Optimization and Control (math.OC)

Abstract: The cyclic feedback interconnection of $n$ subsystems is the basic building block of control theory. Many robust stability tools have been developed for this interconnection. Two notable examples are the small gain theorem and the Secant Criterion. Both of these conditions guarantee stability if an inequality involving the geometric mean of a set of subsystem indices is satisfied. The indices in each case are designed to capture different core properties; gain in the case of the small gain theorem, and the degree of output-strict-passivity in the Secant Criterion. In this paper we identify entire families of other suitable indices based on mappings of the unit disk. This unifies the small gain theorem and the Secant Criterion, as well as a range of other stability criteria, into a single condition.

Abstract: We present a unified viewpoint of proximal point method (PPM), primal-dual hybrid gradient (PDHG) and alternating direction method of multipliers (ADMM) for solving convex-concave primal-dual problems. This viewpoint shows the equivalence of these three algorithms upto a norm change, and it leads to a four-line simple proof of their $\mathcal O(1/k)$ ergodic rates.

Abstract: This paper considers the distributionally robust chance constrained Markov decision process with random reward and ambiguous reward distribution. We consider individual and joint chance constraint cases with Kullback-Leibler divergence based ambiguity sets centered at elliptical distributions or elliptical mixture distributions, respectively. We derive tractable reformulations of the distributionally robust individual chance constrained Markov decision process problems and design a new hybrid algorithm based on the sequential convex approximation and line search method for the joint case. We carry out numerical tests with a machine replacement problem.