Time-Varying Optimization with Optimal Parametric Functions

In this paper, we consider a formulation of nonlinear constrained optimization problems. We reformulate it as a time-varying optimization using continuous-time parametric functions and derive a dynamical system for tracking the optimal solution. We then re-parameterize the dynamical system to express it based on a linear combination of the parametric functions. Calculus of variations is applied to optimize the parametric functions, so that the optimality distance of the solution is minimized. Accordingly, an iterative dynamic algorithm, named as OP-TVO, is devised to find the solution with an efficient convergence rate. We benchmark the performance of the proposed algorithm with the prediction-correction method (PCM) from the optimality and computational complexity point-of-views. The results show that OP-TVO can compete with PCM for the optimization problem of interest, which indicates it can be a promising approach to replace PCM for some time-varying optimization problems. Furthermore, this work provides a novel paradigm for solving parametric dynamical system.