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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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