Transcription of Chapter 4: Unconstrained Optimization - McMaster University
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Chapter 4: Unconstrained Optimization - McMaster University
Chapter 4: Unconstrained Optimization Unconstrained Optimization problemminxF(x)ormaxxF(x) Constrained Optimization problemminxF(x)ormaxxF(x)subject tog(x) = 0and/orh(x)<0orh(x)>0 Example: minimize the outer area ofa cylinder subject to a fixed functionF(x) = 2 r2+ 2 rh, x=[rh]Constraint:2 r2h=V1 Outline: Part I: one-dimensional Unconstrained Optimization Analytical method Newton s method Golden-section search method Part II: multidimensional Unconstrained Optimization Analytical method Gradient method steepest ascent (descent) method Newton s method2 PART I: One-Dimensional Unconstrained Optimization Techniques1 Analytical approach (1-D)minxF(x)ormaxxF(x) LetF (x) = 0and findx=x . IfF (x )>0,F(x ) = minxF(x),x is a local minimum ofF(x); IfF (x )<0,F(x ) = maxxF(x),x is a local maximum ofF(x); IfF (x ) = 0,x is a critical point ofF(x)Example1:F(x) =x2,F (x) = 2x= 0,x = (x ) = 2>0.
5 Steepest Ascent (Descent) Method Idea: starting from an initial point, find the function maximum (minimum) along the steepest direction so that shortest searching time is required. Steepest direction: directional derivative is maximum in that direction — gradi-ent direction. f() = ¢µ + @f @y ¢ =[] ¢]
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