# Test problems

List of test problems available in BlueM.Opt. Some of them are taken from Moré et al. (1981)[1].

## Test problems

### Ackley function

Ackley function

The Ackley function is a non-convex function used as a performance test problem for optimization algorithms. It was proposed by David Ackley in his 1987 PhD dissertation[2].

On a 2-dimensional domain it is defined by:

\displaystyle{ \begin{align} f(x,y) = -20&{}\exp\left[-0.2\sqrt{0.5(x^2+y^2)}\,\right] \\ & {} -\exp\left[0.5\left(\cos 2\pi x + \cos 2\pi y \right)\right] + e + 20 \end{align} }[3]

Its global optimum point is

$\displaystyle{ f(0,0) = 0. }$

### Beale problem

Beale function (created with SensiPlot)
Beale problem being solved with PES (animation)

Finding the minimum of the Beale function[4].

• Parameters: 2
• Objective functions: 1

$\displaystyle{ f(x,y)=(1.5-x(1-y))^2+(2.25-x(1-y^2))^2+(2.625-x(1-y^3))^2 }$

Global minimum: f(3, 0.5) = 0

### Box

Box

Multicriteria test problem (circle) with two contraints

### CONSTR

CONSTR

Multicriteria test problem (convex) with two constraints

### Deb 1

Multicriteria test problem (convex)

### Dependent parameters

Parameter dependency: Y > X

### Flood Mitigation

Flood Mitigation

Multicriteria Problem Flood Mitigation and Hydropower Generation[5]

### Schwefel 2.4 problem

Find the minimum (xi=1, F(x)=0)

### Sine function

Fit parameters to the sine function

### Zitzler/Deb T1

Multicriteria test problem (convex)

### Zitzler/Deb T2

Multicriteria test problem (concave)

### Zitzler/Deb T3

Zitzler/Deb T3 being solved with PES (animation)

Multicriteria test problem (convex, non-continuous)

### Zitzler/Deb T4

Multicriteria test problem (convex)

## References

1. Moré, J.J., Garbow, B.S. and Hillstrom, K.E. (1981): Testing Unconstrained Optimization Software, ACM Transactions on Mathematical Software (TOMS) 7:1, p. 17-41, doi:10.1145/355934.355936
2. Ackley, D. H. (1987): "A connectionist machine for genetic hillclimbing", Kluwer Academic Publishers, Boston MA.
3. Bäck, Thomas (1996): "Artificial Landscapes". Evolutionary Algorithms in Theory and Practice. Oxford University Press. p. 142. doi:10.1093/oso/9780195099713.003.0008. ISBN 978-0-19-509971-3.
4. Beale, E. M. L. (1958): On an iterative method of finding a local minimum of a function of more than one variable. Technical Report 25, Statistical Techniques Research Group, Princeton University.
5. Sharma, Ajay (2008): Inflow prediction and optimal operation of reservoir system during flood by the combined application of ANN and different Optimization techniques. Master Thesis, Institute of Hydraulic and Water Resources Engineering, Technische Universität Darmstadt.