I want to find $(x_1,\dots,x_n) \in (\mathbb{R}_+^*)^n$ to minimize $$ \sum_{k=1}^{n}{x_k \prod_{i=1}^{k}{(1 + x_i)}} $$ with the following constraint $\displaystyle\sum_{k=1}^{n}{x_k} = 1$.
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What have you tried? – Daron Aug 24 '19 at 12:47
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What tools do you have ? Do you know some differential geometry ? – Maxime Ramzi Aug 24 '19 at 12:50
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I tried Lagrange multipliers but it seems to messy to be solved in that way. I think that the solution satisfies $x_1 \geq x_2 \geq \dots \geq x_n$, but I am not sure, still working on it... – booster Aug 24 '19 at 12:52
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differential geometry ? just a little. But any idea to help me to solve this it will be great ! ^^ I am ready to study diferential geometry if it is necessary to come close to a solution ^^ – booster Aug 24 '19 at 12:55
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Have you solved the problem for $n=2$ or $n=3$? – Dr. Sonnhard Graubner Aug 24 '19 at 12:58
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Sure, for $n=2$ the solution is given by $(2/3,1/3)$, for $n=3$ I obtained a numerical solution only (0.54,0.26,0.20) – booster Aug 24 '19 at 13:01
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Not an answer.
We could try Lagrange multipliers. Take the Lagrangian $$\mathcal L (\textbf{x},\lambda) = \sum_{k=1}^n x_k \prod _{j=1}^k (1+x_j) + \lambda \left (\sum _{k=1}^n x_k -1\right ) =: f(\textbf{x}) + \lambda g(\textbf{x}). $$ Find the candidate solution(s) by finding stationary point(s) i.e $\nabla\mathcal L= 0$. Get a system $$\frac{\partial}{\partial x_i}f = (2x_i+1)\prod _{j=1}^{i-1}(1+x_j) + \sum _{k=i+1}^n \prod_{j=1 \\ j\neq i}^{i+1} (1+x_j) = -\lambda $$ with $g(\textbf{x})=0$, where for $i=1$ the first additive is simply $2x_1+1$. Unsure how to solve this analytically.
AlvinL
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Thank you, I already tried Lagrange multipliers but as I said it seems too messy to be solved in that way analytically. – booster Aug 24 '19 at 16:55