Comparison between Derivative-Free Optimization
Methods for DEB parameter estimation of different species
Fifth international symposium on
Dynamic Energy Budget theory: metabolic organization plays a role in
planetary stewardship
J.V.Morais, A.L.Custódio and G.M.Marques
[email protected]
[email protected]
1 June 2017
Summary
1
Problem definition
2
Nelder-Mead Simplex method
3
Directional Direct-Search methods
4
SID-PSM algorithm
5
DS-Random algorithm
6
Numerical results
7
Conclusions and future research
2
Problem definition
Biological
interpretation
Estimation
(data)
(parameters)
3
Problem definition
Biological
interpretation
Estimation
(parameters)
(data, results &
parameters)
...
Final parameters
4
Problem definition
DEB parameters estimation
𝑛
𝑛𝑖
min
𝒙 ∈ 𝛺
1
𝑑𝑖 =
𝑛𝑖
𝑛𝑖
𝑑𝑖𝑗
𝑗=1
𝑖=1 𝑗=1
(𝑝𝑖𝑗 𝒙 − 𝑑𝑖𝑗 )
𝜔𝑖𝑗 2
𝑑𝑖 + 𝑝𝑖2 (𝒙)
1
𝑝𝑖 (𝑥) =
𝑛𝑖
𝑛𝑖
𝑑𝑖𝑗 - Observed values
𝑝𝑖𝑗 (𝑥)
𝑗=1
2
𝑝𝑖𝑗 - Predicted values
𝑥 - Parameters
5
Derivative-Free Optimization Methods
The problem is to minimize a nonlinear function of several variables
6
Derivative-Free Optimization Methods
• The derivatives of this function are not available
The methods are known as Derivative-Free Optimization methods (DFO)
7
Nelder-Mead Simplex method
8
Nelder-Mead Simplex method
- Suited for unconstrained
Derivative-free Optimization problems
- Simplex can become flat or needle
shaped, causing convergence to non
stationary points
- Final result depends on starting point
McKinnon function
10
Directional Direct-Search methods
11
Directional Direct-Search methods – Poll step
12
Directional Direct-Search methods
13
Directional Direct-Search methods - Scalling
Sensitive to the dimension of the variables
1.
𝑥
𝑥𝑖𝑛𝑖
2. Log
𝑥
𝑥𝑖𝑛𝑖
14
SID-PSM method (A.L Custódio and L.N Vicente - 2007)
Search step
Poll step
- Optional
- Oportunistic polling
- Unnecessary for establishing
convergence properties
- Ordering of the poll directions
- Used to improve the numerical
efficiency of the method
- Based on quadratic polynomial
interpolation models
15
SID-PSM method
Convergence to some form of stationarity from arbitrary starting points
Why?
1. Convergence of step size parameters to zero
2. Control of the geometry of the sample sets with the use of positive
bases in the poll step
16
DS-Random method (C.W Royer and L.N Vicente - 2015)
Search step
No search step
Poll step
Random sets
−𝛻 f(x)
Such that:
−𝛻 f(x)
17
Problem definition
𝑛
𝑛𝑖
min
𝒙 ∈ 𝛺
𝑖=1 𝑗=1
(𝑝𝑖𝑗 𝒙 − 𝑑𝑖𝑗 )
𝜔𝑖𝑗 2
𝑑𝑖 + 𝑝𝑖2 (𝒙)
2
Constraints (Ω):
- Lower and upper bounds
𝐸𝐻𝑏 > 0
(maturity at birth)
- Linear inequalities
𝑝
𝐸𝐻𝑏 > 𝐸𝐻
(maturity at puberty > maturity at birth)
- Black-box constraint
𝑘𝑣𝐻𝑏
<
𝑓
𝑙𝑏2 (𝑙𝑏
𝑓+𝑔
+ 𝑔) (condition required for reaching birth)
18
Nelder-Mead Simplex method - Constraints
Extreme barrier approach
Consequences:
- Rapid degeneration of simplex vertices
- Unexplored area near to boundary
19
Directional Direct-Search methods - Constraints
Bounds
Linear inequalities
Black-box constraint – Extreme barrier approach
20
Results
Academic Problems (Dimension = 8)
1E+04
Number of function evaluations
1E+02
f(xfinal) - f(xoptimum)
1E+00
1E-02
1E-04
1E-06
1E-08
1E-10
1E-12
1E+02
1E+00
1E-14
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P1
P2
P4
P5
P6
P7
P8
P9
P10
P11
P12
Problem
Problem
NM Simplex
𝒇(𝒙𝒇𝒊𝒏𝒂𝒍 ) − 𝒇(𝒙𝒐𝒑𝒕𝒊𝒎𝒖𝒎 )
5,90
𝒏𝒆𝒗𝒂𝒍𝒔
P3
0%
SID-PSM
0,08
DS-RANDOM
2,36
-49,57 %
-18,62%
𝑓(𝑥𝑓𝑖𝑛𝑎𝑙 ) − 𝑓(𝑥𝑜𝑝𝑡𝑖𝑚𝑢𝑚 )
𝑛𝑒𝑣𝑎𝑙𝑠 𝑚𝑒𝑡ℎ𝑜𝑑 − 𝑛𝑒𝑣𝑎𝑙𝑠 𝑁𝑀 𝑆𝑖𝑚𝑝𝑙𝑒𝑥
𝑛𝑒𝑣𝑎𝑙𝑠 𝑁𝑀 𝑆𝑖𝑚𝑝𝑙𝑒𝑥
21
Results
Number of function evaluations
Academic Problems (Dimension = 20)
f(xfinal) - f(xoptimum)
1E+03
1E+01
1E-01
1E-03
1E-05
1E-07
1E-09
1E+04
1E+02
1E+00
P1
P2
P3
P4
P5
P6
P7
P8
P1
Problem
P2
P3
P4
P5
P6
P7
P8
Problem
NM Simplex
SID-PSM
DS-RANDOM
𝒇(𝒙𝒇𝒊𝒏𝒂𝒍 ) − 𝒇(𝒙𝒐𝒑𝒕𝒊𝒎𝒖𝒎 )
709,79
0,28
90,25
𝒏𝒆𝒗𝒂𝒍𝒔
0%
-81,46 %
-77,67%
𝑓(𝑥𝑓𝑖𝑛𝑎𝑙 ) − 𝑓(𝑥𝑜𝑝𝑡𝑖𝑚𝑢𝑚 )
𝑛𝑒𝑣𝑎𝑙𝑠 𝑚𝑒𝑡ℎ𝑜𝑑 − 𝑛𝑒𝑣𝑎𝑙𝑠 𝑁𝑀 𝑆𝑖𝑚𝑝𝑙𝑒𝑥
𝑛𝑒𝑣𝑎𝑙𝑠 𝑁𝑀 𝑆𝑖𝑚𝑝𝑙𝑒𝑥
22
Numerical Results
First scaling
Best lossfunction value
𝒇𝒙𝒇𝒊𝒏𝒂𝒍
Channa punctata
𝒏𝒆𝒗𝒂𝒍𝒔
𝒇𝒙𝒇𝒊𝒏𝒂𝒍
Pleuroxus aduncus
𝒏𝒆𝒗𝒂𝒍𝒔
𝒇𝒙𝒇𝒊𝒏𝒂𝒍
Pleuroxus striatus
𝒏𝒆𝒗𝒂𝒍𝒔
𝒇𝒙𝒇𝒊𝒏𝒂𝒍
Credipula fornicata
𝒏𝒆𝒗𝒂𝒍𝒔
𝒇𝒙𝒇𝒊𝒏𝒂𝒍
Pleuronectes platessa
𝒏𝒆𝒗𝒂𝒍𝒔
NM Simplex
1,9665E+00
2,0020E+03
1,0213E+00
3,0430E+03
3,6570E+00
3,1970E+03
3,8032E+00
2,4000E+03
1,4233E+01
3,0000E+03
Also best
SID-PSM
5,2173E-01
1,6002E+04
1,0218E+00
1,6000E+04
9,5702E-01
1,6008E+04
3,4061E+00
1,8088E+04
5,8704E+00
3,0008E+04
Best number of functions evaluations
DS-RANDOM
3,1470E-01
1,6000E+04
2,9992E+00
1,3745E+04
1,1759E+00
1,1789E+04
3,5250E+00
2,4000E+04
3,0093E+00
3,0000E+04
Dimension
8
8
8
12
15
23
Numerical Results
Second scaling
Best lossfunction value
𝒇𝒙𝒇𝒊𝒏𝒂𝒍
Channa punctata
𝒏𝒆𝒗𝒂𝒍𝒔
𝒇𝒙𝒇𝒊𝒏𝒂𝒍
Pleuroxus aduncus
𝒏𝒆𝒗𝒂𝒍𝒔
𝒇𝒙𝒇𝒊𝒏𝒂𝒍
Pleuroxus striatus
𝒏𝒆𝒗𝒂𝒍𝒔
𝒇𝒙𝒇𝒊𝒏𝒂𝒍
Credipula fornicata
𝒏𝒆𝒗𝒂𝒍𝒔
𝒇𝒙𝒇𝒊𝒏𝒂𝒍
Pleuronectes platessa
𝒏𝒆𝒗𝒂𝒍𝒔
NM Simplex
1,9665E+00
2,0020E+03
1,0213E+00
3,0430E+03
3,6570E+00
3,1970E+03
3,8032E+00
2,4000E+03
1,4233E+01
3,0000E+03
Also best
SID-PSM
1,9621E-03
1,7520E+03
1,9999E+00
1,2710E+03
2,9997E+00
1,7240E+03
5,0293E+00
2,4004E+04
2,2216E+00
2,4330E+03
Best number of functions evaluations
DS-RANDOM
7,9500E-01
1,4784E+04
4,2446E-01
6,9800E+02
6,4884E-01
3,8070E+03
2,8043E+00
2,4000E+04
2,0264E+00
6,1560E+03
Dimension
8
8
8
12
15
24
Simplex 2.0
Minimum
Numerical Results
First scaling
Best lossfunction value
Channa punctata
Pleuroxus aduncus
Pleuroxus striatus
Credipula fornicata
Pleuronectes platessa
𝒇𝒙𝒇𝒊𝒏𝒂𝒍
𝒏𝒆𝒗𝒂𝒍𝒔
𝒇𝒙𝒇𝒊𝒏𝒂𝒍
𝒏𝒆𝒗𝒂𝒍𝒔
𝒇𝒙𝒇𝒊𝒏𝒂𝒍
𝒏𝒆𝒗𝒂𝒍𝒔
𝒇𝒙𝒇𝒊𝒏𝒂𝒍
𝒏𝒆𝒗𝒂𝒍𝒔
𝒇𝒙𝒇𝒊𝒏𝒂𝒍
𝒏𝒆𝒗𝒂𝒍𝒔
Also best
NM Simplex
SID-PSM
1,9665E+00
2,0020E+03
1,0213E+00
3,0430E+03
3,6570E+00
3,1970E+03
3,8032E+00
2,4000E+03
1,4233E+01
3,0000E+03
1,9621E-03
1,7520E+03
1,9999E+00
1,2710E+03
2,9997E+00
1,7240E+03
5,0293E+00
2,4004E+04
2,2216E+00
2,4330E+03
Best number of functions evaluations
DS-RANDOM “Simplex 2.1”
7,9500E-01
1,4784E+04
4,2446E-01
6,9800E+02
6,4884E-01
3,8070E+03
2,8043E+00
2,4000E+04
2,0264E+00
6,1560E+03
6,3811E-31
1,2401E+04
9,8659E-05
1,5507E+04
5,3102E-05
1,5543E+04
2,4933E+00
2,1161E+04
1,2404E+00
1,3398E+04
26
Numerical Results
Second scaling
Best lossfunction value
Channa punctata
Pleuroxus aduncus
Pleuroxus striatus
Credipula fornicata
Pleuronectes platessa
fvalue
fevals
fvalue
fevals
fvalue
fevals
fvalue
fevals
fvalue
fevals
Also best
Best number of functions evaluations
NM Simplex
SID-PSM
DS-RANDOM
1,9665E+00
2,0020E+03
1,0213E+00
3,0430E+03
3,6570E+00
3,1970E+03
3,8032E+00
2,4000E+03
1,4233E+01
3,0000E+03
1,9621E-03
1,7520E+03
1,9999E+00
1,2710E+03
2,9997E+00
1,7240E+03
5,0293E+00
2,4004E+04
2,2216E+00
2,4330E+03
7,9500E-01
1,4784E+04
4,2446E-01
6,9800E+02
6,4884E-01
3,8070E+03
2,8043E+00
2,4000E+04
2,0264E+00
6,1560E+03
“Simplex
Turbo”
6,3811E-31
1,2401E+04
9,8659E-05
1,5507E+04
5,3102E-05
1,5543E+04
2,4933E+00
2,1161E+04
1,2404E+00
1,3398E+04
27
Results
Pars_init_mypet
𝐩
𝐄𝐇
Pars_init_Channa_punctata
Optimum
SIMPLEX
SID-PSM
DS-RANDOM
SIMPLEX 2.0
z
𝐛
𝐄𝐇
28
Conclusions
• Possible existence of different local minimums
• SID-PSM and DS-RANDOM present better performance than Nelder-Mead
Simplex:
• in academic problems, both for final objective function value and total
number of function evaluations
• in species problems, in terms of lossfunction value
• “Simplex 2.0” presents the best lossfunction values
29
Future research
• Explore global derivative-free optimization
• Adapt “Simplex 2.0” approach to SID-PSM method
• Develop the Nelder-Mead Simplex for constrained optimization
30