  In this paper we describe a parameterized family of first-order distributed
optimization algorithms that enable a network of agents to collaboratively
calculate a decision variable that minimizes the sum of cost functions at each
agent. These algorithms are self-healing in that their correctness is
guaranteed even if they are initialized randomly, agents drop in or out of the
network, local cost functions change, or communication packets are dropped. Our
algorithms are the first single-Laplacian methods to exhibit all of these
characteristics. We achieve self-healing by sacrificing internal stability, a
fundamental trade-off for single-Laplacian methods.
