Network Optimization via Smooth Exact Penalty Functions Enabled by Distributed Gradient Computation

This paper proposes a distributed algorithm for a network of agents to solve\nan optimization problem with separable objective function and locally coupled\nconstraints. Our strategy is based on reformulating the original constrained\nproblem as the unconstrained optimization of a smooth (continuously\ndifferentiable) exact penalty function. Computing the gradient of this penalty\nfunction in a distributed way is challenging even under the separability\nassumptions on the original optimization problem. Our technical approach shows\nthat the distributed computation problem for the gradient can be formulated as\na system of linear algebraic equations defined by separable problem data. To\nsolve it, we design an exponentially fast, input-to-state stable distributed\nalgorithm that does not require the individual agent matrices to be invertible.\nWe employ this strategy to compute the gradient of the penalty function at the\ncurrent network state. Our distributed algorithmic solver for the original\nconstrained optimization problem interconnects this estimation with the\nprescription of having the agents follow the resulting direction. Numerical\nsimulations illustrate the convergence and robustness properties of the\nproposed algorithm.\n