what is a hyper-heuristic? - university of stirling - single point metaheuristics.pdfoptima by...

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(ITNPDB8) Evolutionary and Heuristic Optimisation

Lecture 3: Single Point Methods Gabriela Ochoa http://www.cs.stir.ac.uk/~goc/

Computing Science and Mathematics,

School of Natural Sciences

University of Stirling, Stirling, Scotland

Outline

1. Optimisation methods

– Heuristics and metaheuristcis

– Single point (trajectory) algorithms

– Population-based (evolutionary) algorithms

Gabriela Ochoa, [email protected] 2

Video for TSP: https://www.youtube.com/watch?v=SC5CX8drAtU

http://www.cs.stir.ac.uk/~goc/



: https:/www.youtube.com/watch?v=SC5CX8drAtU



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Optimisation problems

• Definition (minimisation problem): couple (S,f)

– Given • a search space S,feasible solutions

• an objective function f: S R

– Find s* in S such that • f(s*)≤ f(s) for all s in S

• s* is the global optimum

• Large scale/complex optimisation problems in many areas of science and industry: Transportation, Logistics, Finance, Design, Biology, Environment


Classical representations


Metaheuristics: From Design to Implementation By El-Ghazali Talbi (2009)

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Towards effective search techniques

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

– exploring the search space

• Extreme examples: – Hill-climbing 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.


Towards effective search techniques

• Aim is to design search algorithms that can – escape local optima

– balance exploration and exploitation

– make the search independent from the initial configuration

• Successful single point search algorithms – Iterated local search

– Simulated annealing

– Tabu search

– Other methods


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Hill climbing algorithm (First Improvement, random mutation)


procedure hill-climbing

begin

s = random initial solution solution

repeat

evaluate solution (s)

s’ = random neighbour of s

if evaluation(s’) is better than evaluation(s)

s = s’

until stopping-criteria satisfied

return s

end

Hill climbing algorithm (Best Improvement)


procedure best-hill-climbing

begin

s = random initial solution solution

repeat

evaluate solution (s)

s’ = best solution in the Neighbourhood

if evaluation(s’) is better than evaluation(s)

s = s’

until s is a local optimum

return s

end

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Hill-climbing example


Binary representation, 1-flip move operator, best-improvement.

Multiple restarts hill-climbing • Simple strategy to improve hill-climbing

• Have an additional loop, repeat hill-climbing for a number of times from a different starting solution


procedure Multiple restart hill-climbing

begin

s = randomly generated solution

best = s

repeat

s = hill_climbing(s)

if evaluation(s) is better than evaluation(best)

best = s



return best

end

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Iterated local search

Procedure Iterated Local Search (ILS) determine initial candidate solution s Perform local search on s while NOT termination_criterion { r = s perform perturbation on s perform local search on s based on acceptance criterion keep s or revert to s = r }

• Key idea: use two stages – Subsidiary local search for efficiently reaching local optima (intensification)

– Perturbation stage, for effectively escaping local optima (diversification)

• Acceptance criterion: to control diversification vs. intensificaction

Key idea rediscovered several times with different names (80s &90s). Term iterated local search proposed HR Lourenço, OC Martin, T Stützle(2003). Iterated local search. Handbook of metaheuristics, 320-353, Springer . Google cites: 964


Iterated Local Search • Simple strategy to improve hill-climbing

• Have an additional loop, repeat hill-climbing for a number of times from a perturbed solution


procedure Iterated local search

begin


s = hill-climbing(s)

best = s

repeat

s = perturb(s)

s = hill-climbing(s)

if evaluation(s) is better than evaluation(best)

best = s


return best

end

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Move operators Binary representation

Hill-climbing stage

• 1-flip: Solutions generated by flipping a single bit in the given bit string

– Example: 1 1 0 0 1 → 0 1 0 0 1,

Perturbation: a stronger move

• 2-flip: Solutions generated by flipping two bits in the given bit string

– Example: 1 1 0 0 1 → 0 1 0 1 1

• k-flip: can be generalised for larger k, k < n


Travelling salesman

14 Gabriela Ochoa,

[email protected]

Name Cities Description Optimum Year Concorde (secs)

lin318 318 Drilling problem 42029 1980 9.74

att532 532 Cities in the US 27686 1982 109.52

Example instances

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TSP operators

Chained Lin-Kernighan

• An iterated local search approach

– Perturbation kick: double-bridge move (form of 4-exchange)

– Local search: Lin-Kernighan local search heuristic

• Lin-Kernighan heuristic (1973)

– Based k-exchanges: remove k different links and reconnect them a

new way to achieve a legal tour. – LK applies 2, 3 and higher-order k-exchanges

– Adaptive, k is increased until a stopping criterion is met

16 O. C Martin, Otto, Felten. 1992. Large-Step Markov Chains for the TSP Incorporating Local Search Heuristics. Operations Research Letters 11: 219–24.

Double-bridge move

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TSP State-of-the-art • TSP home and Concorde Solver, U. Waterloo, Canada

• TSPLIB, Gerhard Reinelt, Universität Heidelberg

• Lin-Kernighan Helsgaun LKH

• World TSP images and animations

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Progress in Solving TSP instances (log scale) • 1954 Dantzig, Fulkerson, Johnson

(cutting-plane): 49 • 1971 Held, Karp: 57, 64 • 1975 Miliotis: 80 • 1977 Grotschel: 12 • 1980 Crowder, Padberg: 318 • 1987 Padberg, Rinaldi: 2,392 • 1990 > Concorde (Applegate, Bixby,

Chvátal, Cook)

Simulated annealing • Key idea: provides a mechanism to escape local

optima by allowing moves that worsen the objective function value

• Annealing: the physical process of heating up a solid and then cooling it down (slowly) until it crystallizes – candidate solutions → states of physical system

– objective function → thermodynamic energy

– globally optimal solutions → ground states

– parameter T → physical temperature

Gabriela Ochoa, [email protected]

Kirkpatrick, S., Gelatt, C. D., & Vecchi, M. (1983). Optimization by simulated annealing. Science, 220, 671–680.

Google Scholar citations: 31,477

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http://www.math.uwaterloo.ca/tsp/index.html

http://www.math.uwaterloo.ca/tsp/concorde/index.html

http://www.akira.ruc.dk/~keld/research/LKH/

http://www.math.uwaterloo.ca/tsp/world/pictures.html

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Simulated annealing – Algorithm

1. Start with a random solution s

2. Choose some “nearby” solution s’

3. If the new solution is better (i.e. f(s’) ≤ f(s)) , take it as the current solution (= accept it)

4. If it is worse, accept it with a probability that depends on the deterioration f(s)-f(s’) and a global parameter T (the temperature)

Metropolis acceptance criterion

Cooling schedule: a mechanism for reducing the temperature


Simulated annealing – pseudo-code

procedure simulated annealing

begin

create an initial solution

repeat

evaluate solution

if accepted then update current solution

if time to change temperature then

decrease temperature

if stopping-criteria NOT satisfied then

generate new solution


end

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Simulated annealing - design Four choices to make:

1. Representation of solutions

2. Definition of cost function

3. Neighbourhood

4. Cooling schedule: four parameters must be specified • initial temperature,

• temperature decrement rule,

• number of iterations to be performed at each temperature,

• stopping criteria

Simplest and common temperature decrement rule: Ti+1 = αTi Where α is a constant close to, but smaller than, 1. This exponential cooling scheme (ECS) was first proposed Kirkpatrick et al. α = 0.95

Tabu search

• It behaves like a best-improvement Hill-Climbing algorithm

• But it accepts non-improving (worse) solutions in order to escape from local optima

– When all the neighbouring solutions are non-improving (worse or equal)

• Deterministic algorithm


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Tabu search

Procedure Tabu Search (TS) determine initial candidate solution s while NOT termination criterion { determine set N’ of non-tabu neighbours of s choose a best improving candidate solution s’ in N’ update Tabu attributes based on s’ s := s’ }

• Associate Tabu attributes with candidate solutions or solution components. Maintains a list of prohibitions.

• Forbid steps to search positions recently visited based on Tabu attributes (characteristics of the moves)

F. Glover (1989). Tabu Search - Part 1. ORSA Journal on Computing 1 (2): 190–206. Google cites: 5,675 F. Glover (1990). Tabu Search - Part 2. ORSA Journal on Computing 2 (1): 4-32. Google cites: 3,684 R. Battiti, G. Tecchiolli (1994) The reactive tabu search . ORSA journal on computing 6 (2): 126-140.

The word 'tabu' comes from Tongan, a language of Polynesia, used by the locals to indicate things that cannot be touched because they are sacred.


Escaping from local optima


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Summary

• Aim is to design search algorithms that can – escape local optima – balance exploration and exploitation – make the search independent from the initial

configuration

• Successful single point search algorithms – Multiple restart hill-climbing – Iterated local search – Simulated annealing – Tabu search – Other methods


Next Lecture: Population-based, Evolutionary Algorithms

what is a hyper-heuristic? - university of stirling - single point metaheuristics.pdfoptima by...

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