Convex Optimization

• Convexity
  • bowl-shapedness: function >= tangent
    • ie. f(u) >= f(w) + <del f(w), u - w>
                   = f(w) + f'(w) (u - w)
  • chord across bowl: chord >= function
  • 2nd gradient of function is psd; ▽2f(w) or, f''(w) >= 0
    • 2nd gradient > 0 : strict convexity
• Global minimum
  • convex & the critical point (ie. gradient = 0)