Convex Optimization · Problem 9.3

Equality constrained minimization — Boyd & Vandenberghe

Mathematics — Optimization · initial-point-sublevel-set-condition

Problem

Initial point and sublevel set condition. Consider the function

f(x) = x_1^2 + x_2^2

with domain

\mathbf{dom}\, f = \{(x_1, x_2) \mid x_1 > 1\}

. ((a)) What is

p^\star

? ((b)) Draw the sublevel set

S = \{x \mid f(x) \leq f(x^{(0)})\}

for

x^{(0)} = (2, 2)

. Is the sublevel set

S

closed? Is

f

strongly convex on

S

? ((c)) What happens if we apply the gradient method with backtracking line search, starting at

x^{(0)}

? Does

f(x^{(k)})

converge to

p^\star