Derivative-Free Optimization

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1 Derivative-Free Optimization Chapter 7 from Jang

2 Outline Simulated Annealing (SA) Downhill simplex search Random search Genetic algorithms (GA) 2

3 The Big Picture Model space Adaptive networks Neural networks Fuzzy inf. systems Approach space Derivative-free optim. Soft Computing Derivative-based optim. 3

4 Simulated Annealing Analogy 4

5 Simulated Annealing Terminology: Objective function E(x): function to be optiimized Move set: set of next points to explore Generating function: to select next point Acceptance function h( E, T): to determine if the selected point should be accept or not. Usually h( E, T) = 1/(1+exp( E/(cT)). Annealing (cooling) schedule: schedule for reducing the temperature T 5

6 Simulated Annealing Flowchart: Select a new point xnew in the move sets via generating function Compute the obj. function E(xnew) Set x to xnew with prob. determined by h( E, T) Reduce temperature T 6

7 Simulated Annealing Example: Travel Salesperson Problem (TSP) How to transverse n cities once and only once with a minimal total distance? 7

8 Simulated Annealing Move sets for TSP Inversion Translation Switching

9 Simulated Annealing A 100-city TSP using SA Initial random path During SA process Final path 9 MATLAB file: tsp.m

10 Simulated Annealing 100-city TSP with penalities when crossing the circle Penalty = 0 Penalty = 0.5 Penalty =

11 Random Search Properties: Intuitive Simple Analogy: Get down to a valley blindfolded Two heuristics: Reverse step Bias direction 11

12 Random Search Flowchart: Select a random dx f(x+b+dx)<f(x)? no yes x = x + b + dx b = 0.2 b dx f(x+b-dx)<f(x)? no yes x = x + b - dx b = b dx b = 0.5 b 12

13 Random Search Example: Find the min. of the peaks function z = f(x, y) = 3*(1-x)^2*exp(-(x^2) - (y+1)^2) - 10*(x/5 - x^3 - y^5)*exp(-x^2-y^2) -1/3*exp(-(x+1)^2 - y^2). 13 MATLAB file: go_rand.m

14 Downhill Simplex Search Simplex: a set of n+1 points in n-dim. space A triangle in a 2D space A tetrahedron in a 3D space Concept of downhill simplex search Repeatedly replaces the highest points with a lower one Consecutive successful replacements lead to the enlargement of the simplex Consecutive unsucessful replacements lead to the shrinkage of the simplex 14

15 Downhill Simplex Search Flowchart Figure 7.9 in page 188 Behavior The simplex can adapt itself to the objective function landscape (just like an amoeba), and eventually converges to a nearby local minimum. Program The search procedure is implemented as a function fmins.m that comes with MATLAB. 15

16 Downhill Simplex Search Example: Find the min. of the peaks function z = f(x, y) = 3*(1-x)^2*exp(-(x^2) - (y+1)^2) - 10*(x/5 - x^3 - y^5)*exp(-x^2-y^2) -1/3*exp(-(x+1)^2 - y^2). 16 MATLAB file: go_simp.m

17 Genetic Algorithms Motivation Look at what evolution brings us? -Vision - Hearing - Smelling -Taste - Touch - Learning and reasoning Can we emulate the evolutionary process with today's fast computers? 17

18 Genetic Algorithms Terminology: Fitness function Population Encoding schemes Selection Crossover Mutation Elitism 18

19 Genetic Algorithms Binary encoding Chromosome (11, 6, 9) Crossover Gene Mutation Crossover point Mutation bit

20 Genetic Algorithms Flowchart Current generation Selection Elitism Crossover Mutation Next generation 20

21 Genetic Algorithms Example: Find the max. of the peaks function z = f(x, y) = 3*(1-x)^2*exp(-(x^2) - (y+1)^2) - 10*(x/5 - x^3 - y^5)*exp(-x^2-y^2) -1/3*exp(-(x+1)^2 - y^2). 21

22 Genetic Algorithms Derivatives of the peaks function dz/dx = -6*(1-x)*exp(-x^2-(y+1)^2) - 6*(1-x)^2*x*exp(-x^2-(y+1)^2) - 10*(1/5-3*x^2)*exp(-x^2-y^2) + 20*(1/5*x-x^3-y^5)*x*exp(-x^2-y^2) - 1/3*(-2*x-2)*exp(-(x+1)^2-y^2) dz/dy = 3*(1-x)^2*(-2*y-2)*exp(-x^2-(y+1)^2) + 50*y^4*exp(-x^2-y^2) + 20*(1/5*x-x^3-y^5)*y*exp(-x^2-y^2) + 2/3*y*exp(-(x+1)^2-y^2) d(dz/dx)/dx = 36*x*exp(-x^2-(y+1)^2) - 18*x^2*exp(-x^2-(y+1)^2) - 24*x^3*exp(-x^2-(y+1)^2) + 12*x^4*exp(-x^2-(y+1)^2) + 72*x*exp(-x^2- y^2) - 148*x^3*exp(-x^2-y^2) - 20*y^5*exp(-x^2-y^2) + 40*x^5*exp(- x^2-y^2) + 40*x^2*exp(-x^2-y^2)*y^5-2/3*exp(-(x+1)^2-y^2) - 4/3*exp(- (x+1)^2-y^2)*x^2-8/3*exp(-(x+1)^2-y^2)*x d(dz/dy)/dy = -6*(1-x)^2*exp(-x^2-(y+1)^2) + 3*(1-x)^2*(-2*y-2)^2*exp(- x^2-(y+1)^2) + 200*y^3*exp(-x^2-y^2)-200*y^5*exp(-x^2-y^2) + 20*(1/5*x-x^3-y^5)*exp(-x^2-y^2) - 40*(1/5*x-x^3-y^5)*y^2*exp(-x^2- y^2) + 2/3*exp(-(x+1)^2-y^2)-4/3*y^2*exp(-(x+1)^2-y^2) 22

23 Genetic Algorithms GA process: Initial population 5th generation 10th generation MATLAB file: go_ga.m 23

24 Genetic Algorithms Performance profile 24 MATLAB file: go_ga.m

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