1. Meta-Heuristic Algorithms 16B1NCI637
Raju Pal
Assistant Professor
JIIT, Sector 128, Noida

2. Course Objectives
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3. Why?
Used for solving many real world, complex or large problems where classical methods does not work
The number of application areas are substantially increasing.
These methods are very easy and simple to be applied and very simple to be learn.
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4. Optimization Introduction
Meta-heuristic is a subfield of stochastic optimization.
Stochastic optimization is a optimization method in which input parameters are subject to randomness.
Stochastic = probabilistic
A process of making something (as a design of a system, or decision) as fully perfect, functional, or effective as possible
Almost everywhere it is used viz., science, economics, finance ... every area of engineering and industry
Objectives is to save time, money, energy, resources, and to maximize efficiency, performance, quality
Optimization
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5. Examples of Optimization
Routing and scheduling: transport routes, bus/train/airlines routes, project scheduling ...
Planning: utility of resources, time, money, and optimal output, services, performance
Design optimization: structural optimization, shape optimization, parameter optimization ... better products
Optimal control: optimal control of dynamic systems (e.g., spacecraft, moon lander, car, airplanes...)
Economics: portfolio, financial derivatives, ..., banking.
Optimization is Like Treasure Hunting
How to find a treasure, a hidden 1 million dollars?
What is your best strategy?
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6. Mathematical Representation of Optimization Problem
Components xi of x are called design or decision variables, and they can be real continuous, discrete, or a mix of these two
The functions where are called the objective functions or simply cost functions
The space spanned by the decision variables is called the design space or search space
The space formed by the objective function values is called the solution space or response space
The equalities for h and inequalities for g are called constraints
where,
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7. Optimization Methods Classical (deterministic) optimization methods
Stochastic (probabilistic) optimization methods
Classical optimization methods include, for example
gradient descent (following the steepest slope)
Newton’s method
penalty methods,
Lagrange multiplier methods etc.
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8. Types of problems (or functions)
Classical methods are particularly useful in convex problems e.g.  quadratic function  and the exponential function, where any local minimum also is a global minimum.
Linear functions are convex, so linear programming problems are convex problems.
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9. Deterministic Method Demerits Merits
Give exact solutions
Do not use any stochastic technique
Rely on the thorough search of the feasible domain.
Demerits
Not Robust- can only be applied to restricted class of problems.
Often too time consuming or sometimes unable to solve real world problems
Classical methods are less useful in cases with
non-differentiable objective functions
objective functions whose values can only be obtained as a result of a (lengthy) simulation
varying number of variables (as in optimization of neural networks).
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10. Stochastic optimization
Merits
Applicable to wider set of problems i.e. function need not be convex, continuous or explicitly defined
Use the stochastic or probabilistic approach i.e. random approach
Demerits
Converges to the global optima probabilistically
Some times get stuck at local optima.
Many (but not all) stochastic optimization methods are inspired by Nature
Nature is all about adaptation, which can be seen as a kind of optimization
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11. Nature Inspired Algorithms
Nature provide some of the efficient ways to solve problems
Algorithms imitating processes inspired from nature
Problems
Aircraft wing design
Bullet train
Robotic Spy Plane
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12. The No Free Lunch Theorem
One of the more interesting developments in optimization theory was the publication of the No Free Lunch (NFL) theorem (Wolpert and Macready, 1995; Wolpert and Macready, 1997).
This theorem states that the performance of all optimization (search) algorithms, amortized over the set of all possible functions, is equivalent.
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13. Nature Inspired Algorithms for Optimizations
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14. Evolution

15. Evolutionary Algorithms
Natural selection - A guided search procedure
Individuals suited to the environment survive, reproduce and pass their genetic traits to offspring (Survival of the fittest)
Offsprings created by reproduction, mutation, etc.
Populations adapt to their environment. Variations accumulate over time to generate new species
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16. Evolutionary Algorithms
Terminologies
Individual - carrier of the genetic information (chromosome). It is characterized by its state in the search space, its fitness (objective function value).
Population - pool of individuals which allows the application of genetic operators.
Fitness function - The term “fitness function” is often used as a synonym for objective function.
Generation - (natural) time unit of the EA, an iteration step of an evolutionary algorithm.
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17. Evolutionary Algorithms
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18. Evolutionary Algorithms
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19. Evolutionary Algorithms
Selection
The survival of the fittest, which means the highest quality chromosomes and/characteristics will stay within the population
Motivation is to preserve the best (make multiple copies) and eliminate the worst
Crossover
Create new solutions by considering more than one individual
The recombination of two parent chromosomes (solutions) by exchanging part of one chromosome with a corresponding part of another so as to produce offsprings (new solutions)
Search for new and hopefully better solutions
Mutation
The change of part of a chromosome (a bit or several bits) to generate new genetic characteristics
Keep diversity in the population
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20. Concept of Exploration vs Exploitation
Search for promising solutions
Generate solutions with enough diversity and far from the current solutions
Mutation operators
The search is typically on a global scale
Exploitation
Preferring the good solutions
Generate new solutions that are better than existing solutions
Crossover and Selection operator
This process is typically local
Excessive exploration – Random search.
Excessive exploitation – Premature convergence
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21. Evolutionary Algorithms
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22. Evolutionary Algorithms
Classical gradient based algorithms
Convergence to an optimal solution usually depends on the starting solution.
Most algorithms tend to get stuck to a locally optimal solution.
An algorithm efficient in solving one class of optimization problem may not be efficient in solving others.
Algorithms cannot be easily parallelized.
Evolutionary algorithms
Convergence to an optimal solution is designed to be independent of initial population.
A search based algorithm. Population helps not to get stuck to locally optimal solution.
Can be applied to wide class of problems without major change in the algorithm.
Can be easily parallelized.
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23. Dependency on the starting solution for gradient-based algorithms
Newton’s Method
The iteration procedure starts from an initial guess x0 and continues until a certain criterion is met
Let a nonlinear function
Function can be written as
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24. Newton’s Method
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25. Fitness Landscapes
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