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This article presents a simplicial branch and duality bound algorithm for globally solving the sum of convex-convex ratios problem with nonconvex feasible region. To our knowledge, little progress has been made for globally solving this problem so far. The algorithm uses a branch and bound scheme where the Lagrange duality theory is used to obtain the lower bounds. As a result, the lower-bounding subproblems during the algorithm search are all ordinary linear programs that can be solved very efficiently. It has been proved that the algorithm possesses global convergence. Finally, the numerical experiments are given to show the feasibility of the proposed algorithm.

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This paper empirically tests the impact mechanism of digital economy development on manufacturing total factor productivity, using data from Chinese manufacturing enterprises from 2011 to 2020, and based on the theoretical framework of the impact of digital economy development on manufacturing total factor productivity. The development of the digital economy has been found to have a significant positive impact on the total factor productivity of the manufacturing industry. The heterogeneity effect demonstrates that the digital economy in coastal areas has a significant effect on the improvement of manufacturing total factor productivity, with the eastern coastal area having the strongest effect; the digital economy in the Yellow River’s middle reaches, the Yangtze River’s middle reaches, and the southwest also having a significant effect, with the effect in the southwest region being more significant; and the digital economy in the northwest and northeast having no effect. Possible reasons include larger bottlenecks in the western region’s labor force structure, technology level, and management capabilities, which may lead to the inability of enterprises to effectively absorb the dividends of digital change and apply the scenarios, thus affecting the release of their productivity effects.

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