Diapositiva 1

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Clase 2 Conceptos Básicos de Búsqueda, Ascenso
de Colina
Gabriela Ochoa http//www.ldc.usb.ve/gabro/
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Contenido

• Repaso Optimización y Búsqueda
• Fitness Landscape
• Conceptos Básicos
• Casos de Estudio SAT, TSP y NLP
• Vecindad y Busqueda Local
• Ascenso de Colina (Hillclimbing)

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Optimization

• Objetivo encontrar el óptimo (o los óptimos)
  globales de cualquier problema
• No essential difference between maximizing and
  minimizing
• Theres no one single optimization approach that
  is good for all types of optimization problems
  (see NFL theorems)
• better to classify based on special features of
  problems

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Problemas de Optimización

• Optimización Numérica
• Variables de decisión son números reales
• Función objetivo tiene expresión algebraica
  (varias variables)
• Optimización Combinatoria
• Variables de decisión son discretas
• Soluciones suelen presentarse en la forma de
  permutaciones
• Función objetivo expresión mas compleja
  sumatorias, productorias, etc.
• Problemas de grafos, agente viajero,
  particionamiento

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Fitness Landscape (2 traits)
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Basic Concepts

• Representation Encodes alternative candidate
  solutions for manipulation
• Objective describes the purpose to be fulfilled
• Evaluation Function returns a specific value
  that indicates the quality of any particular
  solution given the representation

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Search Problem

• Definition Given a search space S and its
  feasible part F in S, find x ? F such that
• eval(x) eval(y), for all y ? F (minimization)
• The point x that satisfies the above condition is
  called global solution
• The terms search problem and optimization
  problem are considered synonymous. The search
  for the best solution is the optimization problem

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Boolean satisfiability problem (SAT)

• An instance of the problem is defined by a
  Boolean expression written using only AND, OR,
  NOT, variables, and parentheses.
• The question is given the expression, is there
  some assignment of TRUE and FALSE values to the
  variables that will make the entire expression
  true?
• SAT is of central importance in various areas of
  computer science, including theoretical computer
  science, algorithmics, artificial intelligence,
  hardware design and verification.

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Computational Complexity of SAT

• SAT is NP-complete. In fact, it was the first
  known NP-complete problem, as proved by Stephen
  Cook in 1971
• The problem remains NP-complete even if all
  expressions are written in conjunctive normal
  form with 3 variables per clause (3-CNF)
• (x11 OR x12 OR x13) AND
• (x21 OR x22 OR x23) AND
• (x31 OR x32 OR x33) AND ...
• where each x is a variable, with or without a NOT
  in front of it, and each variable can appear
  multiple times in the expression.

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Problem Formulation (SAT)

• Let us consider a problem of size 30 (i.e. 30
  variables)
• Representation
• 1 True, 0 False, Binary String of length 30
• Search Space
• 2 choices for each variable, taken over 30
  variables, generates 230 possibilities
• Objective
• To find the vector of bits such that the compound
  Boolean statement is satisfied (made true)
• Evaluation Function?
• Not enough information to take the objective

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Travelling salesman problem (TSP)

• Given a number of cities and the costs of
  travelling from one to the other, what is the
  cheapest roundtrip route that visits each city
  and then returns to the starting city?
• An equivalent formulation in terms of graph
  theory is Find the Hamiltonian cycle with the
  least weight in a weighted graph.

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Problem Formulation (TSP)

• Let us consider a problem of size 30 (i.e. 30
  cities)
• Representation Permutation of natural numbers
  1, ,30 where each number corresponds to a city
  to be visited in sequence
• Search Space Permutations of all cities.
  Symmetric TSP, circuit the same regardless the
  starting city (n-1)!/2
• Objective Minimize the total distance traversed,
  visiting each city once, and returning to the
  starting city. Min Sum(dist(x,y))
• Evaluation Function Map each tour to its
  corresponding total distance

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Neighbourhoods and local Optima

• Region of the search space that is near to some
  particular point in that space

N(x)
. x
S
A search space S, a potential solution x, and its
neighbourhood N(x)
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Defining Neighbourhoods 1

• Define a distance function dist on the search
  space S
• Dist S x S ? R
• N(x) y ? S dist(x,y) e
• Examples
• Euclidean distance, for search spaces defined
  over continuous variables
• Hamming distance, for search spaces definced over
  binary strings (e.g. SAT)

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Defining Neighbourhoods, TSP

• Use a mapping m, that defines a neighbourhood
  for any point x ? S
• 2-swap mapping generates a new set of potential
  solutions from a given solution x
• Solutions are generated by swapping two cities
  from a given tour
• Every solution has n(n-1)/2 neighbours
• Example 2 4 5 3 1 ? 2 3 5 4 1,

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Defining Neighbourhoods, SAT

• 1-flip mapping generates a new set of potential
  solutions from a given solution x
• Solutions are generated by flipping a single bit
  in the given bit string
• Every solution has n neighbours
• Example 1 1 0 0 1 ? 0 1 0 0 1 ( 1st bit)

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Defining Neighbourhoods, Real Numbers

• Gaussian Distribution for each variable defines
  a neighbourhood
• Mean the current point, Std. dev. 1/6 of the
  range of the variable
• x (x1, , xn), where li xi ui
• x xi N(0,si), where si (ui - li)/6
• N(0,si) is an independent random Gaussian number
  with mean zero and std. dev. si

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Local Optimum

• A potential solution x ? S is a local optimum
  with respect to the neighbourhood N, if and only
  if
• eval(x) eval(y), for all y ? N(x)

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Métodos de Ascenso de Colina - 1

• Usan una técnica de mejoramiento iterativo
• Comienzan a partir de un punto (punto actual) en
  el espacio de búsqueda
• En cada iteración, un nuevo punto es seleccionado
  de la vecindad del punto actual
• Si el nuevo punto es mejor, se transforma en em
  punto actual, sino otro punto vecino es
  seleccionado y evaluado
• El método termina cuando no hay mejorías, o
  cuando se alcanza un numero predefinido de
  iteraciones

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Hillclimbing Methods - 2

• May converge to local optima
• usually have to start search from various
  starting points
• Initial starting points may be chosen,
• randomly
• according to some regular pattern
• based on other information (e.g. results of a
  prior search)

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Hillclimbing Methods - 3

• Variations of hillclimbing algorithms differ in
  the way a new string is selected for comparisons
  with the current string
• One version of simple (iterated) hillclimbing
  method is the steepest ascent hillclimbing

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Hillclimbing Methods - 4

• Example problem
• The search space is a set of binary strings v of
  length 30
• The objective function f (to be maximized)
• f(v)11one(v)-150
• where one(v) returns the number of ones in v.
• e.g. v1(110111101111011101101111010101)
• f(v1) 1122 - 150 92

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Hillclimbing Methods - 5

• procedure iterated hillclimber
• begin
• t ? 0
• repeat
• local ? FALSE
• select a curent string vc at random
• evaluate vc
• repeat
• form 30 new strings in the neigborhood of
  vc by
• flipping single bits of vc
• select vn from the set of new strings with
  the
• largest value of the objective function f
• if f(vc) lt f(vn) then vc ? vn
• else local ? TRUE
• until local
• t ? t1
• until tMAX
• end

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Hillclimbing Methods - 6

• success/failure of each iteration depends on
  starting point
• success defined as returning a local or a global
  optimum
• in problems with many local optima a global
  optimum may not be found

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Hillclimbing Methods - 7

• Weaknesses
• Usually terminate at solutions that are local
  optima
• No information as to how much the discovered
  local optimum deviates from the global (or even
  other local optima)
• Obtained optimum depends on starting point
• Usually no upper bound on computation time

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Hillclimbing Methods - 8

• Advantages
• Very easy to apply (only a representation, the
  evaluation function and a measure that defines
  the neigborhood around a point is needed)

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Search Techniques Revisited - 1

• Effective search techniques provide a mechanism
  to balance exploration and exploitation
• exploiting the best solutions found so far
• exploring the search space

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Search Techniques Revisited - 2

• Hillclimbing methods exploit the best available
  solution for possible improvement but neglect
  exploring a large portion of the search space
• Random search (points in the search space are
  sampled with equal probability) explores the
  search space thoroughly but misses exploiting
  promising regions.