Question

Suppose that f(x) is a convex function with continuous first partials defined on a convex set C in R". Prove that a point x* in C is a global minimizer of f(x) on C if and only if Vf(x*)-(x - x*)2 0 for all x in C.

Answer 1

er Qiven that> ett un ctTM), with anti nous irst partials det ned ⓜau Convex Set c 'r) Rn Here, up bare to prove that a point ø χ c is a gotal minimizer o ) o c; here 'C indiates Convex partials ot Set R l) aus here, f is the Crve· CST tinous partials Set RD.

trom e900 and here point inc is aobal inirmi zer hence psoved

Answer 2

y is continous on (x-x^*) for the convex set because it is not greater than f(x); that it is a global minimizer for f(x). f(x) is not a condition for optimality. i could say its for a duality factor principle for the function f(x).

For a zero factor principle, f(x-x^*) is stationary for optimality. No two sets can be stationary, y can skew either of the two sets. Therefore the condition for optimality has failed as for a single factor.