, the global optimal solution of the problem (LMP) can be obtained. It's worth mentioning that a new region reducing method and a new linear relaxed programming with a higher degree of tightness are also constructed to improve the computational efficiency in the presented algorithm. The global convergence of the presented algorithm is established, and its computational complexity is estimated. Finally, the numerical tests indicate that the presented algorithm has the higher computational efficiency than the known algorithms.
Highlights
Consider the following linear multiplicative programming problem: p (LMP) : min f (x) =(diT x + gi) i=1s.t. x ∈ X = {x | Ax ≤ b}, where p ≥ 2, cTi x + ei and diT x + gi, i = 1, . . . , p, are all bounded affine functions defined on Rn, A ∈ Rm×n is an arbitrary real matrix, b ∈ Rm is an arbitrary real vector, X is a nonempty bounded polyhedron set
IMPROVING THE COMPACTNESS OF LINEAR RELAXED PROGRAMMING PROBLEM Let UB be the currently known upper bound, in order to improve the computational efficiency of the proposed algorithm, for any rectangle F = [L, U ] ⊆ F0, we aim at replacing the linear relaxed programming problem (LPF ) by a new linear relaxed programming problem (LP1F ), which has a higher degree of tightness than the problem (LPF )
Gi(x, βi) must be less than or equal to the currently known i=1 upper bound UB, from the constructing process of the linear relaxed programming problem (LPF ), by adding the linear p constraint ri ≤ UB into the constrained condition of the i=1 problem (LPF ), without losing any global optimal solution of the problem (LMP), we obtain a new linear relaxed programming problem (LP1F ), which has a higher degree of tightness than the problem (LPF ), so that we can improve the convergent speed of the presented algorithm in this paper
Summary
The above algorithms have globally solved the problem (LMP) Most of these algorithms require that cTi x + ei ≥ 0 and diT x + gi ≥ 0 in the problem, except for Refs. A new outcome space branch and bound algorithm for the problem (LMP) is presented in this paper. The global optimal solution of the problem (LMP) can be obtained by subsequently subdividing the outcome space Rp of the affine functions diT x + gi, i = 1, .
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