An Introduction to Optimal Control Problems in Life Sciences and Economics by Sebastian Aniţa Viorel Arnăutu & Vincenzo Capasso

Author:Sebastian Aniţa, Viorel Arnăutu & Vincenzo Capasso
Language: eng
Format: epub
Publisher: Birkhäuser Boston, Boston, MA

SC2 : | Ψ(u(k + 1)) − Ψ(u(k)) | < ε.

SC3 : .

SC4 : ρk < ε1.

Let us remark that SC1 and SC3 are equivalent because

The test SC4 says that the gradient steplength ρ is very small which means that numerically no descent can be obtained along the direction − w(k), where w(k) = Ψu(u(k)). We can therefore assert that SC4 is numerically equivalent to SC2. The policy for our program is to stop when one convergence criterion is satisfied. Inasmuch as the gradient Ψu(u(k)) was just computed we apply SC1 and we continue the program by

% ===========

% PART 4

% ===========

normg = sqrt(sum(w. ∧ 2)) % compute the gradient norm

ii = ii + 1 ;

grad(ii) = normg ;

% verify SC1

if normg < eps

disp(’CONVERGENCE by GRADIENT’)

flag1 = 1 ;

break

end

If the stopping criterion is satisfied (i.e., normg < eps), we give to the convergence indicator flag1 the corresponding value 1 and we use the break statement which moves out the gradient loop started in PART 2 by the statement for iter = 1:maxit. We proceed now to Step S5 of the gradient algorithm, a more difficult one, which requires the computation of the gradient steplength ρk. The formula in Step S5 is useless in practice because usually the minimum there cannot be calculated. We approximate it by a robust and efficient algorithm, namely the Armijo method (for more details see [A66], [AN03, Section 2.3] and [Pol71, Appendix C.2]). We have already computed Ψ(u(k)). The last known value for the steplength ρ is ρk − 1. We also know b ∈ (0. 5, 0. 8) the parameter to fit ρ (by the input of the variable bro). Let us denote for what follows Ψ(u) = ψ(u, xu) in order to emphasize the current state x also. Here

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