Dynamical System Approaches to Solve Convex Optimization Problems

  • 19



Name of the Speaker: Mr. Rejitha Raveendran (EE17D016)
Guide: Dr. Arunkumar D Mahindrakar
Venue/Online meeting link:
Date/Time: 19th December 2022, 10.00 AM

Optimization is ubiquitous in science and many areas of engineering, especially robotics, power distribution networks, signal processing, guidance and navigation of aerial and autonomous vehicles and machine learning. The majority of these applications are subjected to real-time constraints, depending on certain characteristics of the system and typically vary with respect to time. For example, the guidance of a robot in a dynamic environment, traffic engineering in computer networks, economic dispatch problems in power generating units, and neural network learning are some examples where time-varying (TV) optimization is inevitable.

Several methods exist to solve time-invariant convex optimization problems, including Newton’s method, subgradient methods, interior point method and primal-dual dynamics. In this work, a projected primal-dual dynamical system is proposed to solve an inequality constrained nonsmooth convex optimization problem. Then the proposed projected dynamical system is employed to solve the extended Fermat-Torricelli Problem (eFTP) of finding a point in R n , from which the sum-of-distances to a finite number of nonempty, closed and convex sets is minimum. Along with the point that minimizes the sum-of-distances, the eFTP also solves for the points in each convex set at which the minimum sum-of-distances is achieved.

In most real-time applications, either the objective function or the constraints of an optimization problem may change with respect to time, and it leads to an optimal point/set of optimal points at each time instant. Consequently, the optimal points of the problem at each time instant form an optimizer trajectory and hence tracking the optimizer trajectory calls for the need to solve the TV optimization problem. A second-order continuous-time gradient-flow approach is proposed in this work to track the optimizer trajectory of unconstrained and constrained TV convex optimization problems with a strongly convex objective function. First, to track the optimizer trajectory of unconstrained and equality constrained TV optimization problems, finite-time and fixed-time prediction-correction based dynamical system approaches are presented in this work that guarantees the tracking of the optimizer trajectory within a finite time. Later, a projected primal-dual dynamical system based on the prediction-correction technique is proposed to track the optimizer trajectory of an inequality constrained TV convex optimization problem. The trajectories of the proposed projected dynamical system track the optimizer trajectory of the inequality constrained TV optimization problem asymptotically. The uniform asymptotic stability of the proposed dynamical system to the optimizer trajectory is shown via Lyapunov-based analysis. Finally, the TV version of the eFTP is considered to illustrate the applicability of the prediction-correction based projected primal-dual dynamical system proposed in the thesis.