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The bargaining problem is a cooperative game in which all participants agree to form a coalition, instead of competing with each other, to get a higher payoff. Therefore, a key issue to address is determining the payoff for each participant in this coalition. The Nash bargaining solution indicates that for two participants, the problem of maximizing the payoff for each player can be modeled as the linear multiplicative programming problem (LMP). This highlights the importance of establishing efficient algorithms for solving (LMP). In this talk, we focus on developing various branch and bound methods for (LMP). To this end, a new bounding technique is proposed by integrating two linear relaxation methods, then a linear relaxation branch and bound algorithm is presented. Also, we establish a novel second order cone relaxation for (LMP), thus the process of solving (LMP) can be translated into solving a series of second order cone programs. Additionally, a simplicial branch and bound algorithm is designed to solve (LMP) based on a new convex quadratic relaxation and simplicial branching process. Finally, we analyze the convergence and complexity of the developed algorithms, and numerical results demonstrate their efficiency.
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