Fixed-Time Dynamical System Approach for Solving Time-Varying Convex Optimization Problems

A time-varying (TV) optimization problem arises in many real-time applications, where the objective function or constraints change continuously with time. Consequently, the optimal points of the problem at each time instant form an optimal trajectory and hence tracking the optimal trajectory calls for the need to solve the TV optimization problem. A second-order continuous-time gradient-flow approach is proposed in this paper to track the optimal trajectory of TV convex optimization problems in fixed-time irrespective of the initial conditions. Later on we present a second-order nonsmooth dynamical system to solve the TV convex optimization problem in fixed time that does not require the exact information about the time rate of change of the cost function gradient. It makes the non-smooth dynamical system robust to the temporal variation in the gradient of the cost function. Two numerical examples are considered here for the simulation-based validation of the proposed approaches.