In this paper, we first consider a modified Levenberg–Marquardt method for solving nonlinear equations and show that the method converges superlinearly to a solution of the nonlinear equations under suitable conditions. Then, we reformulate the C-/M-/S-stationarity conditions of mathematical program with complementarity constraints as nonlinear equations so that we may employ the modified Levenberg–Marquardt method to solve these stationarity systems. Preliminary numerical experiments show that the new approach is promising.

This paper considers a sales mode selection problem between resale and agency modes on e-commerce platforms for a manufacturer with traditional retail channel, direct selling channel, and e-commerce platform channel. By considering the factors price competition, market shares, and commission rate, we construct two leader-follower models with the manufacturer as a leader and traditional retailer and e-commerce platform as followers. To obtain optimal solutions, we discuss the conditions under which the upper and lower models are convex and then give optimal strategies for all members in the network. Through numerical experiments, we analyze the impact of price competition intensity, market shares, and commission rate on mode selection strategies and the changing trend of each member’s optimal pricing and profit under different sales modes. The numerical results reveal the following revelations: If the market share of the traditional retail channel is lower than the direct selling channel, the manufacturer should choose the agency mode when the market share of the direct selling channel and price competition are lower or when the market share of the direct selling channel together with the price competition and the commission rate is higher; otherwise, the manufacturer should choose the resale mode. If the market share of the direct selling channel is lower than the traditional retail channel, the manufacturer should choose the agency mode when the price competition is weak and choose the resale mode when the price competition is strong. Under certain conditions, a win–win situation can be achieved no matter how the manufacturer chooses.