Subdifferential

The subdifferential describes the local behavior of a convex function. Previously we defined the subdifferential via the nonvertical hyperplanes supporting the epigraph. The subgradient inequality is an equivalent, traditional definition. For a convex function f(x), a change in x changes a subgradient x* in a like direction, rather than in the opposite direction. Whereas differential calculus deals with derivatives, subdifferential calculus deals with subdifferentials. In subdifferential calculus, one works with rules for subdifferentials, which form only a subset of the rules for differential calculus. For example, whereas the chain rule is a key tool of differential calculus, unfortunately this rule does not hold for subdifferential calculus. In general the composition of two convex functions need not be convex, so the chain rule cannot apply in general. For convex F, define the value function $$V\left ( \boldsymbol {y}\right ) =\inf _{\boldsymbol {x}}F\left ( \boldsymbol {x},\boldsymbol {y}\right ) \!,$$ Under general conditions, the following envelope inclusion holds: if choice x is minimizing for perturbation y, then $$\partial V\left ( \boldsymbol {y}\right ) \subset \partial _{{{\sf {Y}}}}F\left ( \boldsymbol {x},\boldsymbol {y}\right ) \!,$$ a generalization of the envelope theorem.