Heuristic Optimization Athens 2004

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Heuristic OptimizationAthens 2004

• Department of Architecture and Technology
• Universidad Politécnica de Madrid
• Víctor Robles
• vrobles_at_fi.upm.es

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Teachers

• Universidad Politécnica de Madrid
• Víctor Robles (coordinator)
• María S. Pérez
• Vanessa Herves
• Universidad del País Vasco
• Pedro Larrañaga

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Course outline and Class hours

• Day 1 / 1000-1400 / Víctor
• Introduction to optimization
• Some optimization problems
• About heuristics
• Greedy algorithms
• Hill climbing
• Simulated Annealing

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Course outline and Class hours

• Day 2 / 930-1330 / Víctor, María
• Learn by practice Simulated Annealing
• Genetic Algorithms
• Day 3 / 930-1330 / Vanessa
• Learn by practice Genetic Algorihtms
• Day 4 / 1000-1330 / Pedro
• Estimation of Distribution Algorithms

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Introduce yourself

• Name
• University / Country
• Optimization experience
• Expectations of the course

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Optimization

• A optimization problem is a par
• being the search space (all the possible
  solutions), and a function,
• The solution is optime if,
• Combinatorial optimization
• Systematic search

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State space landscape

• Objective function defines state space landscape

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Some optimization problems

• TSP Travel Salesman Problem
• The assignment problem
• SAT Satistiability problem
• The 0-1 knapsack problem
• Important tasks
• Find a representation of possible solutions
• To be able to evaluate each of the possible
  solutions ? fitness function or objective
  function

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TSP

• A salesman has to find a route which visits each
  of n cities, and which minimizes the total
  distance travelled
• Given an integer and a n x n matrix
  
• where each is a nonnegative integer. Which
  cyclic permutation of integers from 1 to n
  minimizes the sum ?

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TSP representations

• Binary representation
• Each city is encoded as a string of log2n bits.
• Example 8 cities ? 3 bits
• Path representation
• A list is represented as a list of n cities. If
  city i is the j-th element of the list, city i is
  the j-th city to be visited
• Adjancecy representation
• City j is listed in position i if the tour leads
  from city i to city j
• (3 5 7 6 4 8 2 1) ? tour 1-3-7-2-5-4-6-8

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The assignment problem

• A set of n resources is available to carry out n
  tasks. If resource i is assigned to task j, it
  cost
• units.
• Find an assignment that
  minimizes
• Solution permutition of the numbers

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SAT

• The satisfiability problem consists on finding a
  truth assignment that satisfied a well-formed
  Boolean expression
• Many applications VLSI test and verification,
  consistency maintenance, fault diagnosis, etc
• MAX-SAT Find an assignment which satisfied the
  maximum number of clauses

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SAT

• Data sets in conjunctive normal form (cnf)
• Example
• Literals
• Clauses
• Representation? Fitness function?

...
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The 0-1 knapsack problem

• A set of n items is available to be packed into a
  knapsack with capacity C units. Item i has value
  vi and uses up ci units of capacity. Determine
  the subset of items which should be packed to
  maximize the total value without exceding the
  capacity
• Representation? Fitness function?

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Heuristics

• Faster than mathematical optimization (branch
  bound, simplex, etc)
• Well developed ? good solutions for some problems
• Special heuristics
• Greedy algorithms systematic
• Hill-climbing (based on neighbourhood search)
  randomized

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Greedy algorithms

• Step-by-step algorithms
• Sometimes works well for optimization problems
• A greedy algorithm works in phases. At each
  phase
• You take the best you can get right now, without
  regard for future consequences
• You hope that by choosing a local optimum at each
  step, you will end up at a global optimum

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Example Counting money

• Suppose you want to count out a certain amount of
  money, using the fewest possible bills and coins
• A greedy algorithm would do this would beAt
  each step, take the largest possible bill or coin
  that does not overshoot
• Example To make 6.39, you can choose
• a 5 bill
• a 1 bill, to make 6
• a 25 coin, to make 6.25
• A 10 coin, to make 6.35
• four 1 coins, to make 6.39
• For US money, the greedy algorithm always gives
  the optimum solution

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A failure of the greedy algorithm

• In some (fictional) monetary system, krons come
  in 1 kron, 7 kron, and 10 kron coins
• Using a greedy algorithm to count out 15 krons,
  you would get
• A 10 kron piece
• Five 1 kron pieces, for a total of 15 krons
• This requires six coins
• A better solution would be to use two 7 kron
  pieces and one 1 kron piece
• This only requires three coins
• The greedy algorithm results in a solution, but
  not in an optimal solution

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Practice

• Develop a greedy algorithm for the knapsack
  problem. Groups of 2 persons

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Local search algorithms

• Based on neighbourhood system
• Neighbourhood system
• being X the search space, we define the
  neighbourhood system N in X
• Examples TSP (2-opt), SAT, assignment and knap

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Local search algorithms

• Basic principles
• Keep only a single (complete) state in memory
• Generate only the neighbours of that state
• Keep one of the neighbours and discard others
• Key features
• No search paths
• Neither systematic nor incremental
• Key advantages
• Use very little memory (constant amount)
• Find solutions in search spaces too large for
  systematic algorithms

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TSP 2 opt
A
B
C
D
New distance Old dist dist(A-D) dist(B-C)
dist(A-B) dist (C-D)
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Neighbourhood search (Reeves93)

• (Initialization)
• Select a starting solution
• Current best and
• (Choice and termination)
• Choice . If choice
  criteria cannot be satisfied or if other
  termination criteria apply, then the method stops
• (Update)
• Re-set , and if
  , perform step 1.ii. Return to Step 2

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Hill climbing

• Diferent procedures depending on choice criteria
  and termination criteria
• Hill climbing only permit moves to neighbours
  that improve the current
• (Choice and termination)
• Choose such that
• and terminate if no such can be
  found

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Hill-climbing 8-Queens problem

• Complete state formulation
• All 8 queens on the board,one per column
• Neighbourhood move one queen to a different
  place in the same column
• Fitness function number of pairs of queens that
  are attacking each other

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8-Queens problemfitness values of neighbourhood
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8-Queens problemLocal minimun
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Problematic landscapes

• Local maximum a peek that is higher than all its
  neighbours, but not a global maximum
• Plateau an area where the elevation is constant
• Local maximum
• Shoulder
• Ridge a long, narrow, almost plateau-like
  landscape

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Random-Restart Hill-Climbing

• Method
• Conduct a series of hill-climbing searches from
  randomly generated initial states
• Stop when a goal is found
• Analysis
• Requires 1/p restarts where p is the probability
  of success
• (1 success 1/p failures)

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Hill-Climbing Performance on the 8-Queen Problem

• From randomly generated start state
• Success rate
• 86 - gets stuck
• 14 - solves problem
• Average cost
• 4 steps to success
• 3 steps to get stuck

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Hill-Climbing with Sideways Moves

• Sideways moves moves at same fitness
• Must limit number of sideways moves!
• Performance on the 8-Queen problem
• Success rate
• 6 - get stucks
• 94 - solves problem
• Average cost
• 21 steps to succeed
• 64 steps to get stuck

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Hill-climbing Further variants

• Stochastic hill-climbing
• Choose at random from among uphill moves
• First-choice hill-climbing
• Generate neighbourhood in random order
• Move to first generated that represents an uphill
  move

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Practice

• Develop a hill-climbing algorithm for the
  knapsack problem. Groups of 2 persons

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Shape of State Space Landscape

• Success of hill-climbing depends on shape of
  landscape
• Shape of landscape depends on problem formulation
  and fitness function
• Landscapes for realistic problems often look like
  a worst-case scenario
• NP-hard problems typically have exponential
  number of local-maxima
• Despite the above, hill-climbers tend to have
  good performance

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Simulated annealing

• Failing on neighbourhood search
• Propensity to deliver solutions which are only
  local optima
• Solutions depend on the initial solution
• Reduce the chance of getting stuck in a local
  optimum by allowing moves to inferior solutions
• Developed by Kirkpatrick 83 Simulation of the
  cooling of material in a heat bath could be used
  to search the feasible solutions of an
  optimization problem

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Simulated annealing

• If a move from one solution to another
  neighbouring but inferior solution
• results in a change in value ,
  the move to is still accepted if
• T (temperature) control parameter
• uniform random number

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Simulated annealing Intuition

• Minimization problem imagine a state space
  landscape on table
• Let ping-pong ball from random point ? local
  minimum
• Shake table ?ball tends to find different minimum
• Shake hard at first (high temperature) but
  gradually reduce intensity (lower temperature)

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Simulated annealing Algorithm

• current problem.initialSate
• for t1 to
• T schedule(t)
• if T0 then return
• a random neighbour of
• if then
• else with probability

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Simulated annealing Simple example

• Maximize
• x coded as a 5-bit binary integer in 0,31
• maximum (01010) ? f4100
• With greedy we can find 3 local maxima

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Simulated annealing Simple example
The temperature is not high enough to move out of
this local optimum
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Simulated annealing Simple example
Optimum found!!!
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Simulated annealing Generic decisions

• Initial temperature
• Should be suitable high. Most of the initial
  moves must be accepted (gt 60)
• Cooling schedule
• Temperature is reduced after every move
• Two methods

a close to 1 b close to 0
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Simulated annealing Generic decisions

• Number of iterations
• Other factors
• Reannealing
• Restricted neighbourhoods
• Order of searching