1. 5.2.3 Optimization, Search and
Knowledge Component 5: Information Processing
Computer-Aided Engineering
5.2.3 Optimization, Search and
Exploration 3
2nd Edition

2. Module Information Intended audience Novice Key words
Stochastic methods, global search, simulated annealing, genetic algorithms
Author
Ian Smith, EPFL, Switzerland
Reviewers (1st Edition)
Esther Obonyo, U of Florida, USA
Ni-Bin Chang, U of Central Florida, USA
Frederic Bosche, U of Waterloo, Canada
Rafal Kicinger, George Mason U, USA 2

3. What there is to learn
The quiz at the end will help summarize strengths and weaknesses of stochastic methods. General ideas are:
Stochastic methods should only be used when other methods are not appropriate. Use the simplest method that can be justified by the shape and form of the objective function.
If exhaustive evaluation is feasible, this should be done.
Several runs should be evaluated.
The number of tuning variables influences the usability of methods by novices.

4. Introduction to Stochastic Methods
Outline
Introduction to Stochastic Methods
Simulated Annealing
PGSL
Genetic Algorithms

5. Introduction
Stochastic optimization approaches use random numbers to generate values used in iterations.
While deterministic methods results in the same solution every time for the same problem, stochastic methods may produce different results if the sequence of random numbers generated is different.
Therefore, the starting seed value of the random number generating algorithm may be important.

6. Stochastic Methods Examples of stochastic search methods are:
Simulated Annealing (SA)
Probabilistic Global Search Lausanne (PGSL)
Genetic Algorithms (GA)
These methods are introduced in the following slides.

7. Stochastic Methods (cont'd.)
The methods mentioned above are different in terms of:
the strategies for finding new solutions from the current solutions
point-to-point generation of solutions versus generating sets of solutions

8. Utility Stochastic methods are most useful when
The objective function has multiple minima (or maxima)
The solution space topology is complex. For example, solution spaces may be bounded by non-linear constraints that create holes where no solution exists. Stochastic search methods are more reliable that other methods in such cases.

9. Introduction to Stochastic Methods
Outline
Introduction to Stochastic Methods
Simulated Annealing
PGSL
Genetic Algorithms

10. Simulated Annealing
Simulated Annealing (SA) is a point-to-point search method based on the analogy of annealing of metals (Kirkpatrick et al., 1983). During crystallization of metals, temperature controls the arrangement of atoms in their lowest-energy state.
Analogously, when the temperature is high, the search may move to a point that is worse than the previous point.
When it has cooled down to the freezing temperature, iteration stops.

11. Simulated Annealing: Algorithm
Note
A temperature, T, versus iteration, i, profile that descends to a freezing point is generated before starting. An essential feature of this method is that the temperature is either constant or gradually reducing as the number of iterations increases.
The probability of making the transition from the current state to a candidate new state is a function of the energies of the two states, and of a global time-varying parameter T. It is this feature that prevents the method from becoming stuck in a local minimum.

12. Algorithm (cont’d.) Steps
Generate an initial solution, Si and evaluate the value of the objective function – this is called the energy of the solution, Ei
Select an initial temperature, Ti

13. Algorithm (cont'd.) Steps 3. While it has not “frozen”:
From the current solution Si, generate a new solution, Si+1 stochastically
Evaluate the energy of the new solution, Ei+1
Evaluate the change in energy, E = Ei+1- Ei
If E  0, accept the new solution. If E > 0, accept new solution with transition probability e-E/kT (k is a constant)
Update the temperature to Ti+1 and iterate
4. Return the best solution found.

14. Strengths and Weaknesses
Reliable for non-linear objective functions, chaotic and noisy data and many non-linear constraints.
Weaknesses
A lot of choices are required to turn it into a reliable algorithm. Much tuning may be necessary.
Can be computationally expensive
Solution space coverage may be incomplete.
Example application in civil engineering
Optimization of truss topology

15. Introduction to Stochastic Methods
Outline
Introduction to Stochastic Methods
Simulated Annealing
PGSL
Genetic Algorithms

16. Probabilistic Global Search Lausanne (PGSL)
The PGSL algorithm (Raphael and Smith, 2003) involves sampling of the search space using a probability distribution function (PDF).

17. PGSL (cont’d.)
Steps
Generate solutions according to a PDF and evaluate them using an objective function.
Update the PDF such that probabilities are increased for regions containing good solutions.
The PDF is updated using four search strategies, see next slide.

18. PGSL (cont'd.) Assumption
Better solutions are more likely to be found in the neighborhood of good solutions.
The algorithm consists of four nested cycles
Sampling cycle – a certain number of points are generated randomly according to the current PDF
Probability updating cycle – probabilities are in increased in regions containing good solutions and decreased in regions containing less attractive solutions

19. PGSL (cont'd.)
Focusing cycle – searching is focused on the interval containing the best solution after a number of probability updating cycles (see next slide)
Sub-domain cycle – search space is progressively narrowed by selecting a sub-domain of smaller size centered on the best point after each focusing cycle

20. PGSL (cont'd.)
PDF of a variable value over many probability updating cycles
PDF during focusing cycles

21. Strengths and Weaknesses
Good for smooth domains and continuous variables.
Good convergence and scalability
Little tuning required
Weaknesses
May not be best for non-ordinal discrete variables
Example application in civil engineering
Structural design

22. Introduction to Stochastic Methods
Outline
Introduction to Stochastic Methods
Simulated Annealing
PGSL
Genetic Algorithms

23. Genetic Algorithms
Genetic Algorithms (GA) use the principles of natural evolution to explore a population of solutions.
In most applications, solutions are described by a series of bits.
Example: Weather {Cold, cloudy, rainy}
011 = Cloudy and rainy
Here, if a bit is 1, the corresponding value is true (0 if false)

24. Genetic Algorithms (cont’d.)
Steps are
Generation of an initial population
Reproduction
Selection (and/or Duplication)
Crossover (and/or Mutation)
Evaluation

25. Strengths and Weaknesses
Good for non-ordinal discrete variables
Accommodates discontinuous functions and constraints
Searches a population for good coverage; sets of solutions are generated in a single run
Weaknesses
Models with many parameters may be computationally expensive
Convergence is slow when all the models exhibit similar fitness
Tuning of parameters may be difficult

26. Applications Example applications in civil engineering
Structural design
Design of road networks
Active control of complex structures

27. Review Quiz
List the strengths, weaknesses and ideal conditions for reliable application of (1) Simulated Annealing, (2) PGSL and (3) Genetic Algorithms.

28. Answers to Review Quiz
List the strengths, weaknesses and the ideal condition for effective application of (1) Simulated Annealing, (2) PGSL and (3) Genetic Algorithms.
Method
Ideal application
Strengths
Weaknesses
Simulated Annealing (SA)
Where good heuristics are available for point-to-point movement
Suitable for non-linear models, noisy data and non-linear constraints
Tuning required; complete domain may not be covered
PGSL
Smooth domains and continuous variables
Good convergence, scalability; little tuning required
May not be best for non-ordinal discrete variables
Genetic Algorithms (GAs)
Good for non ordinal discrete variables
Widely known and tested
Tuning required; slow convergence when models have similar fitness

29. Further Reading
“Global Optimization methods for engineering applications: a review”, J.S. Arora, O.A. Elwakeil, A.I. Chahande, C.C. Hsieh, Structural Optimization, 9, , 1995
B. Raphael and I.F.C. Smith, A direct stochastic algorithm for global search, J of Applied Mathematics and Computation, Vol 146, No 2-3, 2003, pp