This paper presents an Algorithm for the numerical solution of the Optimal Control model constrained by Partial Differential Equation using the Alternating Direction Method of Multipliers (ADMM) accelerated with a parameter factor in the sense of Nesterov. The ADMM tool wasapplied to a partial differential equation-governed optimization problem of the one-dimensional heat equation type. The constraint and objective functions of the optimal control model were discretized using the Crank-Nicolson and Composite Simpson’s Methods respectively into a derived discrete convex optimization form amenable to the ADMM. The primal-dual residuals were derived to ascertain the rate of convergence of themodel for increasing iterates. An existing example was used to test the efficiency and degree of accuracy of the algorithm and the results were favorable when compared the existing method.
This study presents a computational technique developed for solving linearly constraint optimal control problems using the Gradient Flow Method. This proposed method, called the Modified Gradient Flow Method (MGFM), is based on the continuous gradient flow reformulation of constrained optimization problem with three-level implicit time discretization scheme. The three-level splitting parameters for the discretization of the gradient flow equations are such that the sum of the parameters equal to one (\theta1 + \theta2 +\theta3=1). The Linear and quadratic convergence of the scheme were analyzed and were shown to have first order scheme when each parameter exist in the domain [0, 1] and second order when the third parameter equal to one. Numerical experiments were carried out and the results showed that the approach is very effective for handling this class of constrained optimal control problems. It also compared favorably with the analytical solutions and performed better than the existing schemes in terms of convergence and accuracy
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