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Results tagged with nonlinear-optimization
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user 11919
Nonlinear objectives, nonlinear constraints, non-convex objective, non-convex feasible region.
21
votes
Accepted
Prove that this expression is greater than 1/2
Let
$$f(x,y):=4x^{2}+4y^{2}-4xy-4y+1 + \frac{4}{\pi^2}\Bigl(
\sin^{2}(\pi x)+ \sin^{2}(\pi y) + \sin^{2}(\pi y-\pi x) \Bigr).$$
I will show that
$$\min_{0\leq x\leq y\leq 1}f(x,y)=\min_{0\leq x\leq 1/ …
- 105k
11
votes
Accepted
Min problem on integers
Let us denote
$$\sigma_\ell:=\sum_{i=1}^\ell q_i\qquad\text{and}\qquad\tau_\ell:=\sum_{i=\ell+1}^s\frac{1}{q_i}.$$
Then
$$\prod_{\ell=1}^{s-1}\left(\frac{q_\ell}{q_{\ell+1}}\cdot\frac{\sigma_{\ell+1}} …
- 105k
12
votes
Smallest root of a degree 3 polynomial
This proof is similar to River Li's, but it requires no machine calculation. Also, we will show more, namely that the original polynomial has three real roots, the smallest of which is at most
$$u:=\f …
- 105k
3
votes
Does this polynomial have a real zero less than or equal to $1/2$?
I will show more, namely that the polynomial in question has three real roots, the smallest of which is at most
$$d:=\frac{5-\sqrt{15}}{5}=0.22540333\dotsc$$
This estimate is sharp, because in case th …
- 105k