Minimum of $\sum\limits_{k=0}^n { n \choose k} (1-x)^{n-k} x^k a_{n-k}b_k$ over $x \in [0,1] $

Question

Given \begin{align} f(x)=\sum_{k=0}^n { n \choose k} (1-x)^{n-k} \cdot x^k \cdot a_{n-k} \cdot b_k \end{align}

Find \begin{align} \min _{x \in [0,1] } f(x) \end{align}

We can assume that $a_k$ and $b_k$ are positive and monotone increasing sequences.

Note, that I am not interested in the minimizer $x^{*}$ but rather a minimum value of $f(x^{*})$.

EDIT

Based on the comments of A.S., we can redefine the problem:

Given

$$\begin{align} g(x)=\sum_{k=0}^n { n \choose k} (1-x)^{n-k} \cdot x^k \cdot c_k \end{align}$$

for some $c_k>0$, find

$$\begin{align} \min _{x \in [0,1] } g(x) \end{align}$$

This problem is related to the question I posted here.

Answer 1

The function $g$ is a linear combination of Bernstein polynomials. As these polynomials are a basis of the space of polynomials, your question is equivalent to asking for the minimum value of an arbitrary polynomial of degree $n$, which is impossible to answer in this general form.

For Bernstein polynomials you can obtain upper and lower bounds of the function values from the coefficients $c_k$: $$ \min_k c_k \le g(x) \le \max_k c_k \quad\forall x\in[0,1]. $$ The proof of this is simple: $$ g(x) =\sum_{k=0}^n {n \choose k}(1-x)^{n-k}x^kc_k \le \max_k c_k \cdot \sum_{k=0}^n {n \choose k}(1-x)^{n-k}x^k = \max_k c_k \cdot 1, $$ which also shows that the bound is sharp if the $c_k$'s are all of the same value.

Answer 2

This is by no means an answer, but obviously more than a comment.

I have begun to make some trials with Matlab.

Here is my modest program with a graphical result.

clear all;close all;hold on; n=5;C=rand(1,n); D=linspace(0,1,n); E=C+i*D;plot(E,'color','r'); t=0:0.01:1;s=1-t;M=0; for k=0:n-1; M=M+nchoosek(n-1,k)*(t.^k).*(s.^(n-1-k))*E(k+1); end; plot(M)

I have assumed WLOG that all $c_k \in [0,1]$ but nothing else about the $c_k$s (the fact that this sequence is the product of two sequences, one increasing, the other decreasing seems to have a very moderate impact...). In red, the control "arm", with vertices the control points with coordinates $(c_k,k/n)$. Only the abscissas are interesting. In this case, the abscissa of the minimum abscissa of the (blue) curve is strongly influenced by $c_3$, the smallest of all $c_k$s.