I'm trying to minimize (numerically) the following expression for $F$ regarding the parameters $\alpha$ and $H$ inside the nested summation of $n$ terms: $$ F = \sum_{i=1}^{n}{\left[3H_i\left(R + \sum_{j=1}^{i}{H_j\tan\alpha_j}\right)^2 + H_i^3\tan^2\alpha_i - 3H_i^2\tan\alpha_i\left(R + \sum_{j=1}^{i}{H_j\tan\alpha_j}\right)\right]} $$ where $\alpha\in[0,\pi/2)$, $R$ is a constant, and the sum of the $H_i$ ( or $H_j$) must be equal to a given constant, say $A$, in a manner that the interval of $H_i$ parameters are given such that: $$ H_1\in[0,A] $$ $$ H2\in[H_1,A] $$ $$ H_3\in[H_2,A] $$ $$ ... $$
Which gives me $2n-1$ terms to minimize since $H_n$ will be given by $A-\sum_{i=1}^{n-1}H_n$.
My first approach was to adopt a "brute force" algorithm to minimize the parameters, but it proved very inefficient and a little bit confusing to implement given the restrictions on the $H_i$ parameters. My question is that if anyone could provide me a hint on a better method that I could adopt in order to minimize this type of expression since I'm new to minimization techniques. Thank you in advance!
P.S.: I'm aware that $\alpha_i=0$ minimizes the expression, but this is just a piece of a larger problem where the solution is not trivial.