How to show the optimization/ODE fixed point iteration steps converge?

Question

1. I have $\vec{C} = G(\vec{\beta})$ by solving a system of ODE numerically. Thanks for the help of Robert the ODE can be found in this link: Solving a system of ODE

2. Also $\vec{\beta}$ should satisfy $$A{\vec{\beta}}\le f(\vec{\beta}, \vec{C})$$ and $$\max 19\beta_1+0.5\beta_2+16\beta_3.$$ where $A$ is a given matrix and $f$ is some given function.

I am thinking of solving this process using iteration. I have a initial approximation $\vec{\beta^0}$, then for $k=1,2,3...$ solve Part $1$ using $\vec{C^k} = G(\vec{\beta^{k-1}})$ then solving part $2$ optimization using $$A{\vec{\beta^{k+1}}}\le f(\vec{\beta^k}, \vec{C^k})$$ and $$\max 19\beta_1^{k+1}+0.5\beta_2^{k+1}+16\beta_3^{k+1}.$$

But I am worried this step will not converge as $k\to\infty$. My questions is if this method will converge? if it is not, how to solve the optimization/ODE system to make it converge to the true solution?

Any help is appreciated! Many thanks!

Answer 1

The task standing has unknown significant parameters. The quantity and localization of maxima are unknown too. Also, the optimization methods are not detalized. In such conditions, the convergence of iterations cannot be guaranteed.

This situation can be improved if to make optimization as accurate as possible.

Let us consider the possible ways for that.

$\color{brown}{\textbf{The choice of initial point.}}$

1. The greatest value of linear function in the area can be achieved only in the bounds of the area.
   This mean that the constraints $A\vec\beta\le f(\vec\beta, C)$ can be used in the rigorous variant $$A\vec\beta = f\left(\vec\beta, \vec C\right).\tag1$$ Thus, the task is to maximize the scalar production $\vec w \vec \beta,$ where $$\vec w=\begin{pmatrix}19\\0.5\\16\end{pmatrix},\tag2$$ under the constraint $(1).$
2. The obtained task can be solved by Lagrange multipliers method, which performs it as calculation of uncondiional maxima of the function $$\varphi\left(\vec \lambda,\vec \beta\right) = \vec w \vec \beta + \vec\lambda\left(A\vec\beta-f\left(\vec\beta,G\left(\vec\beta\right)\right)\right).\tag3$$ The maxima of $\varphi$ achieves only in its stationary points.
3. The stationary points of $\varphi$ can be defined from the system $$\dfrac{\partial \varphi}{\partial \vec \beta} = 0,\quad \dfrac{\partial \varphi}{\partial \vec \lambda} = 0,$$ or \begin{cases} \vec w + \left(A-\dfrac{df}{d\vec \beta}\right)^T \vec \lambda = 0\\[4pt] \dfrac{df}{d\vec \beta} = \dfrac{\partial f}{\partial \vec \beta} +\dfrac{\partial f}{\partial G}\dfrac{dG}{d\vec \beta}\\ A\vec\beta = f\left(\vec\beta, G\left(\vec\beta\right)\right).\tag4 \end{cases} The maxima should be selected among of the all stationary points. Each of them can be choosen as the initial point for the iterations.
   This approach allows to localize the initial points near the possible maxima.

$\color{brown}{\textbf{Iterations.}}$

1. Iterations can use detalized model of the optimization task.
2. The optimization task does not require the full soluiion $\vec C.$ In particular, the derivatives can be obtained from the Part 1 immediately.
3. Convergency of the iterations in the proposed model basically depends from the stability of the system $(1)$ solutions near the maxima points.