1. Evolutionary Computational Intelligence
Ferrante Neri
26/05/2019
Evolutionary Computational Intelligence
Lecture 1:
Basic Concepts
Ferrante Neri
University of Jyväskylä
26/05/ :59:09
Lecture 1:Basic Concepts

2. Introductory Example: Radio Tuning
Ferrante Neri
26/05/2019
Introductory Example: Radio Tuning
Position (e.g. angular) of the Radio Knob : candidate solution
We want to have a clear signal that is:
Maximize the signal
Minimize the background noise
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Lecture 1:Basic Concepts

3. Lecture 1:Basic Concepts
Optimization Problem
Candidate solution
Decision (or design) variables
Variable bounds define the decision space
objective function or fitness function (the behavior taken by the fitness over the decision space is called fitness landscape)
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Lecture 1:Basic Concepts

4. Real-World Optimization Problems
Optimization Problems are often rather easily formulated but very hard to be solved when the problem comes from an application. In fact, some features characterizing the problem can make it extremely challenging.
These features are summarized in the following:
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Lecture 1:Basic Concepts

5. Highly nonlinear fitness function
Usually optimization problems are characterized by nonlinear function. In real world optimization problems, the physical phenomenon, due to its nature (e.g. in the case of saturation phenomenon or for systems which employ electronic components), cannot be approximated by a linear function neither in some areas of the decision space.
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Lecture 1:Basic Concepts

6. Highly multimodal fitness landscape
It often happens that the fitness landscape contains many local optima and that many of these have an unsatisfactory performance (fitness value)
These fitness landscapes are usually rather difficult to be handled since the optimization algorithms which employ gradient based information in detecting the search direction could easily converge to a suboptimal basin of attraction
Basin of attraction: set of points of the decision space such that ,initial conditions chosen, dynamically evolve to a particular attractor
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Lecture 1:Basic Concepts

7. Optimization in Noisy Environment
Uncertainties in optimization can be categorized into three classes.
Noisy fitness function. Noise in fitness evaluations may come from many different sources such as sensory measurement errors or randomized simulations.
Approximated fitness function. When the fitness function is very expensive to evaluate, or an analytical fitness function is not available, approximated fitness functions are often used instead. These approximated models implicitly introduce a noise which is the difference between the approximated value and real fitness value, which is unknown.
Robustness. Often, when a solution is implemented, the design variables or the environmental parameters are subject to perturbations or changes (e.g. control problems).
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Lecture 1:Basic Concepts

8. Computationally expensive problems
Optimization problems can be computationally expensive because of two reasons:
high cardinality decision space (usually combinatorial)
computationally expensive fitness function (e.g. design of on-line electric drives)
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Lecture 1:Basic Concepts

9. Real-World Problems and Classical Methods
When such features are present in an optimization problem, the application of exact methods is usually unfeasible since the hypotheses are not respected.
Moreover the application of classical deterministic algorithms is also questionable since their use could easily lead to suboptimal solutions (e.g. a hill climber for highly multimodal functions) or return completely unreliable results (e.g. a deterministic optimizer in noisy environments).
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Lecture 1:Basic Concepts

10. Lecture 1:Basic Concepts
Rosenbrock Algorithm 1960
From chemical application
Well defined decision space
No analytical expression and no derivatives
Modification of a steepest descent method
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Lecture 1:Basic Concepts

11. Lecture 1:Basic Concepts
Rosenbrock Algorithm
The search is executed along each direction variable (orthogonal search)
The search is continued by enlarging the step size for successful directions and reducing for unsuccessful directions
The search is stopped when the trial was successful in all the directions
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Lecture 1:Basic Concepts

12. Lecture 1:Basic Concepts
Rosenbrock Algorithm
Under these conditions, a new set of directions is determined by means of Gram-Schmidt procedure and the search is start over
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Lecture 1:Basic Concepts

13. Lecture 1:Basic Concepts
Rosenbrock Algorithm
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Lecture 1:Basic Concepts

14. Hooke Jeeves Algorithm (1961)
exploratory radius h, an initial candidate
Initial solution x
n × n direction exploratory matrix U (e.g. diag (w(1),w(2), …w(i)...,w(n), where w (i) is the width of the range of variability of the i-th variable)
U(i,:) is the i-th row of the matrix
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Lecture 1:Basic Concepts

15. Hooke Jeeves Algorithm
Exploratory Move: samples solutions x (i)+hU(i, :) (”+” move) with i = 1, 2, , n and thesolutions x (i)−hU(i, :) (”-” move) with i = 1, 2, , n only along those directions which turned out unsuccessful during the ”+” move
Directions are analyzed
Separately!
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Lecture 1:Basic Concepts

16. Hooke Jeeves Algorithm
Pattern Move: The pattern move is an aggressive attempt of the algorithm to exploit promising search directions.
Rather than centering the following exploration at the most promising explored candidate solution the HJA tries to move further. The algorithm makes a double step and centers the subsequent exploratory move. If this second exploratory fails: step back and exploratory.
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Lecture 1:Basic Concepts

17. Hooke Jeeves Algorithm
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18. Nelder Mead Algortihm (1965)
works on a setof n + 1 solutions in order to perform the local search
it employs an exploratory logic based on a dynamic construction of a polyhedron (simplex)
n + 1 solutions x0, x1, , xn sorted in descending order according to their fitness values (i.e. x0 is the best), the NMA attempts to improve xn
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19. Lecture 1:Basic Concepts
Nelder Mead Algorithm
The centroid xm is calculated:
1st step: reflection
If the reflected point outperforms x0,replacement occurs
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Lecture 1:Basic Concepts

20. Lecture 1:Basic Concepts
Nelder Mead Algorithm
2nd step: expansion. if the reflection was successful, (i.e. reflected point better than x0) it calculates
if the expansion is also successful new replacement occurs
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Lecture 1:Basic Concepts

21. Lecture 1:Basic Concepts
Nelder Mead Algorithm
If xr did not improve upon x0,
If f (xr) < f (xn−1) then xr replaces xn.
If this trial is also unsuccessful,
if f (xr) < f (xn), xr replaces xn and the 3rd step: outside contraction.
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Lecture 1:Basic Concepts

22. Lecture 1:Basic Concepts
Nelder Mead Algorithm
If xr does not outperform neither xn then the 4th step: inside contraction:
if the contraction was successful then xc replaces xn
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Lecture 1:Basic Concepts

23. Lecture 1:Basic Concepts
Nelder Mead Algorithm
If there is no way to improve x0 the 5th step shrinking: n new points a sampled and the process is started over
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Lecture 1:Basic Concepts

24. Lecture 1:Basic Concepts
Comparative Analysis
The three algorithms do not require derivatives and do not require explicit analytical expression
Rosenbrock and Hooke Jeeves are fully deterministic while Nelder Mead has some randomness
Rosenbrock and Nelder Mead move in the space along all the directions simultaneously (e.g. diagonal in 2D) while Hooke Jeeves moves along one direction at once
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25. Fundamental Points in Comparative Analysis
Rosenbrock and Hooke Jeeves have a mathematically proved convergence while Nelder Mead doesn’t!
Rosenbrock and Hooke Jeeves have “local properties” while Nelder Mead has “global properties”
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26. Two-Phase Nozzle Design (Experimental)
Experimental design optimisation: Optimise efficieny.
... evolves...
Initial design
Final design: 32% improvement in efficieny.
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Lecture 1:Basic Concepts