Given a C 2 L-smooth function the Lipschitz condition is: | | ∇ f ( x ) − ∇ f ( y ) | | ≤ L | | x − y | | Are these conditions only true for convex C 2 function? What will change If f is C 2 and concave?

Question

Given a       C    2   L-smooth function the Lipschitz condition is:      |        |    ∇  f  (  x  )  −  ∇  f  (  y  )      |        |    ≤  L      |        |    x  −  y      |        |  Are these conditions only true for convex       C    2   function? What will change If   f is       C    2   and concave?

Kalmukujobvg · Accepted Answer

This condition is completely independent of convexity. As long as we can connect any two points in your domain to be able to use the argument, the two conditions are equivalent for       C    2   functions. This is true for convex domains. No assumption on   f is needed, apart from regularity.