-
Notifications
You must be signed in to change notification settings - Fork 530
Commit
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
optimize/functions: implement the many local minima functions in the …
…virtual library of simulation experiments
- Loading branch information
Showing
3 changed files
with
341 additions
and
7 deletions.
There are no files selected for viewing
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Original file line number | Diff line number | Diff line change |
---|---|---|
@@ -0,0 +1,15 @@ | ||
// Copyright ©2017 The Gonum Authors. All rights reserved. | ||
// Use of this source code is governed by a BSD-style | ||
// license that can be found in the LICENSE file. | ||
|
||
// Package functions provides objective functions for testing optimization | ||
// algorithms. | ||
// | ||
// We encourage outside contributions of additional test functions that exhibit | ||
// properties not already covered in the testing suite or that have | ||
// significance due to prior use as benchmark cases. | ||
package functions // import "gonum.org/v1/gonum/optimize/functions" | ||
|
||
const ( | ||
badInputDim = "functions: wrong input dimension" | ||
) |
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Original file line number | Diff line number | Diff line change |
---|---|---|
@@ -0,0 +1,325 @@ | ||
// Copyright ©2017 The Gonum Authors. All rights reserved. | ||
// Use of this source code is governed by a BSD-style | ||
// license that can be found in the LICENSE file. | ||
|
||
package functions | ||
|
||
import "math" | ||
|
||
// This file implements functions from the Virtual Library of Simulation Experiments. | ||
// https://www.sfu.ca/~ssurjano/optimization.html | ||
// In many cases gradients and Hessians have been added. In some cases, these | ||
// are not defined at certain points or manifolds. The gradient in these locations | ||
// has been set to 0. | ||
|
||
// Ackley implements the Ackley function, a function of arbitrary dimension that | ||
// has many local minima. It has a single global minimum of 0 at 0. Its typical | ||
// domain is the hypercube of [-32.768, 32.768]^d. | ||
// f(x) = -20 * exp(-0.2 sqrt(1/d sum_i x_i^2)) - exp(1/d sum_i cos(2π x_i)) + 20 + exp(1) | ||
// where d is the input dimension. | ||
// | ||
// Reference: | ||
// https://www.sfu.ca/~ssurjano/ackley.html (obtained June 2017) | ||
type Ackley struct{} | ||
|
||
func (Ackley) Func(x []float64) float64 { | ||
var ss, sc float64 | ||
for _, v := range x { | ||
ss += v * v | ||
sc += math.Cos(2 * math.Pi * v) | ||
} | ||
id := 1 / float64(len(x)) | ||
return -20*math.Exp(-0.2*math.Sqrt(id*ss)) - math.Exp(id*sc) + 20 + math.E | ||
} | ||
|
||
// Bukin6 implements Bukin's 6th function. The function is two-dimensional, with | ||
// the typical domain as x_0 ∈ [-15, -5], x_1 ∈ [-3, 3]. The function has a unique | ||
// global minimum at [-10, 1], and many local minima. | ||
// f(x) = 100 * sqrt(|x_1 - 0.01*x_0^2|) + 0.01*|x_0+10| | ||
// Reference: | ||
// https://www.sfu.ca/~ssurjano/bukin6.html (obtained June 2017) | ||
type Bukin6 struct{} | ||
|
||
func (Bukin6) Func(x []float64) float64 { | ||
if len(x) != 2 { | ||
panic(badInputDim) | ||
} | ||
return 100*math.Sqrt(math.Abs(x[1]-0.01*x[0]*x[0])) + 0.01*math.Abs(x[0]+10) | ||
} | ||
|
||
// CrossInTray implements the cross-in-tray function. The cross-in-tray function | ||
// is a two-dimensional function with many local minima, and four global minima | ||
// at (±1.3491, ±1.3491). The function is typically evaluated in the square | ||
// [-10,10]^2. | ||
// f(x) = -0.001(|sin(x_0)sin(x_1)exp(|100-sqrt((x_0^2+x_1^2)/π)|)|+1)^0.1 | ||
// Reference: | ||
// https://www.sfu.ca/~ssurjano/crossit.html (obtained June 2017) | ||
type CrossInTray struct{} | ||
|
||
func (CrossInTray) Func(x []float64) float64 { | ||
if len(x) != 2 { | ||
panic(badInputDim) | ||
} | ||
x0 := x[0] | ||
x1 := x[1] | ||
exp := math.Abs(100 - math.Sqrt((x0*x0+x1*x1)/math.Pi)) | ||
return -0.0001 * math.Pow(math.Abs(math.Sin(x0)*math.Sin(x1)*math.Exp(exp))+1, 0.1) | ||
} | ||
|
||
// DropWave implements the drop-wave function, a two-dimensional function with | ||
// many local minima and one global minimum at 0. The function is typically evaluated | ||
// in the square [-5.12, 5.12]^2. | ||
// f(x) = - (1+cos(12*sqrt(x0^2+x1^2))) / (0.5*(x0^2+x1^2)+2) | ||
// Reference: | ||
// https://www.sfu.ca/~ssurjano/drop.html (obtained June 2017) | ||
type DropWave struct{} | ||
|
||
func (DropWave) Func(x []float64) float64 { | ||
if len(x) != 2 { | ||
panic(badInputDim) | ||
} | ||
x0 := x[0] | ||
x1 := x[1] | ||
num := 1 + math.Cos(12*math.Sqrt(x0*x0+x1*x1)) | ||
den := 0.5*(x0*x0+x1*x1) + 2 | ||
return -num / den | ||
} | ||
|
||
// Eggholder implements the Eggholder function, a two-dimensional function with | ||
// many local minima and one global minimum at [512, 404.2319]. The function | ||
// is typically evaluated in the square [-512, 512]^2. | ||
// f(x) = -(x_1+47)*sin(sqrt(|x_1+x_0/2+47|))-x_1*sin(sqrt(|x_0-(x_1+47)|)) | ||
// Reference: | ||
// https://www.sfu.ca/~ssurjano/egg.html (obtained June 2017) | ||
type Eggholder struct{} | ||
|
||
func (Eggholder) Func(x []float64) float64 { | ||
if len(x) != 2 { | ||
panic(badInputDim) | ||
} | ||
x0 := x[0] | ||
x1 := x[1] | ||
return -(x1+47)*math.Sin(math.Sqrt(math.Abs(x1+x0/2+47))) - | ||
x0*math.Sin(math.Sqrt(math.Abs(x0-x1-47))) | ||
} | ||
|
||
// GramacyLee implements the Gramacy-Lee function, a one-dimensional function | ||
// with many local minima. The function is typically evaluated on the domain [0.5, 2.5]. | ||
// f(x) = sin(10πx)/(2x) + (x-1)^4 | ||
// Reference: | ||
// https://www.sfu.ca/~ssurjano/grlee12.html (obtained June 2017) | ||
type GramacyLee struct{} | ||
|
||
func (GramacyLee) Func(x []float64) float64 { | ||
if len(x) != 1 { | ||
panic(badInputDim) | ||
} | ||
x0 := x[0] | ||
return math.Sin(10*math.Pi*x0)/(2*x0) + math.Pow(x0-1, 4) | ||
} | ||
|
||
// Griewank implements the Griewank function, a function of arbitrary dimension that | ||
// has many local minima. It has a single global minimum of 0 at 0. Its typical | ||
// domain is the hypercube of [-600, 600]^d. | ||
// f(x) = \sum_i x_i^2/4000 - \prod_i cos(x_i/sqrt(i)) + 1 | ||
// where d is the input dimension. | ||
// | ||
// Reference: | ||
// https://www.sfu.ca/~ssurjano/griewank.html (obtained June 2017) | ||
type Griewank struct{} | ||
|
||
func (Griewank) Func(x []float64) float64 { | ||
var ss float64 | ||
pc := 1.0 | ||
for i, v := range x { | ||
ss += v * v | ||
pc *= math.Cos(v / math.Sqrt(float64(i+1))) | ||
} | ||
return ss/4000 - pc + 1 | ||
} | ||
|
||
// HolderTable implements the Holder table function. The Holder table function | ||
// is a two-dimensional function with many local minima, and four global minima | ||
// at (±8.05502, ±9.66459). The function is typically evaluated in the square [-10,10]^2. | ||
// f(x) = -|sin(x_0)cos(x1)exp(|1-sqrt(x_0^2+x1^2)/π|)| | ||
// Reference: | ||
// https://www.sfu.ca/~ssurjano/holder.html (obtained June 2017) | ||
type HolderTable struct{} | ||
|
||
func (HolderTable) Func(x []float64) float64 { | ||
if len(x) != 2 { | ||
panic(badInputDim) | ||
} | ||
x0 := x[0] | ||
x1 := x[1] | ||
return -math.Abs(math.Sin(x0) * math.Cos(x1) * math.Exp(math.Abs(1-math.Sqrt(x0*x0+x1*x1)/math.Pi))) | ||
} | ||
|
||
// Langermann2 implements the two-dimensional version of the Langermann function. | ||
// The Langermann function has many local minima. The function is typically | ||
// evaluated in the square [0,10]^2. | ||
// f(x) = \sum_1^5 c_i exp(-(1/π)\sum_{j=1}^2(x_j-A_{ij})^2) * cos(π\sum_{j=1}^2 (x_j - A_{ij})^2) | ||
// c = [5]float64{1,2,5,2,3} | ||
// A = [5][2]float64{{3,5},{5,2},{2,1},{1,4},{7,9}} | ||
// Reference: | ||
// https://www.sfu.ca/~ssurjano/langer.html (obtained June 2017) | ||
type Langermann2 struct{} | ||
|
||
func (Langermann2) Func(x []float64) float64 { | ||
if len(x) != 2 { | ||
panic(badInputDim) | ||
} | ||
var ( | ||
c = [5]float64{1, 2, 5, 2, 3} | ||
A = [5][2]float64{{3, 5}, {5, 2}, {2, 1}, {1, 4}, {7, 9}} | ||
) | ||
var f float64 | ||
for i, cv := range c { | ||
var ss float64 | ||
for j, av := range A[i] { | ||
xja := x[j] - av | ||
ss += xja * xja | ||
} | ||
f += cv * math.Exp(-(1/math.Pi)*ss) * math.Cos(math.Pi*ss) | ||
} | ||
return f | ||
} | ||
|
||
// Levy implements the Levy function, a function of arbitrary dimension that | ||
// has many local minima. It has a single global minimum of 0 at 1. Its typical | ||
// domain is the hypercube of [-10, 10]^d. | ||
// f(x) = sin^2(π*w_0) + \sum_{i=0}^{d-2}(w_i-1)^2*[1+10sin^2(π*w_i+1)] + | ||
// (w_{d-1}-1)^2*[1+sin^2(2π*w_{d-1})] | ||
// w_i = 1 + (x_i-1)/4 | ||
// where d is the input dimension. | ||
// | ||
// Reference: | ||
// https://www.sfu.ca/~ssurjano/levy.html (obtained June 2017) | ||
type Levy struct{} | ||
|
||
func (Levy) Func(x []float64) float64 { | ||
w1 := 1 + (x[0]-1)/4 | ||
s1 := math.Sin(math.Pi * w1) | ||
sum := s1 * s1 | ||
for i := 0; i < len(x)-1; i++ { | ||
wi := 1 + (x[i]-1)/4 | ||
s := math.Sin(math.Pi*wi + 1) | ||
sum += (wi - 1) * (wi - 1) * (1 + 10*s*s) | ||
} | ||
wd := 1 + (x[len(x)-1]-1)/4 | ||
sd := math.Sin(2 * math.Pi * wd) | ||
return sum + (wd-1)*(wd-1)*(1+sd*sd) | ||
} | ||
|
||
// Levy13 implements the Levy-13 function, a two-dimensional function | ||
// with many local minima. It has a single global minimum of 0 at 1. Its typical | ||
// domain is the square [-10, 10]^2. | ||
// f(x) = sin^2(3π*x_0) + (x_0-1)^2*[1+sin^2(3π*x_1)] + (x_1-1)^2*[1+sin^2(2π*x_1)] | ||
// Reference: | ||
// https://www.sfu.ca/~ssurjano/levy13.html (obtained June 2017) | ||
type Levy13 struct{} | ||
|
||
func (Levy13) Func(x []float64) float64 { | ||
if len(x) != 2 { | ||
panic(badInputDim) | ||
} | ||
x0 := x[0] | ||
x1 := x[1] | ||
s0 := math.Sin(3 * math.Pi * x0) | ||
s1 := math.Sin(3 * math.Pi * x1) | ||
s2 := math.Sin(2 * math.Pi * x1) | ||
return s0*s0 + (x0-1)*(x0-1)*(1+s1*s1) + (x1-1)*(x1-1)*(1+s2*s2) | ||
} | ||
|
||
// Rastrigin implements the Rastrigen function, a function of arbitrary dimension | ||
// that has many local minima. It has a single global minimum of 0 at 0. Its typical | ||
// domain is the hypercube of [-5.12, 5.12]^d. | ||
// f(x) = 10d + \sum_i [x_i^2 - 10cos(2π*x_i)] | ||
// where d is the input dimension. | ||
// | ||
// Reference: | ||
// https://www.sfu.ca/~ssurjano/rastr.html (obtained June 2017) | ||
type Rastrigin struct{} | ||
|
||
func (Rastrigin) Func(x []float64) float64 { | ||
sum := 10 * float64(len(x)) | ||
for _, v := range x { | ||
sum += v*v - 10*math.Cos(2*math.Pi*v) | ||
} | ||
return sum | ||
} | ||
|
||
// Schaffer2 implements the second Schaffer function, a two-dimensional function | ||
// with many local minima. It has a single global minimum of 0 at 0. Its typical | ||
// domain is the square [-100, 100]^2. | ||
// f(x) = 0.5 + (sin^2(x_0^2-x_1^2)-0.5) / (1+0.001*(x_0^2+x_1^2))^2 | ||
// Reference: | ||
// https://www.sfu.ca/~ssurjano/schaffer2.html (obtained June 2017) | ||
type Schaffer2 struct{} | ||
|
||
func (Schaffer2) Func(x []float64) float64 { | ||
if len(x) != 2 { | ||
panic(badInputDim) | ||
} | ||
x0 := x[0] | ||
x1 := x[1] | ||
s := math.Sin(x0*x0 - x1*x1) | ||
den := 1 + 0.001*(x0*x0+x1*x1) | ||
return 0.5 + (s*s-0.5)/(den*den) | ||
} | ||
|
||
// Schaffer4 implements the fourth Schaffer function, a two-dimensional function | ||
// with many local minima. Its typical domain is the square [-100, 100]^2. | ||
// f(x) = 0.5 + (cos(sin(|x_0^2-x_1^2|))-0.5) / (1+0.001*(x_0^2+x_1^2))^2 | ||
// Reference: | ||
// https://www.sfu.ca/~ssurjano/schaffer4.html (obtained June 2017) | ||
type Schaffer4 struct{} | ||
|
||
func (Schaffer4) Func(x []float64) float64 { | ||
if len(x) != 2 { | ||
panic(badInputDim) | ||
} | ||
x0 := x[0] | ||
x1 := x[1] | ||
den := 1 + 0.001*(x0*x0+x1*x1) | ||
return 0.5 + (math.Cos(math.Sin(math.Abs(x0*x0-x1*x1)))-0.5)/(den*den) | ||
} | ||
|
||
// Schwefel implements the Schwefel function, a function of arbitrary dimension | ||
// that has many local minima. Its typical domain is the hypercube of [-500, 500]^d. | ||
// f(x) = 418.9829*d - \sum_i x_i*sin(sqrt(|x_i|)) | ||
// where d is the input dimension. | ||
// | ||
// Reference: | ||
// https://www.sfu.ca/~ssurjano/schwef.html (obtained June 2017) | ||
type Schwefel struct{} | ||
|
||
func (Schwefel) Func(x []float64) float64 { | ||
var sum float64 | ||
for _, v := range x { | ||
sum += v * math.Sin(math.Sqrt(math.Abs(v))) | ||
} | ||
return 418.9829*float64(len(x)) - sum | ||
} | ||
|
||
// Schubert implements the Schubert function, a two-dimensional function | ||
// with many local minima and many global minima. Its typical domain is the | ||
// square [-10, 10]^2. | ||
// f(x) = (sum_{i=1}^5 i cos((i+1)*x_0+i)) * (\sum_{i=1}^5 i cos((i+1)*x_1+i)) | ||
// Reference: | ||
// https://www.sfu.ca/~ssurjano/shubert.html (obtained June 2017) | ||
type Schubert struct{} | ||
|
||
func (Schubert) Func(x []float64) float64 { | ||
if len(x) != 2 { | ||
panic(badInputDim) | ||
} | ||
x0 := x[0] | ||
x1 := x[1] | ||
var s0, s1 float64 | ||
for i := 1.0; i <= 5.0; i++ { | ||
s0 += i * math.Cos((i+1)*x0+i) | ||
s1 += i * math.Cos((i+1)*x1+i) | ||
} | ||
return s0 * s1 | ||
} |