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$$\begin{array}{ll} \text{maximize} & \displaystyle\prod_{i=1}^r P_i\\ \text{subject to} & x_1 + x_2 + \dots + x_n = 1\\ & x_1, x_2, \dots, x_n \geq 0\end{array}$$

where each $P_i$ is a sum of some $x_k$'s. Is it true that we can always choose all $x_i$'s to be rational so as to maximize this expression?

(It is not true that the $x_i$'s must always be rational. For example, if $n=2$ and the expression is just $x_1+x_2$, then we can choose $x_1=a$ and $x_2=1-a$ for any $a$. But we can also choose $a$ to be rational.)

Also, is it true that each $P_i$ will be at least $\frac 1r$ at the optimum?

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It is not true that $P_i \geq 1/n$ at the optimum. Take $P_1=x_1$ and $P_i=x_2$ for $i \geq 2$. The objective is then $x_1 x_2^{r-1} = (1-x_2)x_2^{r-1}$. This is maximized at $x_2=(r-1)/r$, so $P_1 = 1/r$, which for $r > 2$ is less than 0.5.

Maximizing the product of $P_i$ is equivalent to maximizing the sum of the logarithm of $P_i$, which is concave: $$\max_x \left\{ \sum_k \log(P_k(x)) : e^T x = 1, x \geq 0 \right\}$$ Since this problem is convex, the KKT conditions are necessary and sufficient:

$$\sum_k \frac{P'_k(x)}{P_k(x)}+\lambda + \mu_i = 0 \text{ (stationarity)}$$ $$\lambda(e^Tx-1)=0, \; \mu_i x_i = 0 \text{ (complementary slackness)}$$ $$\sum_i x_i=1, \; x \geq 0 \text{ (primal feasibility)}$$ $$\mu \geq 0 \text{ (dual feasibility)}$$ where the derivative is taken with respect to $x_i$, so each $P'_k(x)$ is either $0$ or $1$. This essentially the same as identifying a subset $S$ of the variables such that: $$\sum_{k : x_i \in P_k} \frac{1}{P_k(x)}+\lambda = 0 \quad \forall i \in S$$ $$\sum_{k : x_i \in P_k} \frac{1}{P_k(x)}+\lambda \leq 0 \quad \forall i \notin S$$ $$\sum_{i \in S} x_i = 1, \; x \geq 0$$ where I abuse the notation $x_i \in P_k$ to indicate that the sum $P_k$ contains $x_i$. I do not see why there is always a rational solution to these equations.

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