  We describe SH-SVL, 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 convergence to the
correct optimizer can be guaranteed even if they are initialized randomly,
agents join or leave the network, or local cost functions change. We also
present simulation evidence that our algorithms are self-healing in the case of
dropped communication packets. Our algorithms are the first single-Laplacian
methods for distributed convex optimization to exhibit all of these
characteristics. We achieve self-healing by sacrificing internal stability, a
fundamental trade-off for single-Laplacian methods.
