Function optimization

1. Function optimization Stelios Papadakis spap@staff.teicrete.gr

2. The problem of optimization • Consider that you are in the middle of a mountain surrounded by hills, with your eyes hermitically closed. • Your mission is to reach the highest/lowest point of the mountain on foot by only using an altimeter for evaluating your position. • a straightforward strategy for accomplishing your mission? • Step-1: memorize your current position • Step-2: make a step to a random direction and measure the altitude of the new position • Step-2.1: if the new altitudeis higher than the previous one then make a step forward to the same direction • Step-2.2: if the new altitude is lower than the previous one then make a step backwards to the opposite direction • Step-2.3: if the altitude of the new point is the same with the previous one then you are in a plateau, probably.

3. Geometrically (minimization) Local optimum Basin of attraction Derivative based methods Global optimum

4. Analyzing the strategy • There exists a mechanism for changing your position (i.e. for generating new candidate solutions) • There exists a method for evaluating your position (Objective function) • The Objective function is a mapping from the dimensional space: to the set of real numbers: , usually • The objective function is the common concept for all optimization methods. • The target is to find the value of usually noted as that maximizes/minimizes the value of the objective function. • Mathematically: for minimization for maximization

5. Analyzing the strategy • A mechanism for generating new solutions • A function for evaluating potential solutions • Methods based on the derivatives of the objective function • Newton, Backpropagation, Steepest ascent • Derivatives are required (problems, limitations) • Local optima trapping (momentum, etc.) • One starting point (multiple starting points) • The landscape is a plateau derivatives are zero • Population based methods • Derivative free • A population of points, evaluated in parallel • The mechanism for generating new solutions • Evaluation of solutions is the same.

6. Population based algorithms (minimization) • Population of solutions, simultaneously evaluated • No derivatives are required • By design capacity for escaping from local optima • For the next time step, a new population is derived from the previous one • fitness for GA • Premature convergence • Slow convergence (hill climbing operators) • Exploration capacity • Exploitation capacity • The ‘no free lunch’ theorem

7. Single and many optima • Multimodal objective functions • Unimodal objective functions

8. The real picture after some random points • We know • We know the objective function • And domain of parameter • The picture of the previous slide requires the evaluation of infinite points • We know if a solution is better than another one • We have to produce a better solution based on these observations • We don’t know • We don’t know something regarding the global optimum • We don’t know if a solution is global or local optimum • Actually, there is no optimization method that guarantees the location of global optimum.

9. Some special function landscapes • The landscape has wide flat areas (plateaus) • The basin of attraction is extremely narrow surrounded by a plateau (Needle in a Haystack) (e.g. Password detection) • The landscape has convex paths to a local optimum while the global optimum is located in the opposite direction (deceptive problems) • Narrow paths to the global optimum (Ridges of the search space) where the fitness function is worse for every point outside the path • These landscapes usually appear to any non convex objective functions • VeryDifficult for population based algorithms. • Unfeasible for derivative based methods

10. Constraint optimization (minimization) • General formulation S.t. A common trick is to convert the constraint optimization problem to an unconstraint one, by using penalty factors/functions. So, optimize the instead of . • : are arbitrarily large positive real numbers. • Penalty factors deteriorate the solutions that violate the constraints. • The penalty factors may accept constant values or increasing values as the optimization proceeds

11. Evolutionary algorithms’ family tree • Evolutionary algorithms (EA) are optimizationmethods • EA are not modelling tools • Modeling tools are Neural networks and fuzzy systems • An evolutionary algorithm could be used to optimize the parameters/structure of a model as any other optimization method

12. A more detailed family tree

13. A brief history Τhere is no absolute agreement between historians, but it does not matters for engineers

14. Genetic Algorithms (GAs) • GAs are evolutionary algorithms • GAs simulates the Darwinian theory of natural evolution/natural selection • GAs are population based optimization algorithms • SGA: Simple Genetic Algorithm (Holland/Goldberg)

15. Anatomy of a SGA • A population of individuals • An individual is an entity that has at least two properties • A chromosome • A fitness value • The chromosome is a string of symbols usually bit string • The fitness value is a real number that corresponds to the chromosome of the individual • Practically, given an individual, its fitness value is computed from its chromosome according to the objective function being optimized

16. The chromosome • A string of symbols according to a selected alphabet • If the alphabet is then the chromosome is a binary string of bits gene Genotype Provided that this part of chromosome express something with specific meaning (e.ga variable) phenotype A useable number (genotype expressed). The integer number that a genotype substring expresses could be the phenotype (here 170) • Genotype space • Phenotype space

17. Encoding of one variable as binary stringalphabet , gene, genotype, allele, phenotype • Decision : The number of bits • The number of bits defines the resolution of variable • Gene: a single bit • Genotype: a part of chromosome that corresponds to a value with specific meaning • Phenotype: a genotype which is expressed (e.g. the value of a variable that participates in the computation of fitness) • Genotype, phenotype space

18. Decoding of one variable from the chromosome • Decision : the domain of the variable is required • a specific integer value should be linearly mapped in the interval • The integer value is linearly mapped to the real value • From the equation of line:

19. Formulating an optimization problem to be solved by EAs • Define the objective function to be optimized • Define the tunable parameters of the objective function and formulate them as a vector • Encoding of tunable variables into the chromosome • Define and memorize the number of bits for each tunable parameter. Consider each parameter as a tuple • The total length of chromosome is bits.

20. Decoding • The rangeof each tunable parameter is required • the part of chromosome that encodes the parameter starts from the bit • and ends to the bit

21. Decoding • : the integer phenotype value of the binary sub-string • : the real phenotype value in the actual range • The mapping is linear • example

22. Evaluation • After decoding all variables the vector of parameters is known • The respective fitness value is the value of the objective function on these specific parameter values • A single individual is evaluated

23. Initialization of SGA • Compute the chromosome length according to the number of parameters and the number of bits per parameter • Select the number of individuals to craft the population • Randomly initialize the chromosome of each individual • Evaluate each individual according to the objective function • The first population is created at or generation 0 • From the initial population we need to produce a new generation with the expectation that the children would be better than their parents (at least in average) • This is the reproduction procedure

24. Reproduction • Selection • Roulette wheel selection • Tournament selection • Proportional selection • Crossover • One point multipoint, uniform • Mutation • One point, multipoint, inversion • Elitism (not an SGA operator, but necessary for practical applications) • Problem specific operators (e.g. hill climbing) usually applied to the elite individual

25. Roulette wheel Selection

26. crossover

27. Uniform crossover

28. Mutation

29. Are the claims of this slide correct?

30. Specific crossover and mutation operators • Floating point crossover and mutation

31. Single/simple arithmetic crossover

32. Whole arithmetic crossover

33. Specific crossover/mutation for combinatorial problems

34. Example: TSP problem

35. Mutation Many other mutation exist (e.g. insertion mutation, scramble mutation etc)

36. Crossover

37. example Many other crossover operators exist, (e.g. PMX crossover, cycle crossover e.t.c.)

38. Other concepts

39. Fitness ranking • Premature convergence (conditions?) • Fitness scaling • Fitness ranking

40. The new Generation (reproduction) Mutation with probability child 1 child 2 child 1 child 2 Initial population Parent 1 Parent 2 Crossover With probability Selection 2 times Selects 2 parents Mutation with probability Place the new children To the new population • Repeat times to produce the new population • Optional apply elitism (no SGA operator but useful in practice) • How to make a function to return 1 with probability • Generate a uniform random number in [0,1] • If that number is less than the return 1 • Else return 0 New population

41. The process of SGA evolution Initialize the population Reproduction produce a new population No Is stopping criterion satisfied? yes End of evolution

42. Example

43. selection

44. Crossover

45. Mutation

46. Another example

47. GA Example Problem Optimize value of over where x is restricted to integers. The maximum value of 1 occurs at x = 128. Note: Function space is identical to fitness space.

48. GA Example Problem • One variable • Use binary alphabet • Therefore, represent each individual as 8-bit binary string • Set number of population members to (artificially low) 8 • Randomize population • Calculate fitness for each member of (initialized) population

49. GA Example Problem

50. GA Example Problem - Reproduction Reproduction is used to form new population of n individuals Select members of current population Use stochastic process based on fitnesses First, compute normalized fitness value for each individual by dividing individual fitness by sum of all fitnesses (fi / 5.083 in the example case) Generate a random number between 0 and 1 n times to form new population (sort of like spinning roulette wheel)