Optimization Functions For this project, 40 test challeng

Homework answers / question archive / Optimization Functions For this project, 40 test challenging benchmark functions (Liang, Qu et al

Optimization Functions For this project, 40 test challenging benchmark functions (Liang, Qu et al

Computer Science

Share With

Optimization Functions

For this project, 40 test challenging benchmark functions (Liang, Qu et al. 2013, Liang, Qu et al. 2013, Wahab, Nefti-Meziani et al. 2015) are collected. You are required to implement nature-inspired algorithm to find the optimal value of any 20 optimization functions of your choice from the list given below for your class project.

There are several characteristics of the test functions that feature the complexity of the functions’ landscape like modality, separability, scalability (Dieterich and Hartke 2012). Modality is defined by the number of ambiguous peaks in the function landscape. A function is called separable if the variables of the solution are independent and can be optimized separately. On the other hand, the inseparable class of functions are highly difficult to solve as the variables are interrelated and affect each other. The functions that are extendable to arbitrary dimensionality are known as scalable function, otherwise are called non-scalable. The scalability often introduces high extent of complexity to the search space such as some test function landscapes become multimodal from unimodal when the number of dimensions is scaled up.

There are some additional features that can make a function landscape even more challenging while searching for minima such as flat surface (less informative area), basins and narrow curved valley. However, some algorithms are found to exploit several shortcomings (Liang, Suganthan et al. 2005) of popularly used test functions such as the global optimum with same values for different variables (Zhong, Liu et al. 2004), global optimum lying in the center of the landscape, at the origin (Zhong, Liu et al. 2004), along the axes (Leung and Wang 2001, Van den Bergh and Engelbrecht 2004), on the bounds (Coello, Pulido et al. 2004) etc. Thereafter, the conventional test functions are rotated and/or shifted before using them to test optimization algorithms (Liang, Suganthan et al. 2005, Qin, Huang et al. 2009, Liang, Qu et al. 2013, Liang, Qu et al. 2013, Yu and Li 2015). In addition to using rotated and shifted functions, we hybridized functions according to (Liang, Qu et al. 2013) to generate highly complex real-world optimization problems to test the extended-capacity of the algorithms. We collected the rotation matrix, shifting vector and shuffled-indices vector for rotation, shifting and hybridization, respectively from a shared link of data used in CEC 2014[1].

Among 40 functions, 27 are classical functions whereas 13 functions are rotated and/or shifted and hybridized to construct modified functions. Depending on the three basic properties (modality, separability, and scalability) mentioned above, we classify the 27 functions under consideration into six groups (Group I through VI). The rest of the 13 modified functions are categorized into three groups according to their complexities. The definition of the functions under different groups with their corresponding search ranges (lb, ub

) of solution variable (X

), global optimum (X*

) and the function value at global optimum, f(X*

) are listed as following:

Group I: Unimodal, Separable and Scalable functions

Sphere function, fSphereX=i=1dxi2

lb, ub=-100, 100d

, X*=[0, …, 0]T

, fSphereX*=0

Cigar or Bent Cigar function, fCigarX=x12+106i=2dxi2

lb, ub=-100, 100d

, X*=[0, …, 0]T

, fCigarX*=0

Discus function, fDiscusX=106x12+i=2dxi2

lb, ub=-100, 100d

, X*=[0, …, 0]T

, fDiscusX*=0

Rotated Hyper-ellipsoid (RHE) function, fRHEX=i=1dj=1ixj2

lb, ub=-100, 100d

, X*=[0, …, 0]T

, fRHEX*=0

Group II: Unimodal, Inseparable and Non-scalable

Zettl function, fZettlX=14x1+x12-2x1+x222

lb, ub=-5, 5d

, X*=-0.0299, 0T

, fZettlX*=-0.003791237

Leon function, fLeonX=100x2-x122+1-x12

lb, ub=-1.2, 1.2d

, X*=[1, 1]T

, fLeonX*=0

Easom function, fEasomX=-cosx1cosx2e[-(x1-π)2-(x2-π)2]

lb, ub=-100, 100d

, X*=[π, π]T

, fEasomX*=-1

Group III: Unimodal, Inseparable and Scalable

Zakharov function, fZakharovX=i=1dxi2+12i=1dixi2+12i=1dixi4

lb, ub=-5, 10d

, X*=[0, …, 0]T

, fZakharovX*=0

Schwefel 1.2 function, fSchwefel1.2X=i=1dj=1ixj2

lb, ub=-100, 100d

, X*=[0, …, 0]T

, fSchwefel1.2X*=0

Schwefel 2.2 function, fSchwefel2.2X=i=1dxi+i=1dxi

lb, ub=-100, 100d

, X*=[0, …, 0]T

, fSchwefel2.2X*=0

Group IV: Multimodal, Separable and Scalable

Rastrigin function

fRastriginX=10d+i=1dxi2-10cos2πxi

lb, ub=-5.2, 5.2d

, X*=[0, …, 0]T

, fRastriginX*=0

Schwefel 2.26 function

fSchwefel2.6X=418.9829d-i=1d xisinxi

lb, ub=-500, 500d

, X*=[420.9687, …, 420.9687]T

, fSchwefel2.6X*=0

Michalewicz function

fMichalewiczX=-i=1dsinxisin2sixi2π

steepness, s=10

lb, ub=[0, π]d

, X*=[2.20, 1.57]T, d=2

fMichalewiczX*=-1.8013

,-4.687658,-9.66015

, d=2, 5, 10

Styblinski-Tang (ST) function

fSTX=12i=1dxi4-16xi2+5xi

lb, ub=[-5, 5]d

, X*=[-2.903534, …, -2.903534]T

fSTX*=-39.16599d

Group V: Multimodal, Inseparable and non-Scalable

Schaffer F2 function

fScafferF2X=0.5+sin2x12-x22-0.51+0.001x12+x222

lb, ub=[-100, 100]d

, X*=[0, 0]T

, fScafferF2X*=0

Schaffer F6 function

fScafferF6X=0.5+sin2x12+x22-0.51+0.001x12+x222

lb, ub=[-100, 100]d

, X*=[0, 0]T

, fScafferF6X*=0

Bird function

fBirdx=x1-x22+sinx1e1-cosx22+ cosx2e1-sinx12

lb, ub=[-2π, 2π]d

, X*=[4.701056, 3.152946]T

, X*=[-1.582142, -3.130247]T

, fBirdx*=-106.7645367198034

Levy 13 function

fLevy13X=sin23πx1+x1-121+sin23πx2

+ x2-121+sin22πx2

lb, ub=[-10, 10]d

, X*=[1, 1]T

, fLevy13X*=0

Carrom Table (CT) function

fCarromTableX=-130e21-x12+x22πcos2x1cos2x2

lb, ub=[-10, 10]d

, X*=[±9.646157, ±9.646157]T

fCarromTableX*=-24.1568155

Group VI: Multimodal, Inseparable and Scalable

Ackley function

fAckelyX=-ae-b1di=1dxi2-e1di=1dcoscxi+a+e1

a=20,b=0.2,c=2π

lb, ub=[-32, 32]d

, X*=[0, …, 0]T

, fAckleyX*=0

Rosenbrock function

fRosenbrockX=i=1d-1100xi+1-xi22+xi-12

lb, ub=[-30, 30]d

, X*=[1, …, 1]T

, fRosenbrockX*=0

Griewank function

fGriewankX=i=1dxi24000-i=1dcosxii+1

lb, ub=-600, 600d

, X*=[0, …, 0]T

, fGriewankX*=0

Sine Envelope Sine Wave (SESW) function

fSESWX=i=1d-1sin2x12+x22-0.51+0.001x12+x222+0.5

lb, ub=[-100, 100]d

, X*=[0, …, 0]T

, fSESWX*=0

Trigonometric function

fTrigonometricX=i=1dm+i1-cosxi-sinxi-j=1dcosxj2

lb, ub=[-1000, 1000]d

, X*=[0, …, 0 ]T

, fTrigonometricX*=0

Levy function

fLevyX=sin2πy1+i=1d-1yi-121+10sin2yi+1+

yn-121+sin22πyn, yi=1+14xi+1

lb, ub=[-50, 50]d

, X*=[-1,…, -1 ]T

, fLevyX*=0

Schaffer F7 function

fScafferF7X=1d-1i=1d-1yi+sin50yi0.2yi2, yi=xi2+xi+12

lb, ub=[-100, 100]d

, X*=[0, …, 0 ]T

, fScafferF7X*=0

Lunacek Function

fLunacekx=mini=1dxi-μ12,1.d+si=1dxi-μ22+

10i=1d1-cos2πxi-μ1

μ1=2.5, s=1-12d+20-8.2,μ2=-μ12-1s

lb, ub=[-10, 10]d

, X*=[μ1,…, μ1 ]T

, fLunacekX*=0

Group VII: Rotated functions

Rotation of the landscape of the functions using an orthogonal rotation matrix (R

) (Salomon 1996) can prevent the optimum from lying along the coordinate axes. Here, we rotate 7 scalable (2 unimodal and 5 multimodal) functions.

Rotated Cigar function, fR-CigarZ=fCigarZ

, Z=R×X

lb, ub=-100, 100d

, X*=[0, …, 0]T

, fR-CigarX*=0

Rotated Discus function, fR-DiscusZ=fDiscusZ

, Z=R×X

lb, ub=-100, 100d

, X*=[0, …, 0]T

, fR-DiscusX*=0

Rotated Rastrigin function

fR-RastriginZ=fRastriginZ

, Z=R×(X×(5.2100))

lb, ub=-5.2, 5.2d

, X*=[0, …, 0]T

, fR-RastriginX*=0

Rotated Ackley function

fR-AckleyZ=fAckleyZ

, Z=R×(X×(32100))

lb, ub=-32, 32d

, X*=[0, …, 0]T

, fR-AckleyX*=0

Rotated Griewank function

fR-GriewankZ=fGriewankZ

, Z=R×(X×(600100))

lb, ub=-600, 600d

, X*=[0, …, 0]T

, fR-GriewankX*=0

Rotated Schaffer F7 function

fR-SchafferF7Z=fSchafferF7Z

, Z=R×X

lb, ub=-100, 100d

, X*=[0, …, 0]T

, fR-SchafferF7X*=0

Rotated Lunacek function

fR-LunacekZ=fLunacekZ

, Z=R×(X×(10100))

lb, ub=-10, 10d

, X*=[μ1,…, μ1 ]T

, fR-LunacekX*=0

Group VIII: Shifted and Rotated functions

Shifting operation moves the global optimum to a new random position (onew

) from the old optimum (oold

) using Eq. 1 to avoid the limitation of having same values of the variables at optima or having optimum at the center.

fZ, Z=R×(X-onew+oold)

(1)

Shifted and Rotated Rastrigin function

fSR-RastriginZ=fRastriginZ

Z=R×((X-onew+oold)×(5.2100))

lb, ub=-5.2, 5.2d

, X*=[0, …, 0]T

, fSR-RastriginX*=0

Shifted and Rotated Ackley function

fSR-AckleyZ=fAckleyZ

Z=R×((X-onew+oold)×(32100))

lb, ub=-32, 32d

, X*=[0, …, 0]T

, fSR-AckleyX*=0

Shifted and Rotated Lunacek function

fSR-LunacekZ=fLunacekZ

Z=R×((X-onew+oold)×(10100))

lb, ub=-10, 10d

, X*=[μ1,…, μ1 ]T

, fSR-LunacekX*=0

Group IX: Hybrid functions

Hybrid function defined by Eq. 2 represents real-world optimization problems where different subcomponents of the variables can have different characteristics. For these functions, the variables are randomly shuffled and divided into subcomponents and then different basic functions are used for different subcomponents. In our hybrid functions, we have further rotated the basic functions.

fZ=f1R1Z1+f2R2Z2+…+fn(RnZn)