We say that a function $f:R^2\longrightarrow R$ satisfies one-sided Lipschitz condition with respect to x with constant $K$ if $$\langle f(x_{1},y)-f(x_{2},y),x_{1}-x_{2}\rangle \leq K||x_{1}-x_{2}||^2.$$ And my question is how can we check if this these functions $f(x,y)=-\text{sign}(x)$ $f(x,y)=\text{sign}(x)$ satisfy one-sided Lipschitz condition or not.
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Did you try computing the left side? – Jun 09 '16 at 18:17
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it is $f(x,y)=\text{sign}(x)$ – J. Mccain Jun 09 '16 at 18:24
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@J.Mccain, then what purpose does $y$ serve here? – Yuriy S Jun 09 '16 at 18:27
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@YuriyS, in this function y does nothing. – J. Mccain Jun 09 '16 at 18:29
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Hint. A useful approximation is this: $$\text{sign} (x)=\lim_{t \to 0} \frac{x}{\sqrt{x^2+t^2}}$$
Do you see how to continue?
Or you can just consider the two cases.
First: $$\text{sign} (x_1)=\text{sign} (x_2) \\ | \text{sign} (x_1)-\text{sign} (x_2)|=0$$
Second: $$\text{sign} (x_1)=-\text{sign} (x_2) \\ | \text{sign} (x_1)-\text{sign} (x_2)|=2$$
Do you see how to continue here?
Yuriy S
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thanks, but why did we start with considering $\text{sign}(x_{1})=\text{sign}(x_{2})$ – J. Mccain Jun 09 '16 at 18:35
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@J.Mccain what is $\text{sign(x)}$? What is this function, which values does it take? – Yuriy S Jun 09 '16 at 18:36
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