A generalized Alternating Direction Method of Multipliers (ADMM) framework is developed for a convex optimization problem involving nonlinear inequality constraints, extending ADMM beyond the classical linearly constrained setting. The method reformulates the nonlinear constraint to preserve separability and enables stable ADMM-type updates. Under standard regularity assumptions, the algorithm achieves global convergence together with an ergodic O(1/k)-type rate for feasibility and optimality measures.
Numerical experiments further demonstrate the practicality of the approach. In a distributed resource allocation problem and a consensus machine learning task, the method achieves faster progress and significant communication reduction compared with the Augmented Lagrangian Method (ALM) and Douglas–Rachford Splitting (DRS). These results show that the proposed framework provides an efficient and scalable alternative for optimization settings involving nonlinear constraint structures.

Zhengjie Xiong is a third year Math PhD student @ RPI. 

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