Can someone help me to solve this optimization problem

Question

let $\lbrace\alpha_{i}\rbrace_{1\leq i \leq m}$ be positive numbers such that $\sum_{i=1}^m\alpha_{i}=n\in\mathbb{N}$ and $n\leq m$. Let $\lbrace p_i\rbrace_{1\leq i\leq m}$ be positive numbers in $]0,1[$. I have the following minimization problem in $(\mathbb{R}^{*+})^m$: $$\begin{array}{ll} \text{minimize} & \displaystyle\max_{i=1,...m}p_ix_iy_i\\ \text{subject to} & \displaystyle\sum_{i=1}^mx_i=\sum_{i=1}^m\alpha_{i}\\ &\displaystyle\sum_{i=1}^rx_i\leq\sum_{i=1}^r \alpha_i\quad \forall\ 1\leq r\leq n\\ &\displaystyle\sum_{i=1}^my_i=\sum_{i=1}^m \frac{1}{\alpha_i} \end{array}$$ PS: $\lbrace\alpha_{i}\rbrace_{1\leq i m}$ are not equals

Can anyone please recommend any sources that are related to my question?

Answer 1

It seems like the following solution gets arbitrarily close to $0$, when $m>1$.

• Pick arbitrary $\epsilon > 0$. (It will need to be sufficiently small for the constraints indexed by $r=1, \dots, n$ to hold, but eventually that will be fine.)
• Set $x_1 = x_2 = \dots =x_{m-1} = \epsilon$ and $x_m = \sum_{i=1}^m \alpha_i - (m-1)\epsilon$.
• Set $y_1 = \sum_{i=1}^m \frac1{\alpha_i} - (m-1)\epsilon$ and $y_2 = y_3 = \dots = y_m = \epsilon$.

As $\epsilon \to 0$, $p_i x_i y_i \to 0$ for every $i$.