v2007.09.17 - Convex Optimization

D.1. DIRECTIONAL DERIVATIVE, TAYLOR SERIES 563υ ∆ =⎡⎢⎣∇ x f(α)→∇ xf(α)1df(α)2⎤⎥⎦f(α + t y)υ T(f(α),α)✡ ✡✡✡✡✡✡✡✡✡f(x)∂HFigure 117: Convex quadratic bowl in R 2 ×R ; f(x)= x T x : R 2 → Rversus x on some open disc in R 2 . Plane slice ∂H is perpendicular tofunction domain. Slice intersection with domain connotes bidirectionalvector y . Slope of tangent line T at point (α , f(α)) is value of directionalderivative ∇ x f(α) T y (1608) at α in slice direction y . Negative gradient−∇ x f(x)∈ R 2 is direction of steepest descent. [283] [161,15.6] [104] Whenvector υ ∈[ R 3 entry ] υ 3 is half directional derivative in gradient direction at αυ1and when = ∇υ x f(α) , then −υ points directly toward bowl bottom.2