1. Local optimization technique
G.Anuradha

2. Introduction
The evaluation function defines a quality measure score landscape/response surface/fitness landscape

3. Hill Climbing start at randomly generated state
move to the neighbour with the best evaluation value
if a strict local-minimum is reached then restart at other randomly generated state.

4. Flowchart of Hill climbing
Select a current solution s
Evaluate s
Select a new solution x from the neighborhood of s
Evaluate x
Is x better than s?
Select x as new current solution s
yes no

5. Stopping condition Either the whole neighborhood has been searched
Or we have exceeded the threshold of allowed attempts
The last solution is the best solution or the current solution is stored and the same procedure is repeated again(iterated hill climbing)

6. Features of hill climbing techniques
Provides local optimum values that depends on starting solution
Can’t be used for finding the global optimum because there is no general procedure for measuring the relative error with respect to global optimum
The success of the algorithm is depended on the initial value choosen

7. Weakness of hill climbing algorithm
Termination on local optimum values
There is no indication of how much the local optimum deviates from global optimum
The optimum value obtained depends on the initial configurations
An upper bound of computation time can’t be provided

8. Stochastic Hill Climber
The problem of getting struck up in the local optima is eliminated to a certain extend in this approach
In this approach new solutions having negative change in the quality measure score is also accepted.
Some basic changes to the ordinary hill climbing is made in stochastic hill climbing approach

9. Flowchart of Hill climbing
Select a current solution s
Evaluate s
Select a new solution x from the neighborhood of s
Evaluate x
Is x better than s?
Select x as new current solution s
yes no

10. Stochastic Hill climbing approach
Select a current solution s
Evaluate s
Select a new solution x from the neighborhood of s
Evaluate x
Select x as a new current solution s with probability P

11. How this probabilistic function works?
There are 3 cases
50% probable: if the new solution x has the same quality measure score as the current solution s
>50% probable: if the new solution x is superior then the probability of acceptance is greater than 50%
<50% : if the new solution is inferior, then the probability of acceptance is smaller than 50%

12. Effect of parameter T
If the new solution x is superior, then the probability of acceptance is closer to 50% for high values of T, or closer to 100% for low values of T
If the new solution x is inferior, then the probability of acceptance is closer to 50% for high values of T or closer to 0% for low values of T

13. How this probabilistic function works? Contd…
The probability of accepting a new solution x also depends on the value of parameter T( T remains constant during the execution of the algorithm)
Superior solution x would have a probability of acceptance of atleast 50%(irrespective of T)
Inferior solution x have a probability of acceptance of at most 50% (0 – 50%)
T is neither too low nor too high for a particular problem
Forerunner of simulated annealing

14. Annealing
Heating steel at a suitable temperature, followed by relatively slow cooling.
The purpose of annealing may be to remove stresses, to soften the steel, to improve machinability, to improve cold working properties, to obtain a desired structure.
The annealing process usually involves allowing the steel to cool slowly in the furnace.

15. Simulated Annealing n Is T low? y STOP Decrease T y Is K large enough?
Set the initial temperature T n
Is T low? y
Select a current solution s
STOP
Evaluate s
Decrease T
K=0 y
Is K large enough?
K=K+1
Select a new solution x in the neighborhood of s n
Evaluate x
Select x as a new current solution s
Select x as a new current solution s with probability p y
X better than s n

16. Analogy between both Annealing Simulated Annealing State
Feasible solution
Energy
Evaluation function
Ground state
Optimal solution
Rapid quenching
Local search
Careful annealing
Simulated annealing

17. When applied to a specific applications some questions come in mind?
SA resembles a random search at higher temperatures and classic hill climber at lower temperatures
When applied to a specific applications some questions come in mind?
What is the representation?
How are neighbors defined?
What is the evaluation function?
How to determine how big is k?
How to cool the system or how to decrease the temperature?
How to determine the stopping condition?

18. Tabu Search
Meta-heuristics search algorithm that guides a local heuristic search procedure to search beyond local optimality
Uses adaptive memory and responsive exploration to explore the search space
Its deterministic in nature, but its possible to add some probabilistic elements to it

19. Flowchart of tabu search
Set the initial memory M
Select a current solution s
Evaluate S
Select a no. of solutions x,y,.. From neighbourhood of s
Evaluate x,y,z……
Select one solution x as new solution s, the decision based on quality measure score and M
Update M

20. Memory component of tabu search
There are 3 ways of computing memory
Recency based memory:- Memory structure gets updated after certain iterations and records the last few iterations
Frequency based memory: the memory structure works for a longer time horizon and measures the frequency of change at each position

21. Evolution
Here’s a very oversimplified description of how evolution works in biology
Organisms (animals or plants) produce a number of offspring which are almost, but not entirely, like themselves
Variation may be due to mutation (random changes)
Variation may be due to sexual reproduction (offspring have some characteristics from each parent)
Some of these offspring may survive to produce offspring of their own—some won’t
The “better adapted” offspring are more likely to survive
Over time, later generations become better and better adapted
Genetic algorithms use this same process to “evolve” better programs

22. Evolutionary Algorithms

23. Genotype and Phenotype
Genes are the basic “instructions” for building an organism
A chromosome is a sequence of genes
Biologists distinguish between an organism’s genotype (the genes and chromosomes) and its phenotype (what the organism actually is like)
Example: You might have genes to be tall, but never grow to be tall for other reasons (such as poor diet)
Similarly, “genes” may describe a possible solution to a problem, without actually being the solution

25. The basic genetic algorithm
Start with a large “population” of randomly generated “attempted solutions” to a problem
Repeatedly do the following:
Evaluate each of the attempted solutions
Keep a subset of these solutions (the “best” ones)
Use these solutions to generate a new population
Quit when you have a satisfactory solution (or you run out of time)

26. Flowchart of evolution algorithm
Create initial population A
Initialize counter t=0
Evaluate all s from A
Select a set of parents from A
Create a set of offspring
Create a new population A from existing parents and offspring
yes
Is t large
t=t+1 no
STOP

27. A really simple example
Suppose your “organisms” are 32-bit computer words
You want a string in which all the bits are ones
Here’s how you can do it:
Create 100 randomly generated computer words
Repeatedly do the following:
Count the 1 bits in each word
Exit if any of the words have all 32 bits set to 1
Keep the ten words that have the most 1s (discard the rest)
From each word, generate 9 new words as follows:
Pick a random bit in the word and toggle (change) it
Note that this procedure does not guarantee that the next “generation” will have more 1 bits, but it’s likely

28. A more realistic example, part I
Suppose you have a large number of (x, y) data points
For example, (1.0, 4.1), (3.1, 9.5), (-5.2, 8.6), ...
You would like to fit a polynomial (of up to degree 5) through these data points
That is, you want a formula y = ax5 + bx4 + cx3 + dx2 +ex + f that gives you a reasonably good fit to the actual data
Here’s the usual way to compute goodness of fit:
Compute the sum of (actual y – predicted y)2 for all the data points
The lowest sum represents the best fit
There are some standard curve fitting techniques, but let’s assume you don’t know about them
You can use a genetic algorithm to find a “pretty good” solution

29. A more realistic example, part II
Your formula is y = ax5 + bx4 + cx3 + dx2 +ex + f
Your “genes” are a, b, c, d, e, and f
Your “chromosome” is the array [a, b, c, d, e, f]
Your evaluation function for one array is:
For every actual data point (x, y), (I’m using red to mean “actual data”)
Compute ý = ax5 + bx4 + cx3 + dx2 +ex + f
Find the sum of (y – ý)2 over all x
The sum is your measure of “badness” (larger numbers are worse)
Example: For [0, 0, 0, 2, 3, 5] and the data points (1, 12) and (2, 22):
ý = 0x5 + 0x4 + 0x3 + 2x2 +3x + 5 is = 10 when x is 1
ý = 0x5 + 0x4 + 0x3 + 2x2 +3x + 5 is = 19 when x is 2
(12 – 10)2 + (22 – 19)2 = = 13
If these are the only two data points, the “badness” of [0, 0, 0, 2, 3, 5] is 13

30. A more realistic example, part III
Your algorithm might be as follows:
Create 100 six-element arrays of random numbers
Repeat 500 times (or any other number):
For each of the 100 arrays, compute its badness (using all data points)
Keep the ten best arrays (discard the other 90)
From each array you keep, generate nine new arrays as follows:
Pick a random element of the six
Pick a random floating-point number between 0.0 and 2.0
Multiply the random element of the array by the random floating-point number
After all 500 trials, pick the best array as your final answer

31. The really simple example again
Suppose your “organisms” are 32-bit computer words, and you want a string in which all the bits are ones
Here’s how you can do it:
Create 100 randomly generated computer words
Repeatedly do the following:
Count the 1 bits in each word
Exit if any of the words have all 32 bits set to 1
Keep the ten words that have the most 1s (discard the rest)
From each word, generate 9 new words as follows:
Choose one of the other words
Take the first half of this word and combine it with the second half of the other word

32. Asexual vs. sexual reproduction
In the examples so far,
Each “organism” (or “solution”) had only one parent
Reproduction was asexual
The only way to introduce variation was through mutation (random changes)
In sexual reproduction,
Each “organism” (or “solution”) has two parents
Assuming that each organism has just one chromosome, new offspring are produced by forming a new chromosome from parts of the chromosomes of each parent

33. Crossover and mutation operators

34. Crossover
Crossover is a genetic operator that combines (mates) two chromosomes (parents) to produce a new chromosome (offspring). 
Types of crossover
One point
Two point
Arithmetic
Heuristic

35. Types of crossover One point crossover
Two point crossover

36. Arithmetic crossover
Offspring1 = a * Parent1 + (1- a) * Parent2 Offspring2 = (1 – a) * Parent1 + a * Parent2
Parent 1: (0.3)(1.4)(0.2)(7.4) Parent 2: (0.5)(4.5)(0.1)(5.6) a=0.7
Offspring1: (0.36)(2.33)(0.17)(6.86) Offspring2: (0.402)(2.981)(0.149)(6.842)

37. Comparison of simple examples
In the simple example (trying to get all 1s):
The sexual (two-parent, no mutation) approach, if it succeeds, is likely to succeed much faster
Because up to half of the bits change each time, not just one bit
However, with no mutation, it may not succeed at all
By pure bad luck, maybe none of the first (randomly generated) words have (say) bit 17 set to 1
Then there is no way a 1 could ever occur in this position
Another problem is lack of genetic diversity
Maybe some of the first generation did have bit 17 set to 1, but none of them were selected for the second generation
The best technique in general turns out to be sexual reproduction with a small probability of mutation