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### Heuristic OptimizationAthens 2004

Department of Architecture and Technology

Universidad Politécnica de Madrid

Víctor Robles

vrobles@fi.upm.es

Teachers

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

Course outline and Class hours

- Day 1 / 10:00-14:00 / Víctor
- Introduction to optimization
- Some optimization problems
- About heuristics
- Greedy algorithms
- Hill climbing
- Simulated Annealing

Course outline and Class hours

- Day 2 / 9:30-13:30 / Víctor, María
- Learn by practice: Simulated Annealing
- Genetic Algorithms
- Day 3 / 9:30-13:30 / Vanessa
- Learn by practice: Genetic Algorihtms
- Day 4 / 10:00-13:30 / Pedro
- Estimation of Distribution Algorithms

Introduce yourself

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

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

State space landscape

- Objective function defines state space landscape

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”

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 ?

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

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

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

SAT

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

...

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?

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

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

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 be:At 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

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

Practice

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

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

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

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

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

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

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

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)

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

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

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

Practice

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

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

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

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

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)

Simulated annealing: Algorithm

current = problem.initialSate

for t=1 to

T = schedule(t)

if T=0 then return

= a random neighbour of

if then

else with probability

Simulated annealing: Simple example

- Maximize

x coded as a 5-bit binary integer in [0,31]

maximum (01010) f=4100

- With ‘greedy’ we can find 3 local maxima

Simulated annealing: Simple example

The temperature is not high enough to move out of this

local optimum

Simulated annealing: Simple example

Optimum found!!!

Simulated annealing: Generic decisions

- Initial temperature

Should be ‘suitable high’. Most of the initial moves must be accepted (> 60%)

- Cooling schedule

Temperature is reduced after every move

Two methods:

a close to 1

b close to 0

Simulated annealing: Generic decisions

- Number of iterations
- Other factors:
- Reannealing
- Restricted neighbourhoods
- Order of searching

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