Global Optimal of Convex Optimization
Last updated
Last updated
convex and continuously differentiable
convex
If is a local minimizer, then is a global min
level sets of convex functions
first and second order optimization: ;
Optimality for constrained problems
➡️ Fact: If convex, then the level sets are convex sets
Proof Sketch: (⇒ only if direction)
Consider the unconstrained differentiable problem
By convexity,
For convex unconstrained problems,
Suppose solves (P)
⇒ solution set
Take any
Show that for , , .
Because is conex,
✅ Theorem. Let be continuously differentiable over , where means convex set, then is convex iff
By convexity of :
Dividing by both sides,
Because continuously differentiable,
⇒
⇒
If is a local min, then
For ( convex) twice continuous differentiable then is convex iff
Example.
⇒ All feasible directions are non-decreasing on .
⇒