It aims to study the production and emission reduction decisions of production enterprises under carbon constraints. In the case of carbon constraints in production, manufacturers have four strategic choices: production within the carbon quota, adopting emission reduction technologies, purchasing carbon emission rights, and using emission reduction technologies and purchasing carbon emission rights. Based on the income model of production enterprises under four different strategies, first, under the condition of maximizing the interests of production enterprises, the optimal profit, optimal production, optimal total carbon emission, and optimal emission reduction rate under different carbon constraints are determined, and summarize its laws. Afterward, in order to further optimize corporate profits, the impact of changes in the carbon reduction scale cost and consumers’ low-carbon preference was studied. Several important conclusions are shown as follows. First, the stricter the carbon constraint policy, the greater the optimal emission reduction rate of enterprises. Second, the adoption of emission reduction technology can effectively reduce the impact of carbon constraint on output. Third, the optimal strategy is to both reduce emissions and purchase carbon emission rights, which can realize environmental economic dividends. Fourth, the lower the cost factor of the carbon reduction scale and the higher the low-carbon preference of consumers, the easier it is for firms to achieve carbon sufficiency in their production.

In this paper, we study the optimality conditions and perform a stability analysis for the second-order cone constrained variational inequalities (SOCCVI) problem. The Lagrange function and Karush–Kuhn–Tucker (KKT) condition of the SOCCVI problem is given, and the optimality conditions for the SOCCVI problem are studied. Then, the second-order sufficient condition satisfying the constrained nondegenerate condition is proved. The strong second-order sufficient condition is defined. And the nonsingularity of Clarke’s generalized Jacobian of the KKT point, the strong regularity of the KKT point, the uniform second-order growth condition, the strong stability of the KKT point, and the local Lipschtiz homeomorphism of the KKT point for the SOCCVI problem are proved to be equivalent to each other. Then, the stability theorem of the SOCCVI problem is obtained.

Big data service providers (BDSPs) play a critical role in supporting the digital transformation of closed-loop supply chains (CLSCs). However, as the number of CLSC members increases, traditional coordination contracts become complex in the big data era, which challenges effective collaboration and contract implementation. To address this issue, this paper investigates the profit coordination problem in a CLSC with a BDSP, with the aim of lowering the contract implementation threshold and facilitating flexible adjustment of contract terms. This study applies the variational inequality method to derive the necessary conditions under which a CLSC with the participation of a BDSP achieves maximum system profit. The results indicate that these necessary conditions are as follows. First, the wholesale price is equal to the unit cost of new products. Second, the optimal payment level is positively correlated with production volume, unit cost savings, the BDSP marketing effort sensitivity coefficient, and the BDSP recycling effort sensitivity coefficient, while it is negatively correlated with the retail price sensitivity coefficient, the recycling price sensitivity coefficient, and the big data service cost coefficient.