CIAO: Collective Intelligence Algorithm for Optimization

;---------------------------------------------------------------
; CIAO: Collective Intelligence Algorithm for Optimization
;---------------------------------------------------------------
; This NetLogo code implements CIAO, a novel metaheuristic
; algorithm that leverages the collective intelligence of
; solvers and users to explore and optimize the solution
; space of a given optimization problem.
;
; Authors: Sergio Rojas-Galeano, Martha Garzón, Lindsay Álvarez
; Copyright (c) 2024 The authors
; License: GNU General Public License (GPLv3)
; See Info tab for full copyright and license.
;---------------------------------------------------------------


; Global variables to store information about the best solution found
globals [
  true-best-patch ; Patch corresponding to the ground-truth optimum
  best-ever       ; Patch corresponding to the best solution found so far
  best-tick       ; Tick when best-ever was found
  best-time       ; Time when best-ever was found
]


; Patch attributes to represent landscape and cost function of the optimization problem
patches-own [
  x     ; Patch x-coordinate, associated with a point in the solution space within the defined bounds
  y     ; Patch y-coordinate, associated with a point in the solution space within the defined bounds
  value ; Patch value corresponding to the cost function evaluated at its (x, y) coordinates
]


; Agent breeds for different types of agents in the simulation
breed [solvers solver]
breed [users user]
breed [solutions solution]


; Solver agents' attributes
solvers-own [
  bx by   ; Knowledge of the best solution location found so far
  mx my   ; Core (mu) component of solver expertise
  sx sy   ; Breadth (sigma) component of solver expertise
  score   ; Solver's rating given by users
]


; User agents' attributes
users-own [
  own-best ; User's best solution value found so far
]


; Execute a single iteration of the main loop.

to go
  ; Reset the timer on the first tick
  if ticks = 0  [ reset-timer ]

  ; Execute the main components of the simulation
  search-solutions
  update-best
  tick

  ; Restart agents at regular intervals if restarts are enabled
;  if (restarts? and ticks mod 50 = 0) [ restart ]
  if (restarts? and ticks mod 25 = 0) [ restart ]

  ; Stopping conditions: maximum number of ticks reached or true-best-patch found
  if (ticks > max-ticks) or (any? true-best-patch with [value = [value] of best-ever]) [ stop ]
end 


; Search for solutions based on user interactions

to search-solutions
  ask users [

    ; Choose a solver and spawn new candidate solutions according to its expertise model
    let my-solver choose-solver
    hatch-solutions k-solutions [
      set xcor random-normal [mx] of my-solver [sx] of my-solver
      set ycor random-normal [my] of my-solver [sy] of my-solver
    ]

    ; Save current location and move to the best among the new solutions
    let old-patch patch-here
    move-to min-one-of solutions [value]

    ; Check if the best among the new solutions is better than the user's own best
    ifelse value < own-best [

      ; If so, update the user's own best and reward the chosen solver using the new location
      set own-best value
      reward-solver my-solver patch-here
    ] [

      ; Otherwise, preserve the previous own best location
      move-to old-patch
    ]

    ; Update solver's expertise and forget candidate solutions
    update-solver my-solver
    ask solutions [ die ]
  ]
end 


; Reward the solver for obtaining a better solution

to reward-solver [the-solver better-patch]
  ask the-solver [
    set score score + 1   ; Increase the solver's reputation
    move-to better-patch  ; Move to the better solution's location
    set bx xcor           ; Update the solver's best solution's x-coordinate
    set by ycor           ; Update the solver's best solution's y-coordinate
  ]
end 


; Update the solver's knowledge as they search for new solutions

to update-solver [the-solver]
  ask the-solver [
    let avg-x 0 let avg-y 0 ; Initialize variables for averaging coordinates

    ifelse greedy? [
      ; If greedy, use only the location of the best among new solutions
      set avg-x [xcor] of min-one-of solutions [value]
      set avg-y [ycor] of min-one-of solutions [value]
    ][
      ; Otherwise, use the average location of new solutions
      set avg-x mean [xcor] of solutions
      set avg-y mean [ycor] of solutions
    ]

    ; Update solver's core expertise based on the learning rate
    set mx (alpha * bx) + ((1 - alpha) * avg-x)
    set my (alpha * by) + ((1 - alpha) * avg-y)
    setxy mx my ; Move the solver to the updated location

    ; Narrow down the solver's expertise dispersion
    set sx sx * exp (-.001 * (ticks mod 50))
    set sy sy * exp (-.001 * (ticks mod 50))
  ]
end 


; Check if any user has improved the best solution found so far

to update-best
  let best-now min-one-of users [value]

  if [value] of best-now < [value] of best-ever [
    ; Record the location, tick, and time when a newer best-ever solution was found.
    ask best-now [ set best-ever patch-here ]
    set best-tick ticks
    set best-time timer
  ]
end 


; Perform the initial setup for the simulation

to setup
  clear-all                  ; Clear the world and all agents
  setup-search-landscape     ; Set up the landscape and cost function
  setup-solvers              ; Set up solver agents
  setup-users                ; Set up user agents
  setup-globals              ; Set up global variables
end 


; Reset the simulation for a new run (without computing the landscape again)

to reset
  clear-all-plots            ; Clear plot monitors from the previous run
  setup-solvers              ; Reinitialize solver agents
  setup-users                ; Reinitialize user agents
  setup-globals              ; Reinitialize global variables
end 


; Set up solver agents

to setup-solvers
  ; Clear previous solver agents
  ask solvers [ die ]

  ; Create solver agents
  create-solvers n-solvers [

    ; Assign visual attributes
    set shape "wolf 7"
    set color 4 + (10 * who)
    set size agent-size

    ; Initialize expertise model and score
    set mx random-xcor
    set my random-ycor
    set sx random-float 50
    set sy random-float 50
    set bx mx
    set by my
    set score 1

    ; Set initial random location
    setxy mx my
  ]

  ; If enabled, change locations with Latin Hypercube Sampling (lhs)
  if lhs? [ latin-hypercube-sampling ]
end 


; Set up user agents

to setup-users
  ; Clear previous user agents
  ask users [ die ]

  ; Create new users
  create-users n-users [
    ; Assign visual attributes
    set shape "dog"
    set size agent-size

    ; Assign initial own best solution
    move-to one-of patches
    set own-best value
  ]
end 


; Assign initial values for global variables

to setup-globals
  set best-tick 0
  set best-time 0
  set best-ever max-one-of patches [value]	
  reset-ticks
end 


; Restart agents to prevent stagnation

to restart
  setup-solvers
  setup-users
end 


; Choose a solver either randomly or guided by reputation

to-report choose-solver
  ifelse (reputation?) [
    ; Choose a solver based on their reputation using a roulette wheel
    let score-list map [ [the-solver] -> [score] of the-solver ] sort solvers
    let id-list map [ [the-solver] -> [who] of the-solver ] sort solvers

    ; Compute cumulative distribution from the histogram of scores
    let hist fput (list (first score-list)) (but-first score-list)
    let aggs reduce [ [cumul next] -> sentence cumul ((last cumul) + next) ] hist

    ; Use a roulette wheel to choose a solver according to the cumulative distribution
    let pockets map [ p -> p / last aggs ] aggs  ; Compute wheel pockets by normalizing cumulative sum
    let ball random-float 1                      ; Roll the ball, then check the winner pocket
    let winner first filter [ [index] -> ball <= item index pockets ] range length pockets
    report solver item winner id-list
  ] [
    ; Otherwise choose any solver at random uniformly
    report one-of solvers
  ]
end 


; Perform Latin Hypercube Sampling for initial solver locations

to latin-hypercube-sampling
  ; Compute the width of location slots
  let width 2 * max-pxcor / n-solvers

  ; Split each dimension into non-overlapping slots (1 per solver) and sample random locations within
  foreach range 2 [ index ->
    let coordinates shuffle n-values n-solvers [ slot -> width * (slot + random-float 1) ]

    ; Assign coordinate locations for each agent in an orderly manner, ensuring bound constraints
    (foreach sort solvers coordinates [
      [the-solver coordinate] ->
        ask the-solver [
          ifelse index = 0 [
            ; x-coordinate
            set mx (- max-pxcor + coordinate)
            set bx mx
            set xcor mx
          ][
            ; y-coordinate
            set my (- max-pycor + coordinate)
            set by my
            set ycor my
          ]
       ]
    ])
  ]
end 

; Reporter to display solver's score

to-report show-scores [the-agent]
  report (word "s_" who "=" score " | ")
end 

; Reporter to display agent's location

to-report locations [the-agent]
  report (word "u_" who ": (" precision x 1 ", " precision y 1 ")|")
end 

; Define view area settings when loading the model

to startup
  set grid-resolution "web (200x200)"

  ; Startup with cell size 1 to prevent view area distortions when resizing (NetLogo bug)
  set cell-size 1
  set agent-size 10
end 

; Routine for quick testing of success rate of a number of repetitions

to quick-test
  show "------------------------------------------------------------------"
  show (word landscape " >> Testing " runs " runs... (successful runs marked as '!!!')")
  setup
  let n 0 let i 1
  repeat runs [
    reset
    clear-output
    let success false
    while [ ticks < max-ticks and not success] [
      go
      set success (any? true-best-patch with [value = [value] of best-ever])
    ]
    show (word "run #" i " >> best-tick: " best-tick ifelse-value success [ "!!! (hit)" ] [ " (miss)" ])
    set n n + ifelse-value success [ 1 ] [ 0 ]
    set i i + 1
  ]
  show (word landscape " >> Success rate (within " max-ticks " max.ticks): " n "/" runs)
end 


;-------------------- DEFINITION OF BENCHMARK OPTIMIZATION PROBLEM LANDSCAPES --------------------
; This procedure computes the landscape of the chosen cost function and visualizes it.
; Source: Jamil, M., & Yang, X. S. (2013). A literature survey of benchmark functions for global
; optimization problems. International Journal of Mathematical Modelling and Numerical Optimisation.
; To add new problems, simply insert the mathematical expression of their cost function as new cases.

to setup-search-landscape
  clear-all

  ; Setup world and patch size
  set-patch-size cell-size
  ifelse grid-resolution = "web (200x200)" [
    (ifelse
      landscape = "damavandi" or
      landscape = "hosaki" or
      landscape = "michalewicz" or
      landscape = "vincent"   [ resize-world 0 200 0 200 ]
      landscape = "zakharov"  [ resize-world -50 150 -50 150 ]
      ; All other optimisation problems are defined over the four quadrants of the solution space
      [ resize-world -100 100 -100 100 ]
    )
  ] [
    (ifelse
      landscape = "damavandi" or
      landscape = "hosaki" or
      landscape = "michalewicz" or
      landscape = "vincent"   [ resize-world 0 1000 0 1000 ]
      landscape = "zakharov"  [ resize-world -250 750 -250 750 ]
      ; All other optimisation problems are defined over the four quadrants of the solution space
      [ resize-world -500 500 -500 500 ]
    )
  ]

  ; Setup range of variable coordinates for each problem
  set xy-bounds (ifelse-value
    landscape = "ackley" [ 32 ]
    landscape = "beale" or
    landscape = "michalewicz" or
    landscape = "parsopoulos" [ 5 ]
    landscape = "bohachesvsky n.1" [ 100 ]
    landscape = "booth" or
    landscape = "cross-in-tray" or
    landscape = "dixon-price" or
    landscape = "holder-table" or
    landscape = "hosaki" or
    landscape = "levy" or
    landscape = "matyas" or
    landscape = "mishra n.3" or
    landscape = "mishra n.5" or
    landscape = "mishra n.6" or
    landscape = "vincent" or
    landscape = "zakharov" [ 10 ]
    landscape = "damavandi" [ 14 ]
    landscape = "eggholder" [ 512 ]
    landscape = "goldstein-price" [ 2 ]
    ; For the following problems, we use the range [-32, 32] instead of the original [-100, 100], to reduce discretization errors
    landscape = "easom" or
    landscape = "schaffer n.4" or
    landscape = "schaffer n.2" [ 32 ]
    ; For any other problem use a [-5, 5] default range
    [ 5 ]
  )

  ; Evaluate the cost function for each patch in the landscape
  ask patches [
    set x pxcor * (xy-bounds / max-pxcor )
    set y pycor * (xy-bounds / max-pycor )

    ; Note: Trigonometric functions require input in degrees, not radians; thus, a conversion factor (180 / pi) was used
    set value
    (ifelse-value
      landscape = "ackley" [
        -20 * exp(-0.2 * sqrt(0.5 * (x ^ 2 + y ^ 2))) - exp(0.5 * (cos((180 / pi) * (2 * pi) * x) + cos((180 / pi) * (2 * pi) * y))) + 20 + e
      ]
      landscape = "beale" [
        ((1.5 - x + (x * y)) ^ 2) + ((2.25 - x + (x * (y ^ 2))) ^ 2) + ((2.625 - x + (x * (y ^ 3))) ^ 2)
      ]
      landscape = "bohachesvsky n.1" [
        (x ^ 2) + 2 * (y ^ 2) - (0.3 * (cos((180 / pi) * 3 * pi * x))) - (0.4 * cos ((180 / pi) * 4 * pi * y)) + 0.7
      ]
      landscape = "booth" [
        (x + (2 * y) - 7) ^ 2 + ((2 * x) + y - 5) ^ 2
      ]
      landscape = "cross-in-tray" [
        -0.0001 * (((abs(sin((180 / pi) * x) * sin((180 / pi) * y) * exp(abs(100 - ((sqrt((x ^ 2) + (y ^ 2)))) / pi)))) + 1) ^ 0.1)
      ]
      landscape = "damavandi" [
;        ifelse-value ( x = 2 ) or  ( y = 2 ) [ 100 ]
;        ifelse-value ( x = 2 ) and ( y = 2 ) [ 0 ]
        ifelse-value ( abs(x - 2) < 0.0001 ) or ( abs(y - 2) < 0.0001 )  [ 100 ]
        ifelse-value ( abs(x - 2) < 0.0001 ) and ( abs(y - 2) < 0.0001 ) [ 0 ]
        [(1 - (abs(((sin((180 / pi) * pi * (x - 2))) * (sin((180 / pi) * pi * (y - 2)))) /  ((pi ^ 2) * (x  - 2) * (y - 2)))) ^ 5  )  * ((2 + (x - 7) ^ 2) + (2 * (y - 7) ^ 2))]
      ]
      landscape = "dixon-price" [
        (x - 1) ^ 2  + 2 * ((2 * y ^ 2) - x) ^ 2
      ]
      landscape = "dropwave" [
        -1 * (((1 + (cos((180 / pi) * 12 *  sqrt((x ^ 2) + (y ^ 2)) ))) / (0.5 * ((x ^ 2) + (y ^ 2)) + 2 )))
      ]
      landscape = "easom" [
        -1 * (cos ((180 / pi) * x) * cos ((180 / pi) * y)) * exp (-((x - pi) ^ 2 + (y - pi) ^ 2))
      ]
      landscape = "eggholder" [ ; note that degrees, not radians, are needed for sin function
        ( (- x) * sin ( (180 / pi) * sqrt (abs (x - (y + 47))))) - (y + 47) * sin ( (180 / pi) * sqrt (abs ((x / 2) + (y + 47))))
      ]
      landscape = "goldstein-price" [
        (1 + ((x + y + 1) ^ 2) * (19 - (14 * x) + (3 * (x ^ 2) - (14 * y) + (6 * x * y) + (3 * (y ^ 2))))) *
          (30 + (((2 * x) - (3 * y) ) ^ 2) * (18 - (32 * x) + (12 * (x ^ 2) + (48 * y) - (36 * x * y) + (27 * (y ^ 2)))))
      ]
      landscape = "himmelblau" [
        ((x ^ 2) + y - 11) ^ 2 + (x + (y ^ 2) - 7) ^ 2
      ]
      landscape = "holder-table" [
        -1 * abs(sin((180 / pi) * x) * cos((180 / pi) * y) * exp(abs(1 - (sqrt(x ^ 2 + y ^ 2) / pi))))
      ]
      landscape = "hosaki" [
        (1 - (8 * x) + (7 * (x ^ 2)) - ((7 / 3) * x ^ 3) + (0.25 * (x ^ 4)) ) * ((y ^ 2) * exp((- y)))
      ]
      landscape = "levy" [
        (sin((180 / pi) * pi * (1 + (x - 1) / 4)) ^ 2)
        + (((1 + (x - 1) / 4) - 1) ^ 2) * (1 + 10 * (sin((180 / pi) * (pi * (1 + (x - 1) / 4)) + 1)) ^ 2)
        + (((1 + (y - 1) / 4) - 1) ^ 2) * (1 + (sin((180 / pi) * (2 * pi * (1 + (y - 1) / 4)))) ^ 2)
      ]
      landscape = "matyas" [
        (0.26 * ((x ^ 2) + (y ^ 2))) - (0.48 * (x * y))
      ]
      landscape = "michalewicz" [
        -1 * (sin((180 / pi) * x) * (sin((180 / pi) * 1 * (x ^ 2)  / pi )) ^ (2 * 10) )
        - (sin((180 / pi) * y) * (sin((180 / pi) * 2 * (y ^ 2)  / pi )) ^ (2 * 10) )
      ]
      landscape = "mishra n.3" [
        sqrt(abs(cos((180 / pi) * sqrt(abs((x ^ 2) + (y )))))) +  0.01 * (x + y)
      ]
      landscape = "mishra n.5" [
        (( (sin((180 / pi) * (cos((180 / pi) * x) + cos((180 / pi) * y)) ^ 2 )) ^ 2
          + (cos((180 / pi) * (sin((180 / pi) * x) + sin((180 / pi) * y)) ^ 2 )) ^ 2  + x) ^ 2)
        + (.01 * x) + (.1 * y)
      ]
      landscape = "mishra n.6" [
        -1 * ln(( (sin((180 / pi) * (cos((180 / pi) * x) + cos((180 / pi) * y)) ^ 2 )) ^ 2
          - (cos((180 / pi) * (sin((180 / pi) * x) + sin((180 / pi) * y)) ^ 2 )) ^ 2  + x) ^ 2)
        + .1 * ((x - 1) ^ 2 + (y - 1) ^ 2)
      ]
      landscape = "parsopoulos" [
        cos((180 / pi) * x) ^ 2 + sin((180 / pi) * y) ^ 2
      ]
      landscape = "rastrigin" [
        20 + ((x ^ 2) -  10 * cos((180 / pi) * (2 * pi) * x )) + ((y ^ 2) -  10 * cos((180 / pi) * (2 * pi) * y))
    	]
      landscape = "rastrigin offset" [
        20 + (((x - 1.123) ^ 2) -  10 * cos((180 / pi) * (2 * pi) * (x - 1.123))) + (((y - 1.123) ^ 2) -  10 * cos((180 / pi) * (2 * pi) * (y - 1.123)))
    	]
      landscape = "rastrigin bipolar" [
        20 + (((x + 1) ^ 2) -  10 * cos((180 / pi) * (2 * pi) * (x + 1))) + (((y - 1) ^ 2) -  10 * cos((180 / pi) * (2 * pi) * (y - 1)))
      ]
      landscape = "rosenbrock" [
        100 * (y - (x ^ 2)) ^ 2 + (1 - x) ^ 2
      ]
      landscape = "schaffer n.2" [
        0.5 + (((sin((180 / pi) * (x ^ 2 - y ^ 2)) ^ 2) - 0.5) / (1 + (0.001 * (x ^ 2 + y ^ 2))) ^ 2)
      ]
      landscape = "schaffer n.4" [
        0.5 + (((cos((180 / pi) * sin((180 / pi) * abs(x ^ 2 - y ^ 2)))) ^ 2 - 0.5) / (1 + (0.001 * (x ^ 2 + y ^ 2))) ^ 2)
      ]
      landscape = "sphere" [
        x ^ 2 + y ^ 2
      ]
    	landscape = "sphere-offset" [
        (x - 3) ^ 2  + (y + 3) ^ 2
      ]
      landscape = "three-hump camel" [
        (2 * (x ^ 2)) - (1.05 *  (x ^ 4)) + ((x ^ 6) / 6) + (x * y) + (y ^ 2)
      ]
      landscape = "vincent" [
        ifelse-value x < 0.25 or y < 0.25 [ 0 ]
;        [ -1 * sin((180 / pi) * 10 * (log(x) 10)) - sin((180 / pi) * 10 * (log(y) 10))]
        [ -1 * sin((180 / pi) * 10 * ln(x)) - sin((180 / pi) * 10 * ln(y)) ]
      ]
      landscape = "zakharov" [
        (x ^ 2 + y ^ 2) + ((0.5 * x) + (0.5 * 2 * y)) ^ 2 + ((0.5 * x) + (0.5 * 2 * y)) ^ 4
      ]
      ; Otherwise, use a random landscape
      [ random-normal 0 500  ]
    )
  ]

  ; Smooth out random landscape for better visualization and search efficiency
  if landscape = "random" [
    ask min-n-of 4 patches [value] [ ask patches in-radius 30 [ set value value - random-float 300 ] ]
    repeat 10 [ diffuse value 1 ]
  ]

  ; Find the true best patch (global minima) based on the chosen landscape
  (ifelse
    ; Functions with 2 global minima
    landscape = "dixon-price" [ set true-best-patch min-n-of 2 patches [value] ]

    ; Functions with 4 global minima
    landscape = "himmelblau" or
    landscape = "cross-in-tray" or
    landscape = "holder-table" or
    landscape = "schaffer n.4" [ set true-best-patch min-n-of 4 patches [value] ]

    ; Functions with 12 global minima
    landscape = "parsopoulos" [ set true-best-patch min-n-of 12 patches [value] ]

    ; Functions with 6^2 global minima
    landscape = "vincent" [ set true-best-patch min-n-of 36 patches [value] ]

    ; All other cost functions have a single global minima
    [ set true-best-patch patch-set min-one-of patches [value] ]
  )

	;; Scale patches color within min and max values limits for visualisation purposes
  let min-val min [value] of patches
  let max-val max [value] of patches
	ask patches  [
    (ifelse
      ; Problems better visualised using linear color scale
      landscape = "ackley" or
      landscape = "bohachesvsky n.1" or
      landscape = "cross-in-tray" or
      landscape = "damavandi" or
      landscape = "dropwave" or
      landscape = "schaffer n.2" or
      landscape = "schaffer n.4" or
      landscape = "vincent"
      [ set pcolor scale-color yellow value min-val max-val]

      ; Problems better visualised using square rooted color scale
      landscape = "easom" or
      landscape = "eggholder" or
      landscape = "holder-table" or
      landscape = "michalewicz" or
      landscape = "mishra n.6" or
      landscape = "parsopoulos" or
      landscape = "zakharov"
      [ set pcolor scale-color yellow value min-val sqrt max-val ]

      ; Problems better visualised using logarithmic scale
      landscape = "beale" or
      landscape = "dixon-price" or
      landscape = "goldstein-price" or
      landscape = "booth" or
      landscape = "rosenbrock"
      [ set pcolor scale-color yellow value min-val log max-val 1.01 ]

      landscape = "rastrigin" or
      landscape = "rastrigin offset" or
      landscape = "rastrigin bipolar"
      [ set pcolor scale-color yellow value min-val log max-val 1.05 ]

      landscape = "himmelblau" or
      landscape = "levy" or
      landscape = "matyas" or
      landscape = "mishra n.3" or
      landscape = "sphere" or
    	landscape = "sphere-offset" or
      landscape = "three-hump camel"
      [ set pcolor scale-color yellow value min-val log max-val 1.1 ]

      landscape = "hosaki" or
      landscape = "mishra n.5"
      [ set pcolor scale-color yellow value min-val log max-val 10.1 ]

      ; For any other problem use a logarithmic default scale
      [ set pcolor scale-color yellow value min-val log abs (max-val + 0.001) 1.05 ]
     )
  ]

  ;; Set spotlight on or off
  if spotlight = true [ watch one-of true-best-patch ]
end 

; Shorter setup procedure for some representative landscape problems

to setup-search-landscapes-short
  ; Set up world and patch size
  resize-world -500 500 -500 500
  set-patch-size cell-size
  display

  ; Create the 2D landscape according to the chosen cost function and bound constraints
  set xy-bounds ifelse-value landscape = "eggholder" [ 512 ] [ 6 ]
  ask patches [
    set x pxcor * (xy-bounds / max-pxcor)
    set y pycor * (xy-bounds / max-pycor)

    set value (ifelse-value
      landscape = "sphere" [
        ifelse-value (x = 5) or (y = 5)
        [ 100 ]
        [ x ^ 2 + y ^ 2 ]
      ]
      landscape = "sphere-offset" [
        (x - 300 * (xy-bounds / max-pxcor)) ^ 2  + (y + 300 * (xy-bounds / max-pxcor)) ^ 2
      ]
      landscape = "rastrigin" [ ; Note that degrees, not radians, are needed for the cos function
        20 + ((x ^ 2) -  10 * cos ((180 / pi) * (2 * pi) * x)) + ((y ^ 2) -  10 * cos ((180 / pi) * (2 * pi) * y))
      ]
      landscape = "rosenbrock" [
        100 * (y - (x ^ 2))^ 2 + (1 - x)^ 2
      ]
      landscape = "himmelblau" [
        ((x ^ 2) + y - 11) ^ 2 + (x + (y ^ 2) - 7)^ 2
      ]
      landscape = "eggholder" [ ; Note that degrees, not radians, are needed for the sin function
        ( (- x) * sin ((180 / pi) * sqrt (abs (x - (y + 47))))) - (y + 47) * sin ((180 / pi) * sqrt (abs ((x / 2) + (y + 47))))
      ]
      [ random-normal 0 500 ] ; The last case is a random landscape
    )
  ]

  if landscape = "random" [
    ; Smooth out the random landscape
    ask min-n-of 4 patches [value] [ ask patches in-radius 30 [ set value value - random-float 300 ] ]
    repeat 10 [ diffuse value 1 ]
  ]

  ; Find the true best location
  ifelse landscape = "himmelblau" [
    ; "himmelblau" exhibits 4 global minima
    set true-best-patch min-n-of 4 patches [value]
  ] [
    ; All other cost functions have a single global minima
    set true-best-patch patch-set min-one-of patches [value]
  ]

  ; Scale patches color within value limits
  let min-val min [value] of patches
  let max-val max [value] of patches
  ask patches [ set pcolor scale-color yellow value min-val log abs max-val 1.05 ]

  ; Set spotlight if switched on
  if spotlight = true [ watch one-of true-best-patch ]
end 

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