Tight convex relaxation-based simplex branch-and-bound algorithm for generalized linear multiplicative programming

Generalized linear multiplicative programming arises in practical contexts such as venture-capital analysis and system stability assessment. The algorithm in Zhanget al. (2026) eliminates the primal variables before constructing lower bounds via convex envelope operations, which may discard useful primal information and produce loose bounds on some instances. To address this issue, an improved global optimization algorithm is developed. Within the same simplex-domain reformulation framework, the proposed method keeps the primal variables and directly relaxes the inherent equality constraints, thereby yielding a new convex lower-bound estimator for each sub-simplex. By combining this relaxation with simplex-based branching in a branch-and-bound framework, global convergence is guaranteed. Theoretical analysis establishes that the algorithm attains a global ϵ -optimal solution for any tolerance ϵ > 0 , and an explicit computational-complexity estimate is provided. Numerical experiments demonstrate the effectiveness and computational advantages of the proposed algorithm.