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SIMULATED ANNEALING
STOCHASTIC OR RANDOMIZED APPROACH FOR ESCAPING LOCAL
OPTIMA
OPTIMIZATION
HILL CLIMBING
GLOBAL OPTIMUMLOCAL OPTIMUM
Let us consider one-dimensional search space as below:HILL CLIMBING WILL REACH LOCAL OPTIMA AND MAY STOP THEREHILLCLIMBING-EXPLOITATION OF GRADIENTSO GLOBAL OPTIMA MAY NOT BE ACHIEVEDLET C-CURRENT NODEN-NEXT NODE
LOOPMOVE FROM C TO N IF BEST(MOVEGEN(C) IS BETTER THAN CEND LOOP;
RANDOM WALK
• RANDOM WALK IS CHOOSING JUST RANDOM NEXT NODE
• IT IS NOTHING BUT EXPLORATION
GOING BEYOND LOCAL OPTIMA-APPROACHES
• TABU SEARCH• STOCHASTIC HILL CLIMBING• SIMULATED ANNEALING
OPTIMIZATION
• OPTIMIZATION IS HAPPENING IN PHYSICAL WORLD AND NATURE
• OPTIMIZATION IN PHYSICAL WORLD REPRESENTS THE WAY ATOMS ARE ARRANGED IN MATERIALS
• FOR EXAMPLE IF ATOMS ARE IN ARRAY –CRYSTALLINE STRUCTURE IS FORMED
OPTIMIZATION
• SO BASIC METHOD OF ACHIEVING A WELL DEFINED PHYSICAL MATERIAL IS BY GRADUALLY1. HAVING A LIQUID FORM OF MATERIAL2. AND SLOWLY COOLING IT DOWN TO FORM
MOLTEN MATERIAL– THIS PROCESS CALLED CASTING IS USED IN
MAKING BRONZE STATUES etc…
Optimization
• Final form of materials- low energy materials –minimum energy state
• Physical process of MINIMIZATION-ANNEALING
• ANNEALING is nothing but CONTROLLED COOLING
OPTIMIZATION
• HILL CLIMBING -> Generate neighbors of a given candidate. Inspect their evaluation Heuristic value and depending on that move to efficient nearest neighbor->It can reach local optima.
• RANDOM WALK-UNGUIDED.It Can choose ANY random points-EXPLORATION
• For optimizing we need a mixture of both
Maximizing and Minimizing Functions
• C-> Current node• N-> next node• eval(c)- Fitness function of c• eval (n)-Fitness function of n• Maximizing function means
Move to n if eval(n)>eval(c)• Minimizing function means
Move to n if eval(n)<eval(c)
Optimization • Take a Random Neighbour• Then we make a decision to move to that neighbour or not• So we take
– ∆E=eval(n)-eval(c), for maximizing• For maximizing problems if ∆E>0 (OR) positive we will
surely make a move with HIGHER PROBABILITY• With negative ∆E,we will still allow moves with lower
probability• So we NEED A FUNCTION to compute probability such that
– ∆E should influence probability– A control variable how much ∆E should influence probability
SIGMOIDAL FUNCTION
• WE WILL CHOOSE THAT FUNCTION TO BE SIGMOIDAL
• P(c,n)=1/(1+e- ∆E/T )
• T-CONTROL VARIABLE
STOCHASTIC HILL CLIMBING
• n<-random neighbor(c)• Evaluate ∆E– ∆E=eval(n)-eval(c)
• Move with Probability– P(c,n)=1/(1+e- ∆E/T )
EXAMPLES-EFFECT of ∆E• LET US ASSUME T=10,eval(c)=107
eval(n) -∆E e- ∆E/T ProbabilityP(c,n)
Comments
80 27 14.88 0.06 Move with small probability
100 7 2.01 0.33 1/3rd chance
107 0 1.0 0.5 Eval(c)=eval(n)Can make move or cannot make
120 -13 0.27 0.78 Move with Higher Probability
150 -43 0.01 0.99 Move with very Higher Probability
EXAMPLES-EFFECT of ∆E
• The previous table illustrates HOW STOCHASTIC HILL CLIMBING RESPONDS TO DIFFERENT VALUES OF ∆E
• HOW TO CHOOSE T is the next Question?• Let us take this case which is better one and
evaluate how T effects this case:eval(n) -∆E e- ∆E/T Probability
P(c,n)Comments
120 -13 0.27 0.78 Move with Higher Probability
How T effects a Sample case when EVAL(N)=120
T e-13/T Probability P Inference
1 0.000002 1.0 SIMILAR TO HILL CLIMBING
5 0.074 0.93
10 0.27 0.78
20 0.52 0.66
50 0.77 0.56
1010 0.9999 0.5 SIMILAR TO RANDOM WALK
1. As energy level increases or T-Temperature value increases it becomes RANDOM WALK
2. IF we want to EXPLORE MORE or want more RANDOMNESS we make Temperature Very high irrespective of ∆E
3. IF we want to follow the GRADIENT we make TEMPERATURE AS LOW.
SIGMOIDAL PLOT
1
∆E=0
T=1
0.5T=1010
SIGMOIDAL FUNCTION
SIMULATED ANNEALING-ALGORITHM
• SET T<-VERY HIGH VALUE• OUTER LOOP – INNER LOOP
• N<-RANDOM NEIGHBOR(C) • Evaluate ∆E
– ∆E=eval(n)-eval(c)• Move with Probability
– P(c,n)=1/(1+e- ∆E/T )
– END INNER LOOP– T<-MONOTONIC DECREASING FUNCTION(T)
• END OUTER LOOP
MONOTONIC DECREASING FUNCTION(T)
• IT IS CALLED COOLOING RATE• SIMPLEST IS T<-T-1
