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Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 1

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum

Leticia Hernando, Alexander Mendiburu and Jose A. Lozano

Intelligent Systems Group University of the Basque Country UPV/EHU, Spain

July, 11th 2018 2018 IEEE Congress on Evolutionary Computation

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 2

Objectives

Analysis of the solutions found in the attraction basins: distance to the local optimum vs. number of steps of the algorithm. The paths defined by a hill-climbing algorithm do not monotonically reduce the distance to the local optimum. Visual examples of the paths built by the algorithm.

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 2

Objectives

Analysis of the solutions found in the attraction basins: distance to the local optimum vs. number of steps of the algorithm. The paths defined by a hill-climbing algorithm do not monotonically reduce the distance to the local optimum. Visual examples of the paths built by the algorithm.

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 2

Objectives

Analysis of the solutions found in the attraction basins: distance to the local optimum vs. number of steps of the algorithm. The paths defined by a hill-climbing algorithm do not monotonically reduce the distance to the local optimum. Visual examples of the paths built by the algorithm.

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 3

Outline

1

Introduction

2

Results

3

Visualization

4

Conclusions

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 4 Introduction

Outline

1

Introduction

2

Results

3

Visualization

4

Conclusions

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 5 Introduction

Combinatorial Optimization Problem

Definition A Combinatorial Optimization Problem consists of finding the points σ∗ that minimize or maximize a function f: σ∗ = arg min

σ∈Ω f(σ)

where Ω is a finite or countable infinite set Permutation-based COP A COP where Ω is the space of permutations of size n Permutation Flowshop Scheduling Problem Quadratic Assignment Problem Linear Ordering Problem

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 5 Introduction

Combinatorial Optimization Problem

Definition A Combinatorial Optimization Problem consists of finding the points σ∗ that minimize or maximize a function f: σ∗ = arg min

σ∈Ω f(σ)

where Ω is a finite or countable infinite set Permutation-based COP A COP where Ω is the space of permutations of size n Permutation Flowshop Scheduling Problem Quadratic Assignment Problem Linear Ordering Problem

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 5 Introduction

Combinatorial Optimization Problem

Definition A Combinatorial Optimization Problem consists of finding the points σ∗ that minimize or maximize a function f: σ∗ = arg min

σ∈Ω f(σ)

where Ω is a finite or countable infinite set Permutation-based COP A COP where Ω is the space of permutations of size n Permutation Flowshop Scheduling Problem Quadratic Assignment Problem Linear Ordering Problem

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 6 Introduction

Combinatorial Optimization Problem

Definition A Combinatorial Optimization Problem consists of finding the points σ∗ that minimize or maximize a function f: σ∗ = arg min

σ∈Ω f(σ)

where Ω is a finite or countable infinite set Permutation-based COP A COP where Ω is the space of permutations of size n Permutation Flowshop Scheduling Problem Quadratic Assignment Problem Linear Ordering Problem

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 7 Introduction

Permutation Flowshop Scheduling Problem

n jobs m machines Each job consists of m operations

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 7 Introduction

Permutation Flowshop Scheduling Problem

n jobs m machines Each job consists of m operations

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 7 Introduction

Permutation Flowshop Scheduling Problem

n jobs m machines Each job consists of m operations

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 7 Introduction

Permutation Flowshop Scheduling Problem

n jobs m machines Each job consists of m operations

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 8 Introduction

Permutation Flowshop Scheduling Problem

1 M1 M2 M3 2 3 4 5 1 2 3 4 5 2 3 4 5 1 flow time 1 flow time 2 flow time 3 flow time 4 flow time 5

Total Flow Time = FT1 + FT2 + FT3 + FT4 + FT5 minimize TFT

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 9 Introduction

Neighborhood

A neighborhood N in a search space Ω is a mapping that assigns a set of neighboring solutions N(σ) ∈ P(Ω) to each solution σ ∈ Ω: N : Ω − → P(Ω) σ − → N(σ)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 10 Introduction

• Neighborhoods. Examples

2-exchange or Swap Swap two items, not necessarily adjacent (2134) (3214) (4231) (1234) (1324) (1432) (1243)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 10 Introduction

• Neighborhoods. Examples

2-exchange or Swap Swap two items, not necessarily adjacent (2134) (3214) (4231) (1234) (1324) (1432) (1243)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 10 Introduction

• Neighborhoods. Examples

2-exchange or Swap Swap two items, not necessarily adjacent (2134) (3214) (4231) (1234) (1324) (1432) (1243)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 10 Introduction

• Neighborhoods. Examples

2-exchange or Swap Swap two items, not necessarily adjacent (2134) (3214) (4231) (1234) (1324) (1432) (1243)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 10 Introduction

• Neighborhoods. Examples

2-exchange or Swap Swap two items, not necessarily adjacent (2134) (3214) (4231) (1234) (1324) (1432) (1243)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 10 Introduction

• Neighborhoods. Examples

2-exchange or Swap Swap two items, not necessarily adjacent (2134) (3214) (4231) (1234) (1324) (1432) (1243)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 10 Introduction

• Neighborhoods. Examples

2-exchange or Swap Swap two items, not necessarily adjacent (2134) (3214) (4231) (1234) (1324) (1432) (1243)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 10 Introduction

• Neighborhoods. Examples

2-exchange or Swap Swap two items, not necessarily adjacent (2134) (3214) (4231) (1234) (1324) (1432) (1243)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 11 Introduction

• Neighborhoods. Examples

Insert Move an item to a different position (2134) (2314) (2341) (1342) (1324) (1234) (3124) (1243) (4123) (1423)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 11 Introduction

• Neighborhoods. Examples

Insert Move an item to a different position (2134) (2314) (2341) (1342) (1324) (1234) (3124) (1243) (4123) (1423)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 11 Introduction

• Neighborhoods. Examples

Insert Move an item to a different position (2134) (2314) (2341) (1342) (1324) (1234) (3124) (1243) (4123) (1423)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 11 Introduction

• Neighborhoods. Examples

Insert Move an item to a different position (2134) (2314) (2341) (1342) (1324) (1234) (3124) (1243) (4123) (1423)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 11 Introduction

• Neighborhoods. Examples

Insert Move an item to a different position (2134) (2314) (2341) (1342) (1324) (1234) (3124) (1243) (4123) (1423)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 11 Introduction

• Neighborhoods. Examples

Insert Move an item to a different position (2134) (2314) (2341) (1342) (1324) (1234) (3124) (1243) (4123) (1423)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 11 Introduction

• Neighborhoods. Examples

Insert Move an item to a different position (2134) (2314) (2341) (1342) (1324) (1234) (3124) (1243) (4123) (1423)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 11 Introduction

• Neighborhoods. Examples

Insert Move an item to a different position (2134) (2314) (2341) (1342) (1324) (1234) (3124) (1243) (4123) (1423)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 11 Introduction

• Neighborhoods. Examples

Insert Move an item to a different position (2134) (2314) (2341) (1342) (1324) (1234) (3124) (1243) (4123) (1423)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 11 Introduction

• Neighborhoods. Examples

Insert Move an item to a different position (2134) (2314) (2341) (1342) (1324) (1234) (3124) (1243) (4123) (1423)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 11 Introduction

• Neighborhoods. Examples

Insert Move an item to a different position (2134) (2314) (2341) (1342) (1324) (1234) (3124) (1243) (4123) (1423)

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 12 Introduction

Distance

σ1 and σ2 are at distance i if, starting from σ1, and moving from neighboring to neighboring solutions, the length of the shortest path to reach σ2 is i. Two neighboring permutations are at distance one. Under the 2-exchange and the insert neighborhoods the maximum distance between two permutations is n − 1.

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 12 Introduction

Distance

σ1 and σ2 are at distance i if, starting from σ1, and moving from neighboring to neighboring solutions, the length of the shortest path to reach σ2 is i. Two neighboring permutations are at distance one. Under the 2-exchange and the insert neighborhoods the maximum distance between two permutations is n − 1.

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 12 Introduction

Distance

σ1 and σ2 are at distance i if, starting from σ1, and moving from neighboring to neighboring solutions, the length of the shortest path to reach σ2 is i. Two neighboring permutations are at distance one. Under the 2-exchange and the insert neighborhoods the maximum distance between two permutations is n − 1.

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 13 Introduction

Local Optima

A solution σ∗ ∈ Ω is a local optimum if f(σ∗) ≤ f(σ), ∀σ ∈ N(σ∗) (local minimum)

12345 21345 13245 12435 12354 f

( ) ( ) ( ) ( ) ( )

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 14 Introduction

Attraction basins of local optima

The attraction basin of a local optimum σ∗: Bσ∗ = {σ ∈ Ω| H(σ) = σ∗}, where H is the operator that associates to each solution σ, the local optimum obtained after applying the algorithm

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 15 Introduction

Deterministic best-improvement local search algorithm

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 16 Introduction

Deterministic best-improvement local search algorithm

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 17 Introduction

Deterministic best-improvement local search algorithm

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 18 Introduction

Deterministic best-improvement local search algorithm

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 19 Introduction

Deterministic best-improvement local search algorithm

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 20 Introduction

Deterministic best-improvement local search algorithm

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 21 Introduction

Deterministic best-improvement local search algorithm

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 22 Introduction

Deterministic best-improvement local search algorithm

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 23 Introduction

Deterministic best-improvement local search algorithm

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 24 Introduction

Deterministic best-improvement local search algorithm

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 25 Introduction

Deterministic best-improvement local search algorithm

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 26 Introduction

Deterministic best-improvement local search algorithm

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 27 Introduction

Deterministic best-improvement local search algorithm

Choose an initial solution σ ∈ Ω repeat σ∗ = σ for each σ′

i ∈ N(σ∗) do

if f(σ′

i ) < f(σ) then

σ = σ′

i

end if end for until σ = σ∗

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 28 Introduction

Deterministic best-improvement local search algorithm

Choose an initial solution σ ∈ Ω repeat σ∗ = σ for each σ′

i ∈ N(σ∗) do

if f(σ′

i ) < f(σ) then

σ = σ′

i

end if end for until σ = σ∗ The attraction basins are sets of paths!

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 29 Results

Outline

1

Introduction

2

Results

3

Visualization

4

Conclusions

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 30 Results

Experimental Design

9 PFSP instances of Taillard’s benchmark. 10 jobs and 5 machines (n=10). The local optima and the attraction basins are calculated considering the 2-exchange and the insert neighborhoods.

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 30 Results

Experimental Design

9 PFSP instances of Taillard’s benchmark. 10 jobs and 5 machines (n=10). The local optima and the attraction basins are calculated considering the 2-exchange and the insert neighborhoods.

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 30 Results

Experimental Design

9 PFSP instances of Taillard’s benchmark. 10 jobs and 5 machines (n=10). The local optima and the attraction basins are calculated considering the 2-exchange and the insert neighborhoods.

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 31 Results

• No. of local optima and sizes of attraction basins

# Loc. Opt inst 1 inst 2 inst 3 inst 4 inst 5 inst 6 inst 7 inst 8 inst 9 2-exch. 225 117 295 11 58 58 158 83 80 insert 43 16 24 3 8 15 31 8 19 500000 1500000

PFSP Size of the attraction basins

inst 1 inst 2 inst 3 inst 4 inst 5 inst 6 inst 7 inst 8 inst 9 2−exchange insert − −

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 32 Results

Distance to local optima vs. number of solutions in attraction basins

0.0 0.2 0.4 0.6 0.8 1.0

• Prop. of solutions in the attraction basin

dist 1 dist 2 dist 3 dist 4 dist 5 dist 6 dist 7 dist 8 dist 9 2−exchange insert − −

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 33 Results

Distance to local optima vs. number of steps of the algorithm

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 34 Results

Distance to local optima vs. number of steps of the algorithm

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 35 Results

Distance to local optima vs. number of steps of the algorithm

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 36 Visualization

Outline

1

Introduction

2

Results

3

Visualization

4

Conclusions

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 37 Visualization

Visualization of paths: PFSP 2-exchange

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 38 Visualization

Visualization of paths: PFSP insert

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 39 Conclusions

Outline

1

Introduction

2

Results

3

Visualization

4

Conclusions

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 40 Conclusions

Conclusions

The algorithm goes for a walk before finding the optimum! Future Work Analysis of larger instances Use this information to design/modify algorithms

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum 40 Conclusions

Conclusions

The algorithm goes for a walk before finding the optimum! Future Work Analysis of larger instances Use this information to design/modify algorithms

Hill-Climbing Algorithm: let’s go for a walk before finding the optimum

Leticia Hernando, Alexander Mendiburu and Jose A. Lozano

Intelligent Systems Group University of the Basque Country UPV/EHU, Spain

July, 11th 2018 2018 IEEE Congress on Evolutionary Computation