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Optimization methods Review

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Optimization methodsReview

Mateusz Sztangret

Faculty of Metal Engineering and Industrial Computer Science

Department of Applied Computer Science and Modelling

Krakow, 03-11-2010 r.

Basic concepts of optimization

Review of optimization methods

- gradientless methods,
- gradient methods,
- linear programming methods,
- non-deterministic methods
Characteristics of selected methods

- method of steepestdescent
- genetic algorithm

Man’s longing for perfection finds expression in the theory of optimization. It studies how to describe and attain what is Best, once one knows how to measure and alter what is Good and Bad… Optimization theory encompasses the quantitative study of optima and methods for finding them.

Beightler, Phillips, Wilde

Foundations of Optimization

Optimization /optimum/ - process of finding the best solution

Usually the aim of the optimization is to find better solution than previous attached

Specification of the optimization problem:

- definition of the objective function,
- selection of optimization variables,
- identification of constraints.

where:

- x is the vector of variables, also called unknowns or parameters;
- f is the objective function, a (scalar) function of x that we want to maximize or minimize;
- giand hi are constraint functions, which are scalar functions of x that define certain equations and inequalities that the unknown vector x must satisfy.

Constrain functions define the set of allowed solution that is a set of points which we consider in the optimization process.

X

Xd

Solution is called global minimum if,

for all

Solution is called local minimum if there is a neighbourhood N of such that

for all

Global minimum as well as local minimum is never exact due to limited accuracy of numerical methods and round off error

f(x)

local minimum

global minimum

x

f(x)

start

start

x

f(x)

Discontinuous function

x

3

f(x)

f

c

x*

x

– c

– f

Start

Set starting point x(0)

i = 0

i = i + 1

Calculatef(x(i))

NO

Stop condition

x(i+1) = x(i) + Δx(i)

YES

Stop

Commonly used stop conditions are as follows:

- obtain sufficient solution,
- lack of progress,
- reach the maximum number of iterations

The are several type of optimization algorithms:

- gradientless methods,
- line search methods,
- multidimensional methods,

- gradient methods,
- linear programming methods,
- non-deterministic methods

- Line search methods
- Expansion method
- Golden ratio method

- Multidimensional methods
- Fibonacci method
- Method based on Lagrange interpolation
- Hooke-Jeeves method
- Rosenbrock method
- Nelder-Mead simplex method
- Powell method

Advantages:

- simplicity,
- they do not require computing derivatives of the objective function.
Disadvantages:

- they find first obtained minimum
- they demand unimodality and continuity of objective function

- Method of steepest descent
- Conjugate gradients method
- Newton method
- Davidon-Fletcher-Powell method
- Broyden-Fletcher-Goldfarb-Shanno method

Advantages:

- simplicity,
- greater effciency in comparsion with gradientless methods.
Disadvantages:

- they find first obtained minimum
- they demand unimodality, continuity and differentiability of objective function

If both the objective function and constraints are linear we can use one of the linear programming method:

- Graphical method
- Simplex method

- Monte Carlo method
- Genetic algorithms
- Evolutionary algorithms
- strategy (1 + 1)
- strategy (μ + λ)
- strategy (μ, λ)

- Particle swarm optimization
- Simulated annealing method
- Ant colony optimization
- Artificial immune system

Advantages:

- any nature of optimised objective function,
- they do not require computing derivatives of the objective function.
Disadvantages:

- high number of objective function calls

Ways of integrating constrains

- External penalty function method
- Internal penalty function method

In some cases solved problem is defined by few objective function. Usually when we improve one the others get whose.

- weighted criteria method
- ideal point method

Method involves the transformation

multicriterial problem into

one-criterial problem by adding

particular objective functions.

In this method we choose

an ideal solution which is

outside the set of allowed

solution and the searching

optimal solution inside

the set of allowed solution

which is closest the

the ideal point. Distance we can

measure using various metrics

Ideal point

Allowed solution

Algorithm consists of following steps:

- Substitute data:
- u0 – starting point
- maxit – maximum number of iterations
- e – require accuracy of solution
- i = 0 – iteration number

- Compute gradient in ui

- Choose the search direction
- Find optimal solution along the chosen direction (using any line search method).
- If stop conditions are not satisfied increased i and go to step 2.

Let’s consider a problem

of finding minimum

of function:

f(u)=u12+3u22

Starting point:

u0=[-2 3]

Isolines

Algorithm consists of following steps:

- Creation of a baseline population.
- Compute fitness of whole population
- Selection.
- Crossing.
- Mutation.
- If stop conditions are not satisfied go to step 2.

Genotype

1 0 1 0 1 0 1 0

0 1 0 1 0 1 0 1

1 1 0 1 0 1 0 0

1 0 1 1 0 1 1 0

0 0 1 0 1 0 1 1

1 1 1 0 0 1 0 0

Objective function value (f(x)=x2)

28900

7225

44944

33124

1849

51984

Baseline population

1 0 1 0 1 0 1 0

0 1 0 1 0 1 0 1

1 1 0 1 0 1 0 0

1 0 1 1 0 1 1 0

0 0 1 0 1 0 1 1

1 1 1 0 0 1 0 0

Parents’ population

1 1 1 0 0 1 0 0

1 1 0 1 0 1 0 0

1 1 1 0 0 1 0 0

0 1 0 1 0 1 0 1

1 0 1 1 0 1 1 0

1 0 1 0 1 0 1 0

Parent individual no 1

1 0 1 0 1

Parent individual no 2

0 1 0 1 0

crossing point

Descendant individual no 1

0 1 0

Descendant individual no 2

1 0 1

Parent individual1 0 1 0 1 0 1 0

Mutation1 0 1 0 1 0 1 0

r>pm

r>pm

r<pm

Mutation 1 0 0 0 1 0 1 0

r<pm

r>pm

r>pm

r>pm

r<pm

Mutation1 0 0 0 1 0 0 0

r>pm

r<pm

Parent individual1 0 1 0 1 0 1 0

Descendant individual1 0 0 0 1 0 0 0

After mutation, completion individuals are recorded in the descendant population, which becomes the baseline population for the next algorithm iteration.

If obtained solution satisfies stop condition procedure is terminated. Otherwise selection, crossing and mutation are repeated.

Thank you for your attention!