Assume $f(x,y) = \langle x,y\rangle$, i.e., the dot product between $x$ and $y$ with $x, y \in \mathbb{R^n}$. The Lipschitz constant $L$ of $f$ is given by:

$ | \langle x_1,y_1 \rangle - \langle x_2, y_2 \rangle| \le L || (x_1,y_1) - (x_2,y_2)||_2, x_1, y_1, x_2, y_2 \in \mathbb{R^n}$. Note that $(x_1,y_1)$ is the concatenation of $x_1$ and $y_1$, i.e., $(x_1,y_1)\in \mathbb{R}^{2n}$

Any hints on how to compute this? I tried to transform the inner product into a matrix product.

  • $\begingroup$ Sorry, the problem used wrong variables. $\endgroup$ – 今天春天 Jul 3 '18 at 12:15

Counterexample : If $a=(1,1),\ A=(2,1),\ b=(L,0)= B$, then $$ |(a,b)-(A,B)|=1$$

And $$ \langle a,b\rangle =L,\ \langle A,B\rangle =2L $$


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