Distributed Robust Optimization in Networked System

Abstract

In this paper, we consider a distributed robust optimization (DRO) problem, where multiple agents in a networked system cooperatively minimize a global convex objective function with respect to a global variable under the global constraints. The objective function can be represented by a sum of local objective functions. The global constraints contain some uncertain parameters which are partially known, and can be characterized by some inequality constraints. After problem transformation, we adopt the Lagrangian primal-dual method to solve this problem. We prove that the primal and dual optimal solutions of the problem are restricted in some specific sets, and we give a method to construct these sets. Then, we propose a DRO algorithm to find the primal-dual optimal solutions of the Lagrangian function, which consists of a subgradient step, a projection step, and a diffusion step, and in the projection step of the algorithm, the optimized variables are projected onto the specific sets to guarantee the boundedness of the subgradients. Convergence analysis and numerical simulations verifying the performance of the proposed algorithm are then provided. Further, for nonconvex DRO problem, the corresponding approach and algorithm framework are also provided.

Distributed Computation for Solving the Sylvester Equation Based on Optimization

“…Then system (2) is semi-stable with respect to D if every solution with initial condition in D converges to a Lyapunov stable equilibrium. Moreover, (2) is said to be globally semi-stable if it is semi-stable with respect to R d . Lemma 1 (Theorem 3.1 in [25]): Let D be an open positively invariant set with respect to (2), V : D → R be a continuously differentiable function, and x(t) be a solution of (2) with May 1, 2019 DRAFT x(0) ∈ D, contained in a compact subset of D. Assume d dt V (x(t)) ≤ 0, for all x ∈ D and define Z = {x ∈ D : d dt V (x) = 0}.…”

“…Moreover, (2) is said to be globally semi-stable if it is semi-stable with respect to R d . Lemma 1 (Theorem 3.1 in [25]): Let D be an open positively invariant set with respect to (2), V : D → R be a continuously differentiable function, and x(t) be a solution of (2) with May 1, 2019 DRAFT x(0) ∈ D, contained in a compact subset of D. Assume d dt V (x(t)) ≤ 0, for all x ∈ D and define Z = {x ∈ D : d dt V (x) = 0}. If every point in the largest invariant subset M ofZ ∩ D is Lyapunov stable equilibrium, whereZ is the closure of Z, then the system (2) is semi-stable with respect to D.…”

Dynamic coordinated scheduling scheme for transmission and distribution system considering uncertainties of distributed generations

“…Among them, the RO theory is a powerful tool to solve the uncertainty problems . Generally, RO problem is described as follows: where x is decision variable, ζ is uncertain parameter, X is a decision variable set, and U is a bounded uncertainty set.…”

“…where (19) is power balance constraint, (20) is the spinning reserve constraint, (21) is purchase electricity power constraint, and 22 is DG power constraint. Uncertainties of DGs are included in equations 20.…”

Distributed convex optimization with coupling constraints over time-varying directed graphs^†

“…Problem (1) is quite general arising in applications. For examples, distributed model predictive control [26], network utility maximization [4], real-time pricing problems for smart grid [33,32,13] can be modeled into this class of problems.…”