In this paper, we consider an approximate analytical method of optimal homotopy asymptotic method-least square (OHAM-LS) to obtain a solution of nonlinear fractional-order gradient-based dynamic system (FOGBDS) generated from nonlinear programming (NLP) optimization problems. The problem is formulated in a class of nonlinear fractional differential equations, (FDEs) and the solutions of the equations, modelled with a conformable fractional derivative (CFD) of the steepest descent approach, are considered to find the minimizing point of the problem. The formulation extends the integer solution of optimization problems to an arbitrary-order solution. We exhibit that OHAM-LS enables us to determine the convergence domain of the series solution obtained by initiating convergence-control parameter Cj′s. Three illustrative examples were included to show the effectiveness and importance of the proposed techniques.
Solving fractional optimal control problems (FOCPs) with an approximate analytical method has been widely studied by many authors, but to guarantee the convergence of the series solution has been a challenge. We solved this by integrating the Galerkin method of optimization technique into the whole region of the governing equations for accurate optimal values of control-convergence parameters
C
j
s
. The arbitrary-order derivative is in the conformable fractional derivative sense. We use Euler–Lagrange equation form of necessary optimality conditions for FOCPs, and the arising fractional differential equations (FDEs) are solved by optimal homotopy asymptotic method (OHAM). The OHAM technique speedily provides the convergent approximate analytical solution as the arbitrary order derivative approaches 1. The convergence of the method is discussed, and its effectiveness is verified by some illustrative test examples.
In a class of solving unconstrained optimization problems, the conjugate gradient method has been proved to be efficient by researchers' due to it's smaller storage requirements and computational cost. Then, a class of penalty algorithms based on three-term conjugate gradient methods was developed and extend to and solution of an unconstrained minimization portfolio management problems, where the objective function is a piecewise quadratic polynomial. By implementing the proposed algorithm to solve some selected unconstrained optimization problems, resulted in improvement in the total number of iterations and CPU time. It was shown that this algorithm is promising.
This paper deals with a recent approximate analytical approach of the
multistage optimal homotopy asymptotic method (MOHAM) for fractional optimal
control problems (FOCPs). In this paper, FOCPs are developed in terms of a
conformable derivative operator (CDO) sense. It is validated that the right CDO
appears naturally in the formulation even when the system dynamics are described
with the left CDO only. The CDO is employed to enlarge the stability region of the
dynamical systems of the optimal control problems (OCPs). The necessary and
transversal conditions are achieved using a Hamiltonian technique. The results
demonstrated that as the fractional-order solution derivative tends to integer-order 1,
the formulations lead to integer-order system solutions. Numerical results and a
comparison with the exact solution and other approximate analytical solutions in
fractional order are given to validate the efficiency of the MOHAM. Some numerical
examples are included to demonstrate the effectiveness and applicability of the new
technique.
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