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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Optimization
Pantelis P. Analytis
March 26, 2018
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
1 Introduction
2 Continuous optimization problems
3 Discrete Problems
4 Collective search for good solutions
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Convex optimization
The Cobb-Douglas production function (Y = ALβKα)Many of the problems studied across fields are convex innature.Solutions can be calculated analytically, hill climbing isguaranteed to converge to the optimal solution.
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Local and global extrema
An entire subfield of operations research is dedicated todeveloping optimization algorithm.
They are often evaluated against a testbed of challengingenvironments.
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Optimizing more challenging functions
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Optimizing more challenging functions
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Gradient descent and stochastic gradient descent
xn+1 = xn − γn∇F (xn), n ≥ 0.
F (x0) ≥ F (x1) ≥ F (x2) ≥ · · ·
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Gradient descent and stochastic gradient descent
F (x , y) = sin(12x
2 − 14y
2 + 3)
cos(2x + 1− ey )
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Human search in rugged landscapes
Rieskamp, Busemeyer, Laine (2003)
Participants made 100 allocation decision between 3assets.
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Rieskamp, Busemeyer, Laine (2003)
GLOS model: each allocation has an expectancy andpeople choose probabilistically among them.
LOCAD: probabilistically test another tile in theneighborhood. Move there is that’s better.
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Rieskamp, Busemeyer, Laine (2003)
GLOS model: each allocation has an expectancy andpeople choose probabilistically among them.LOCAD: probabilistically test another tile in theneighborhood. Move there is that’s better. 11 / 29
Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Optimizing more challenging functions
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Simulated annealing
Let s = s0
T ← temperature(k/kmax)
Pick a random neighbour, snew ← neighbour(s)
P(s, snew ,T ) :
{1 if snew > s
e(snew−s)/Totherwise
Gradually reduce the temperature T
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
The shortest path problem
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
The shortest path problem
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
How does nature solve the shortest path problem?
At each junction the ants select the way to followprobabilistically.The are more likely to select paths with more pheromone.As pheromone evaporates mediacre paths are much lesslikely to be selected. 16 / 29
Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
How does nature solve the shortest path problem?
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
The traveling salesman problem
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
TSP experiment
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Human performance in the TSP
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Human performance in the TSP
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Human performance in the TSP
Numerous experiments. The main paradigm in the humanproblem solving literature.
Humans use an array of heuristics to solve the problem.They often find optimal solutions to small problems, butbehave suboptimaly in larger ones.
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Evolutionary algorithms
Population of individuals
Mutation (local search)
Crossover (e.g one-point crossover: [0110](1100),(1101)[0111]))
Generations - iterations of improvement
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
The Knapsack problem
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Murawski and Bossaerts (2015)
20 participants, solved 8 problems each.
Overall performance came quite close to the optimal,people changed strategies in the course of the experiment.
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Murawski and Bossaerts (2015)
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Collective search, innovation and learning (Lazerand Friedman, 2007)
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Collective search, innovation and learning
Some random search markedly improves performanceLocal search and imitation algorithms get stack to localminima.
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Optimization
Pantelis P.Analytis
Introduction
Continuousoptimizationproblems
DiscreteProblems
Collectivesearch forgood solutions
Collective search, innovation and learning
In some environments sparser networks lead to betterresults.
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