Subderivative

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, subderivatives (or subgradient) generalizes the <a href="/facts/Derivative/7Ito70Dh">derivative</a> to convex functions which are not necessarily <a href="/facts/Differentiable_function/jQqTgLk1">differentiable</a>. The set of subderivatives at a point is called the subdifferential at that point. Subderivatives arise in <a href="/facts/Convex_analysis/SGhUNLMK">convex analysis</a>, the study of <a href="/facts/Convex_functions/IbzG5SLF">convex functions</a>, often in connection to <a href="/facts/Convex_optimization/7D6U4MTN">convex optimization</a>.
Let 
 
 
 
 f
 :
 I
 →
 
 R
 
 
 
 {\displaystyle f:I\to \mathbb {R} }
 
 be a <a href="/facts/Real_number/R02gw5Pb">real</a>-valued convex function defined on an <a href="/facts/Open_interval/e9se3vTe">open interval</a> of the real line. Such a function need not be differentiable at all points: For example, the <a href="/facts/Absolute_value/dwJBpEs5">absolute value</a> function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle f(x)=|x|}
 is non-differentiable when 
 
 
 
 x
 =
 0
 
 
 {\displaystyle x=0}
 
. However, as seen in the graph on the right (where 
 
 
 
 f
 (
 x
 )
 
 
 {\displaystyle f(x)}
 
 in blue has non-differentiable kinks similar to the absolute value function), for any 
 
 
 
 
 x
 
 0
 
 
 
 
 {\displaystyle x_{0}}
 
 in the domain of the function one can draw a line which goes through the point 
 
 
 
 (
 
 x
 
 0
 
 
 ,
 f
 (
 
 x
 
 0
 
 
 )
 )
 
 
 {\displaystyle (x_{0},f(x_{0}))}
 
 and which is everywhere either touching or below the graph of f. The <a href="/facts/Slope/a2L3k0j5">slope</a> of such a line is called a subderivative.

Subderivative open-in-new

Subderivative