Test functions for optimization needs

Transcription

1 Test functions for optimization needs Marcin Molga, Czesław Smutnicki 3 kwietnia 5 Streszczenie This paper provides the review of literature benchmarks (test functions) commonl used in order to test optimization procedures dedicated for multidimensional, continuous optimization task. Special attention has been paid to multiple-etreme functions, treated as the qualit test for resistant optimization methods (GA, SA, TS, etc.) Introduction Qualit of optimization procedures (those alread known and these newl proposed) are frequentl evaluated b using common standard literature benchmarks. There are several classes of such test functions, all of them are continuous: (a) unimodal, conve, multidimensional, (b) multimodal, two-dimensional with a small number of local etremes, (c) multimodal, two-dimensional with huge number of local etremes (d) multimodal, multidimensional, with huge number of local etremes,. Class (a) contains nice functions as well as malicious cases causing poor or slow convergence to single global etremum. Class (b) is mediate between (a) and (c)- (d), and is used to test qualit of standard optimization procedures in the hostile environment, namel that having few local etremes with single global one. Classes (c)-(d) are recommended to test qualit of intelligent resistant optimization methods, as an eample GA, SA, TS, etc. These classes are considered as ver hard test problems. Class (c) is artificial in some sense, since the behavior of

2 optimization procedure is usuall being justified, eplain and supported b human intuitions on D surface. Moreover, two-dimensional optimization problems appear ver rarel in practice. Unfortunatel, practical discrete optimization problems provide instances with large number of dimensions, laing undoubtedl in class (d). For eample, the smallest known currentl benchmark ft for so called job shop scheduling problem has dimension 9, the biggest known - has dimension 98. Therefore, in order to test real qualit of proposed algorithms, we need to consider chiefl instances from class (d). As the shocking contrast, the proposed GA approaches for continuous optimization do not eceed dimension. Notice, polarization (a constant added to function value) has no influence on the result of minimization. Therefore, definitions of functions can differ from these original ones b a constant. All tests are formulated hereinafter as minimization problems, nevertheless can be applied also for maimization problems b simple inverting sign of the function. Test functions In this section we present benchmarks commonl known in the literature.. De Jong s function So called first function of De Jong s is one of the simplest test benchmark. Function is continuous, conve and unimodal. It has the following general definition f() = n i. () i= Test area is usuall restricted to hphercube 5. i 5., i =,..., n. Global minimum f() = is obtainable for i =, i =,..., n.

3 Rsunek : De Jong s function in D, f(, ) = + 3

4 Rsunek : Ais parallel hper-ellipsoid function in D, f(, ) = +. Ais parallel hper-ellipsoid function The ais parallel hper-ellipsoid is similar to function of De Jong. It is also known as the weighted sphere model. Function is continuous, conve and unimodal. It has the following general definition f() = n (i i ). () i= Test area is usuall restricted to hphercube 5. i 5., i =,..., n. Global minimum equal f() = is obtainable for i =, i =,..., n.

5 Rsunek 3: Rotated hper-ellipsoid function in D, f(, ) = + ( + ).3 Rotated hper-ellipsoid function An etension of the ais parallel hper-ellipsoid is Schwefel s function. With respect to the coordinate aes, this function produces rotated hper-ellipsoids. It is continuous, conve and unimodal. Function has the following general definition f() = n i j. (3) i= j= Test area is usuall restricted to hphercube i , i =,..., n. Its global minimum equal f() = is obtainable for i =, i =,..., n. 5

6 Rsunek : Rosenbrock s valle in D, f(, ) = ( ) + ( ). Rosenbrock s valle Rosenbrock s valle is a classic optimization problem, also known as banana function or the second function of De Jong. The global optimum las inside a long, narrow, parabolic shaped flat valle. To find the valle is trivial, however convergence to the global optimum is difficult and hence this problem has been frequentl used to test the performance of optimization algorithms. Function has the following definition n [ f() = (i+ i ) + ( i ) ]. () i= Test area is usuall restricted to hphercube.8 i.8, i =,..., n. Its global minimum equal f() = is obtainable for i, i =,..., n. 6

7 Rsunek 5: An overview of Rastrigin s function in D, f(, ) = + [ cos(π)] + [ cos(π)].5 Rastrigin s function Rastrigin s function is based on the function of De Jong with the addition of cosine modulation in order to produce frequent local minima. Thus, the test function is highl multimodal. However, the location of the minima are regularl distributed. Function has the following definition f() = n + n [ i cos(π i ) ]. (5) i= Test area is usuall restricted to hphercube 5. i 5., i =,..., n. Its global minimum equal f() = is obtainable for i =, i =,..., n. 7

8 Rsunek 6: Zoom on Rastrigin s function in D, f(, ) = + [ cos(π)] + [ cos(π)] 8

9 Rsunek 7: An overview of Schwefel s function in D, f(, ) = sin( sin(.6 Schwefel s function Schwefel s function is deceptive in that the global minimum is geometricall distant, over the parameter space, from the net best local minima. Therefore, the search algorithms are potentiall prone to convergence in the wrong direction. Function has the following definition f() = n i= [ i sin( ] i ). (6) Test area is usuall restricted to hphercube 5 i 5, i =,..., n. Its global minimum f() = 8.989n is obtainable for i =.9687, i =,..., n. 9

10 Rsunek 8: Zoom on Schwefel s function in D, f(, ) = sin( sin(

11 Rsunek 9: An overview of Griewangk s function in D, f(, ) = + cos() cos( ) +.7 Griewangk s function Griewangk s function is similar to the function of Rastrigin. It has man widespread local minima regularl distributed. Function has the following definition f() = n i i= n i= cos( i i ) +. (7) Test area is usuall restricted to hphercube 6 i 6, i =,..., n. Its global minimum equal f() = is obtainable for i =, i =,..., n. The function interpretation changes with the scale; the general overview suggests conve function, medium-scale view suggests eistence of local etremum, and finall zoom on the details indicates comple structure of numerous local etremum.

12 Rsunek : Medium-scale view of Griewangk s function in D, f(, ) = + cos() cos( ) +

13 Rsunek : Zoom on Griewangk s function in D, f(, ) = + cos() cos( ) + 3

14 Rsunek : Sum of different power functions in D, f(, ) = Sum of different power functions The sum of different powers is a commonl used unimodal test function. It has the following definition n f() = i i+. (8) i= Test area is usuall restricted to hphercube i, i =,..., n. Its global minimum equal f() = is obtainable for i =, i =,..., n.

15 Rsunek 3: An overview of Ackle s function in D, f(, ) = sin( sin(.9 Ackle s function Ackle s is a widel used multimodal test function. It has the following definition f() = a ep( b n i n ) ep( n cos(c i )) + a + ep() (9) n i= It is recommended to set a =, b =., c = π. Test area is usuall restricted to hphercube i 3.768, i =,..., n. Its global minimum f() = is obtainable for i =, i =,..., n. i= 5

16 Rsunek : Zoom on Ackle s function in D, f(, ) = sin( sin( 6

17 Rsunek 5: An overview of Langermann s function in D. f(, ) = m i= c i ep( ( a j ) /π ( b j ) /π) cos(π( a j ) + π( b j ) ), m = 5, a = [3, 5,,, 7], b = [5,,,, 9], c = [,, 5,, 3]. Langermann s function The Langermann function is a multimodal test function. The local minima are unevenl distributed. Function has the following definition f() = m c i ep[ π i= n ( j a ij ) ] cos[π j= n ( j a ij ) ] () where (c i, i =,..., m), (a ij, j =,..., n, i =,..., m) are constant numbers fied in advance. It is recommended to set m = 5. j= 7

18 Rsunek 6: Medium-scale view on Langermann s function in D. f(, ) = m i= c i ep( ( a j ) /π ( b j ) /π) cos(π( a j ) + π( b j ) ), m = 5, a = [3, 5,,, 7], b = [5,,,, 9], c = [,, 5,, 3] 8

19 Rsunek 7: Zoom on Langermann s function in D. f(, ) = m i= c i ep( ( a j ) /π ( b j ) /π) cos(π( a j ) + π( b j ) ), m = 5, a = [3, 5,,, 7], b = [5,,,, 9], c = [,, 5,, 3] 9

20 . Michalewicz s function The Michalewicz function is a multimodal test function (owns n! local optima). The parameter m defines the steepness of the valles or edges. Larger m leads to more difficult search. For ver large m the function behaves like a needle in the hastack (the function values for points in the space outside the narrow peaks give ver little information on the location of the global optimum). Function has the following definition f() = n i= [ ] m sin( i ) sin( i i π ) () It is usuall set m =. Test area is usuall restricted to hphercube i π, i =,..., n. The global minimum value has been approimated b f() =.687 for n = 5 and b f() = 9.66 for n =. Respective optimal solutions are not given.

21 Rsunek 8: Michalewicz s function in D for m =, f(, ) = sin()(sin( π )m sin()(sin( π )m

22 Rsunek 9: Michalewicz s function in D for m =, f(, ) = sin()(sin( π )m sin()(sin( π )m

23 Branins s function Rsunek : Branins s function The Branin function is a global optimization test function having onl two variables. The function has three equal-sized global optima, and has the following definition f(, ) = a( b + c d) + e( f) cos( ) + e. () It is recommended to set the following values of parameters: a =, b = 5. c = 5, d = 6, e =, f =. Three global optima equal f( π 8π, ) = are located as follows: (, ) = ( π,.75), (π,.75), (9.78,.75). π, 3

24 Easom s function Rsunek : Easom s function The Easom function is a unimodal test function, where the global minimum has a small area relative to the search space. The function was inverted for minimization. It has onl two variables and the following definition f(, ) = cos( ) cos( ) ep( ( π) ( π) ) (3) Test area is usuall restricted to square,. Its global minimum equal f() = is obtainable for (, ) = (π, π).

25 Rsunek : Zoom on Easom s function 5

26 e Rsunek 3: Goldstein-Price s function. Goldstein-Price s function The Goldstein-Price function is a global optimization test function. It has onl two variables and the following definition f(, ) = [ + ( + + ) ( ] [3 + ( 3 ) ( ]. () Test area is usuall restricted to the square,. Its global minimum equal f() = 3 is obtainable for (, ) = (, ). 6

27 Rsunek : Si-hump camel back function.5 Si-hump camel back function The Si-hump camel back function is a global optimization test function. Within the bounded region it owns si local minima, two of them are global ones. Function has onl two variables and the following definition f(, ) = (. + 3 ) + + ( + ). (5) Test area is usuall restricted to the rectangle 3 3,. Two global minima equal f() =.36 are located at (, ) = (.898,.76) and (.898,.76). 7

28 Rsunek 5: Zoom on si-hump camel back function 8

29 .6 Fifth function of De Jong This is a multimodal test function. The given form of function has onl two variables and the following definition where 5 f(, ) = {. + [j + ( a j ) 6 + ( a j ) 6 ] }, (6) (a ij ) = j= ( The function can also be rewritten as follows f(, ) = {.+ i= j= [5(i+)+j +3+( 6j) 6 +( 6i) 6 ] }, (7) Test area is usuall restricted to the square , ) 9

30 Rsunek 6: An overview of fifth function of De Jong 3

31 Rsunek 7: Medium-scale view on the fifth function of De Jong 3

32 Rsunek 8: Zoom on fifth function of De Jong 3

33 Rsunek 9: An overview of drop wave function.7 Drop wave function This is a multimodal test function. The given form of function has onl two variables and the following definition f(, ) = + cos( + ) (8) ( + ) + Test area is usuall restricted to the square 5. 5.,

34 Rsunek 3: Medium-scale view on drop wave function 3

35 Rsunek 3: Zoom on drop wave function 35

36 Shubert s function Rsunek 3: An overview of Shubert s function This is a multimodal test function. The given form of function has onl two variables and the following definition f(, ) = 5 i cos((i + ) + ) i= 5 i cos((i + ) + ), (9) Test area is usuall restricted to the square 5. 5., i= 36

37 Rsunek 33: Medium-scale view on Shubert s function 37

38 Rsunek 3: Zoom on Shubert s function 38

39 Rsunek 35: An overview of Shekel s foholes in D. f(, ) = m i= [( a j ) + ( b j ) + c j ].9 Shekel s foholes This is a multimodal test function. It has the following definition f() = m n ( [( j a ij ) + c j ]), () i= j= where (c i, i =,..., m), (a ij, j =,..., n, i =,..., m) are constant numbers fied in advance. It is recommended to set m = 3. 39

40 Rsunek 36: Zoom on Shekel s foholes in D. f(, ) = m i= [( a j) + ( b j ) + c j ]

41 . Deceptive functions A deceptive problem is a class of problems in which the total size of the basins for local optimum solutions is much larger than the basin size of the global optimum solution. Clearl, this is a multimodal function. The general form of deceptive function is given b the following formulae f() = [ n β n g i ( i )], () where β is an fied non-linearit factor. It has been defined in the literature at least three tpes of deceptive problems, depending the form of g i ( i ). A comple deceptive problem (Tpe III), in which the global optimum is located at i = α i, where α i is a unique random number between and depending on the dimension i. To this aim the following form of auiliar functions has been proposed α i + if 5 i α 5 i 5 α g i ( i ) = i if α 5 i < i α i () 5( α i ) i= α i + if α i < i +α i 5 +α if i < 5 i α i + 5 The two other tpes of deceptive problems (Tpes I and II) are special cases of the comple deceptive problem, with α i = (Tpe I), or α i = or at random (Tpe II) for each dimension i, i =,..., n. Clearl formulae () should be suitable adjusted for tpe I and II. For all three tpes of g i ( i ), the region with local optima is 5 n times larger than the region with a global optimum in the n-dimensional space. The number of local optima is n for Tpe I and Tpe II deceptive problems and 3 n for Tpe III.

42 Rsunek 37: Deceptive function of Tpe III in D. α =.3, α =.7, β =.

43 Rsunek 38: Deceptive function of Tpe III in D. α =.3, α =.7, β =.5 3