Minimizing a sum of powers

Question

For positive integers $ b_1,b_2,...,b_T $, I am trying to solve the following optimization problem: $$ \min_{\substack{p_1,\ldots,p_T\\ p_1+\cdots+p_T=1\\}} \sum_{i=1}^T {p_i} ^ {b_i} .$$

The solution I would like to find is an expression for $ p_i $ in terms of $ b_i $. For example, something like $ p_i \sim 1/b_i $.

I have tried a few different things, including experimenting with the simple case where $ T = 2 $, but I have been having some trouble. If anyone can tell me how to solve this, give me any guidance, or even tell me whether this problem is tractable, I would really appreciate it. Thanks!

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Update:

I have now tried using Lagrange multipliers to solve this optimization problem. However, I'm not entirely sure that this is correct.

The Lagrangian is:

$$ L(p,\lambda) = \sum_{i} {p_i}^{b_i} - \lambda \bigl(\sum_{i}{p_i -1} \bigr) $$

Next I calculate the gradient

$$ \frac{dL}{dp_i} = b_i p_i^{b_i - 1} - \lambda$$ $$ \frac{dL}{d\lambda} = \sum_i p_i - 1 $$

I set these to zero to calculate the fixed points. But lambda is only in my first equation? I end up with:

$$ p_i = \bigl( \frac{\lambda}{b_i} \bigr)^{\frac{1}{(b_i - 1)}}$$

So then, to solve for lambda, I should substitute this expression in for the 2nd equation:

$$ \sum_i p_i = \sum_i \bigl ( \frac{\lambda}{b_i} \bigr ) ^{\frac{1}{(b_i - 1)}} = 1 $$

Answer 1

If it would help to have an explicit solution on hand to test hypotheses, solvers have no difficulty with individual instances. For example, $$ \min \; p_1^2 + p_2^3 + p_3^2 + p_4^4 $$ leads to $$ (p_1,\, p_2,\, p_3,\, p_4) \approx (0.14058,\, 0.30614,\, 0.14058,\, 0.41270) \;. $$

Answer 2

I'll focus on solving for $\lambda$ in your last equation.

$$\sum_{i}\left(\frac{\lambda}{b_i}\right)^\frac{1}{b_i-1}=1\quad\equiv\quad\sum_{i}\left(\frac{1}{b_i}\right)^\frac{1}{b_i-1}\left(\lambda\right)^\frac{1}{b_i-1}=1\quad \equiv:\quad\sum_i c_i\ \sqrt[b_i-1]{\lambda}=1$$

which clearly shows, that the problem amounts to removing the radicals; that can be done by the nice method of Dr. Vogler, which is described on Stackexchange https://mathoverflow.net/questions/177765/rewrite-sum-of-radicals-equation-as-polynomial-equation.

After the radicals have been removed, the problem has been converted into the easier task of solving a polynomial equation; that problem can then be fed into a symbolic or numeric solver.