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Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M0
Optimization of Mechanical Systems
Prof. Dr.-Ing. Prof. E.h. P. Eberhard (peter.eberhard@itm.uni-stuttgart.de)
M.Sc. Elizaveta Shishova (elizaveta.shishova@itm.uni-stuttgart.de)
Institute of Engineering and Computational Mechanics
Pfaffenwaldring 9, 4th floor
70563 Stuttgart
Lecture information
Course: This semester studies take place online. The recorded lectures will be published on the ILIAS platform according to the course schedule and will remain available for one week.
Language: The lecture is taught in English.
Audience: Students of the interdisciplinary graduate program of study COMMAS and students of the programs of study Mechanical Engineering, Mechatronics, Engineering Cybernetics, Technology Management, Automotive and Engine Technology, Mathematics, etc.
Credits: 3 ECTS (2 SWS)
WT 20/21: 2.11.20 - 13.2.21, Christmas break: 23.12.20 - 06.01.21
Dates of class:
Wednesday, 9.45 - 11.15 a.m., weekly, first class November 4, 2020
Internet: The course web page can be found at
www.itm.uni-stuttgart.de/en/courses/lectures/optimization-of-mechanical-systems
Course material: Handouts can be downloaded at the course web page.
Exercises: Exercises are an incorporated part of the lecture.
Office hours: The questions will be answered during online consultations, which will be orga-nized during the semester when the need arises. To request the consultation please contact the supervisor of the course at elizaveta.shishova@itm.uni-stuttgart.de.
Exam: The exam is scheduled for end of February 2020 or beginning of March 2020. Exact date and room will be announced later. The exam is mandatory for COMMAS students. All students please register with the examination office (Prüfungsamt).
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M1.1
Optimization of Mechanical Systems
- Content of lecture -
1 Introduction
1.1 Motivational Examples
1.2 Formulation of the Optimization Problem
1.3 Classification of Optimization Problems
2 Scalar Optimization
2.1 Optimization Criteria
2.2 Standard Problem of Nonlinear Optimization
3 Sensitivity Analysis
3.1 Mathematical Tools
3.2 Numerical Differentiation
3.3 Semianalytical Methods
3.4 Automatic Differentiation
4 Unconstrained Parameter Optimization
4.1 Basics and Definitions
4.2 Necessary Conditions for Local Minima
4.3 Local Optimization Strategies (Deterministic Methods)
4.4 Global Optimization Strategies (Stochastic Methods)
5 Constrained Parameter Optimization
5.1 Basics and Definitions
5.2 Necessary Conditions for Local Minima
5.3 Optimization Strategies
6 Multicriteria Optimization
6.1 Theoretical Basics
6.2 Reduction Principles
6.3 Strategies for Pareto-Fronts
7 Application Examples and Numerical Tools
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M1.2
Bibliography
Optimization:
S.K. Agrawal and B.C. Fabien: Optimization of Dynamic Systems. Dordrecht: Kluwer Ac-
ademic Publishers, 1999.
M.P. Bendsoe and O. Sigmund: Topology Optimization: Theory, Methods and Applica-
tions. Berlin: Springer, 2004.
D. Bestle: Analyse und Optimierung von Mehrkörpersystemen - Grundlagen und rech-
nergestützte Methoden. Berlin: Springer, 1994.
D. Bestle and W. Schiehlen (Eds.): Optimization of Mechanical Systems. Proceedings of
the IUTAM Symposium Stuttgart, Dordrecht: Kluwer Academic Publishers, 1996.
P. Eberhard: Zur Mehrkriterienoptimierung von Mehrkörpersystemen. Vol. 227, Reihe 11,
Düsseldorf: VDI–Verlag, 1996.
R. Fletcher: Practical Methods of Optimization. Chichester: Wiley, 1987.
R. Haftka and Z. Gurdal: Elements of Structural Optimization. Dordrecht: Kluwer Aca-
demic Publishers, 1992.
E.J. Haug and J.S. Arora: Applied Optimal Design: Mechanical and Structural Systems.
New York: Wiley, 1979.
J. Nocedal and S.J. Wright: Numerical Optimization. New York: Springer, 2006.
A. Osyczka: Multicriterion Optimization in Engineering. Chichester: Ellis Horwood, 1984.
K. Schittkowski: Nonlinear Programming Codes: Information, Tests, Performance. Lect.
Notes in Econ. and Math. Sys., Vol. 183. Berlin: Springer, 1980.
W. Stadler (Ed.): Multicriteria Optimization in Engineering and in the Sciences. New
York: Plenum Press, 1988.
G.N. Vanderplaats: Numerical Optimization Techniques for Engineering Design. Colora-
do Springs: Vanderplaats Research & Development, 2005.
P. Venkataraman: Applied Optimization with MATLAB Programming. New York: Wiley,
2002.
T.L. Vincent and W.J. Grantham: Nonlinear and Optimal Control Systems. New York:
Wiley, 1997.
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M2.1
Optimization of mechanical systems
classical/engineering approach
analytical/numerical approach
intuition, experience of the design engi-
neer and experiments and fiddling with
hardware prototypes at the end of the de-
sign process
intuition, experience of the design engineer
and virtual prototypes based on computer
simulations throughout the whole design
process
sequential process
concurrent engineering
only small changes of the design pos-
sible to influence the system behavior
more fundamental changes might re-
quire the reconsideration of all previous
design steps
“optimal” solution without quantitative
objectives
hardware experiments are costly and
time intensive (only few are possible)
provides many design degrees of free-
dom at beginning of the design process
parameter studies can be executed fast
and easy
systematic way to find optimal solution
with respect to defined criteria
cost efficient
shortening of the development time
initial design for hardware experiments is
already close to optimum
idea concept
design
calculation
prototype
experiments product
idea
concept
CAD design
simulation
prototype
experiments product
optimization
production preparation
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M2.2
Optimization is a part of any engineering design process. Furthermore optimization is also
extensively used in many other disciplines, such as e.g. industrial engineering, logistics or
economics.
The focus of this class is on optimization of mechanical systems using the analyti-
cal/numerical approach. The concepts and methods are presented in a general manner,
such that they can be applied to general optimization problems.
The systematic formulation of an optimization problem requires the answers of three basic
questions:
1. What should be achieved by the optimization?
2. Which changeable variables can influence the optimization goals?
3. Which restrictions apply to the system?
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M3
Iterative solution of the standard problem of parameter optimization
𝑓 𝐩 𝑖 , 𝑔 𝐩 𝑖 , ℎ 𝐩 𝑖
initial design
𝑖 = 0, 𝐩 0
evaluation of the performance
propose a better design
𝐩 𝑖+1 , 𝑖 = 𝑖 + 1
performance satisfying?
yes design “optimal”
no
evaluation of functions
performance functions
constraints
simulation model
calculation of gradients
performance functions
constraints
𝐩 𝑖
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M4
Integrated Modelling and Design Process
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M5
Classification of optimization problems
optimization
topology function parameters
scalar optimization
multi-criteria optimization
min𝐩∈𝑃
𝐟 𝐩
unconstrained min𝐩∈𝑅ℎ
𝑓 𝐩 constrained
min𝐩∈𝑃
𝑓 𝐩
𝑃 = 𝐩 ∈ Rh 𝐠 𝐩 = 𝟎, 𝐡 𝐩 ≤ 𝟎, 𝐩u ≤ 𝐩 ≤ 𝐩0
scalarization
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M6.1
ℓ
x
y
1
2
B
F,uy
uy
Optimization in Engineering Applications
Static Analysis – Truss Framework
A simple truss structure, shown to the right, shall be optimized. The truss consists of two round bars with Young’s modulus E =2,1 ∙ 1011 N m2⁄ and density ρ = 2750 kg m3⁄ . As design variables the radii of the bars r1 and r2 are chosen
𝐩 = [r1r2
] , whereby 2 mm ≤ ri ≤ 5mm, i = 1,2.
Applying a force F = 100 N at point B a displacement 𝐮 is caused, which can be computed using the finite element method
𝐊 ⋅ 𝐮 = 𝐪, with the stiffness matrix
𝐊 =E
ℓ2√2 [
A2 A12√2 + A2A2 A2
] ,
the vector of nodal coordinates 𝐮 = [ux uy]T and the vector of applied forces 𝐪 = [0 F]T. In an optimization the displacement uy shall be minimized. Thus, the scalar objective function reads
ψ(𝐩) = uy =√2
2
Fℓ
E(
4r12 + √2r2
2
πr12r2
2 ).
Evaluating ψ(𝐩) in the feasible design space returns the following results.
It can be seen that by increasing the radii, the displacement is reduced. Thus, if there are no additional con-straint equations, such as mass restriction, the solution of the minimization problem is p1
∗ = p2∗ = 5 mm and
ψ(𝐩∗) = 0.07 mm.
2
3
4
5
0
0.1
0.2
0.3
0.4
0.5
23
45
p2 [mm]
ψ(p
) [m
m]
p1 [mm]
ℓ
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M6.2
Dynamic Analysis – Slider-Crank Mechanism
Not only static but also dynamic problems are analyzed and optimized in engineering. For instance, using
the method of multibody systems the slider-crank mechanism shown below is modeled. The multibody sys-
tems consists of the crank (m1 = 0.24kg, J1 = 0.26 kg m2), the piston rod (m2 = 0.16kg, J2 = 0.0016 kg m
2) as
well as the slider block (m3 = 0.46kg). The crank angle is assumed to rotate at constant angular velocity φ̇ =
8 Hz and, thus, the motion of the mechanism is clearly defined.
Performing a simulation for the time domain t ∈ [0, 3]s, the resulting reaction force between the crank and
the inertial frame, which is defined as
F(p, t) = √Fx2(p, t) + Fy
2(p, t),
can be computed. For two different values p = −0.02 m and p = −0.03 m the resulting reaction forces F(p, t)
are displayed below.
Performing an optimization, F(p, t) shall be minimized. However, in contrast to static problems, first the tran-
sient system response has to be converted into a scalar value. Therefore, the time-dependent resulting reac-
tion force F(p, t) is integrated over the simulation time t. Thus, it holds for the objective function
ψ(p) = ∫ F(p, t)dt
t1
t0
= ∫ √Fx2(p, t) + Fy
2(p, t)dt.
3s
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 1 2 3
F(p
, t)
time t
F(-0.02, t)
F(-0.03, t)
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M6.3
Then, evaluating the objective function ψ(p) for p ∈ [−0.02 − 0.01] m the local minimum can be determined
as p∗ ≈ −0.017 and ψ(p∗) ≈ 0.646.
Dynamic Analysis – Planar 2-Arm Welding Robot
A further example for the optimization of dynamic systems is the planar 2-arm welding robot shown below.
For the welding process the tool center point (TCP) has to follow a semi-circular trajectory (—) within 3 sec-
ond. The joint angles φ and ψ are modeled as rheonomic constraints, i.e. φ = φ(t) and ψ = ψ(t). However,
due to joint elasticity, which is modeled by rotational springs with stiffness 𝑐, there are additional rotations of
the two arms Δφ and Δψ. These additional rotations represent the generalized degrees of freedom of the
system 𝐲 = [Δφ Δψ]T. As a consequence, the actual trajectory of the TCP (- - -) differs from the desired
trajectory.
0.64
0.66
0.68
0.7
0.72
0.74
0.76
0.78
-0.0225 -0.0175 -0.0125 -0.0075
ψ(p
)
p [m]
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M6.4
By varying the design variables p the center of gravity of the second arm is changed and, thereby, the track-
ing error of the TCP shall be reduced. The tracking error F is determined by the Euclidean distance between
the actual position 𝐫a = [xa ya]T and the desired position 𝐫d = [xd yd]
T and is computed as
F(p, 𝐲, t) = √(xa(p, 𝐲, t) − xd(t))2
+ (ya(p, 𝐲, t) − yd(t))2
.
It can be seen, that not only the tracking error F but also that the generalized degrees of freedom Δφ and Δψ
depend on the design variable p.
To obtain a scalar objective function, the tracking error F is integrated over the simulation time
ψ(p) = ∫ F(p, 𝐲, t)dt
t1
t0
= ∫ √(xa(p, 𝐲, t) − xd(t))2
+ (ya(p, 𝐲, t) − yd(t))2
dt
3s
0
.
Evaluating the objective function for p ∈ [−0.02 − 0.01], a local minimum can be graphically determined at
p∗ ≈ −0.04 and ψ(p∗) ≈ 0.0039.
-1
0
1
2
3
4
5
6
7
8
9
10
0 1 2 3
F[m
m]
time t
F(0.4)
F(0.5)
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0 1 2 3
Δφ
, Δψ
time t
Δφ(0.4) Δψ(0.4)
Δφ(0.5) Δψ(0.5)
0.002
0.003
0.004
0.005
0.006
0.007
0.008
-0.2 -0.1 0 0.1 0.2
ψ(p
)
p [mm]
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M7.1
Geometric visualization in 2D
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M7.2
inequality constraints
equality constraints
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M8.1
Matrix Algebra and Matrix Analysis
vector 𝐱 ∈ ℝn : 𝐱 = [x1 … xn] , xi ∈ ℝ ,
matrix 𝐀 ∈ ℝm×n : 𝐀 = [
A11 … A1n⋮ ⋮Am1 … Amn
] , Aij ∈ ℝ .
Basic Operations
operation notation components mapping
addition 𝐂 = 𝐀 + 𝐁 Cij = Aij + Bij ℝm×n ×ℝm×n → ℝm×n
multiplication
with scalar 𝐂 = α 𝐀 Cij = α Aij ℝ ×ℝm×n → ℝm×n
transpose 𝐂 = 𝐀T Cij = Aji ℝm×n → ℝn×m
differentiation
𝐂 =d
dt𝑨
𝐂 =∂𝐱
∂𝐲
Cij =d
dtAij
Cij =∂xi∂yj
ℝm×n → ℝm×n
ℝm ×ℝn → ℝm×n
matrix multiplication 𝐲 = 𝐀 ∙ 𝐱
𝐂 = 𝐀 ∙ 𝐁
yi =∑Aikk
xk
Cij =∑Aikk
Bkj
ℝm×n × ℝn → ℝm
ℝm×n ×ℝn×p → ℝm×p
scalar product
(dot/inner product) α = 𝐱 ∙ 𝐲 α =∑xk
k
yk ℝn × ℝn → ℝ
outer product 𝐀 = 𝐱 ⨂ 𝐲 Aij = xi yj ℝm ×ℝn → ℝm×n
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M8.2
Basic Rules
addition: 𝐀 + (𝐁 + 𝐂) = (𝐀 + 𝐁) + 𝐂
𝐀 + 𝐁 = 𝐁 + 𝐀
multiplication with scalar: α(𝐀 ∙ 𝐁) = (α 𝐀) ∙ 𝐁 = 𝐀 ∙ (α 𝐁)
α(𝐀 + 𝐁) = α 𝐀 + α 𝐁
transpose: (𝐀T)T = 𝐀
(𝐀 + 𝐁)T = 𝐀T + 𝐁T
(α 𝐀T)T = α 𝐀
(𝐀 ∙ 𝐁)T = 𝐁T ∙ 𝐀T
differentiation: d
dt(𝐀 + 𝐁) =
d
dt𝐀 +
d
dt𝐁
d
dt(𝐀 ∙ 𝐁) = (
d
dt𝐀) ∙ 𝐁 + 𝐀 ∙ (
d
dt𝐁)
d
dt𝐟(𝐱) =
∂𝐟
∂𝐱∙d𝐱
dt
matrix multiplication: 𝐀 ∙ (𝐁 + 𝐂) = 𝐀 ∙ 𝐁 + 𝐀 ∙ 𝐂
𝐀 ∙ (𝐁 ∙ 𝐂) = (𝐀 ∙ 𝐁) ∙ 𝐂
𝐀 ∙ 𝐁 ≠ 𝐁 ∙ 𝐀 in general
scalar produkt: 𝐱 ∙ 𝐲 = 𝐲 ∙ 𝐱
𝐱 ∙ 𝐲 ≥ 0 ∀ 𝐱, 𝐱 ∙ 𝐱 = 0 ⇔ 𝐱 = 0
𝐱 ∙ 𝐲 = 0 ⇔ 𝐱, 𝐲 orthogonal
Quadratic Matrices
identity matrix 𝐄 = [1 ⋯ 0⋮ ⋱ ⋮0 ⋯ 1
]
diagonal matrix 𝐃 = diag{Di} = [D1 ⋯ 0⋮ ⋱ ⋮0 ⋯ Dn
]
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M8.3
inverse matrix 𝐀−1 ∙ 𝐀 = 𝐀 ∙ 𝐀−1 = 𝐄
(𝐀 ∙ 𝐁)−1 = 𝐁−1 ∙ 𝐀−1
orthogonal matrix 𝐀−𝟏 = 𝐀T
symmetric matrix 𝐀 = 𝐀T
skew symmetric matrix 𝐀 = −𝐀T
decomposition
𝐀 =𝟏
𝟐(𝐀 + 𝐀T)⏟ 𝐁=𝐁T
+1
2(𝐀 − 𝐀T)⏟ 𝐂=−𝐂T
skew symmetric 3 × 3 matrix �̃� = [
0 −a3 a2a3 0 −a1−a2 a1 0
]
�̃� ∙ 𝐛 =̂ 𝐚 × 𝐛
�̃� ∙ 𝐛 = −�̃� ∙ 𝐚
�̃� ∙ �̃� = 𝐛𝐚 − (𝐚 ∙ 𝐛)𝐄
(�̃� ∙ 𝐛)̃ = 𝐛 𝐚 − 𝐚 𝐛
symmetric, positive definite matrix: 𝐱 ∙ 𝐀 ∙ 𝐱 > 0 ∀ 𝐱 ≠ 𝟎
eigenvalues λα > 0, α = 1(1)n
symmetric, positive semidefinite matrix: 𝐱 ∙ 𝐀 ∙ 𝐱 ≥ 0 ∀ 𝐱
eigenvalues λα ≥ 0, α = 1(1)n
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M9
Deterministic optimization strategies
Optimization algorithms are iterative and efficient strategies work in two-steps:
initial design
𝑖 = 0, 𝐩(0)
evaluation of the performance of the current
design 𝐩(i)
propose a better design
1. search direction 𝐬(i)
2. line search α(i)
no
performance satisfying?
yes design “optimal”
𝐩(i+1) = 𝐩(i) + α𝐬(i)
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M10
Deterministic Optimization Strategies
optimization
strategy
search direction
model
order
information
order
search parallel
to the axes 𝐬(v) = 𝐞v mod h
gradient based
method 𝐬(v) = −∇f (v)
conjugate
gradient
method
𝐬(0) = −∇f (0)
𝐬(v+1) = −∇f (v+1) +‖∇f (v+1)‖
2
‖∇f (v)‖2𝐬(v)
Newton
method 𝐬(v) = −(∇2f (v))
−1∙ ∇f (v)
Example: quadratic criteria function
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M11
Line Search
possible requirements for line search
exact minimization:
f ′(α(v)) = 𝐬(v) ∙ ∇f (v+1) =!
0
- many function evaluations inefficient
sufficient improvement:
in order to avoid infinitesimally small improvements,
some conditions have been proposed,
e.g. Wolfe-Powell conditions
f(α) ≤!
f(0) + αρf ′(0), ρ ∈ (0,1), e.g. ρ = 0.01
f ′(α) ≥!
σf ′(0), σ ∈ (ρ, 1), e.g. σ = 0.1
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M12.1
Simulated Annealing
basic algorithm
acceptance function cooling velocity
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M12.2
generation probability
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M13
Optimization by a Stochastic Evolution Strategy
from: P. Eberhard, F. Dignath, L. Kübler: Parallel Evolutionary Optimization of Multibody
Systems with Application to Railway Dynamics, Vol. 9, No 2, 2003, pp. 143–164.
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M14
initialization
recursive update equation
find best particle and
best solution 𝐩swarmbest,k
terminate?
𝐩swarmbest,k
yes no
Particle Swarm Optimization
simulation of social behavior of bird flock (introduced by Kennedy & Eberhart in 1995)
recursive update equation algorithm
𝐩ik position of particle i at time k
∆𝐩ik velocity of particle i at time k
r1, r2 ∈ U[0,1] evenly distributed numbers
w, c1, c2 control parameters
𝐩ik+1 = 𝐩i
k + ∆𝐩ik+1
∆𝐩ik+1 = w∆𝐩i
k + c1r1,ik 𝐩i
best,k − 𝐩ik + c2r2,i
k 𝐩swarmbest,k − 𝐩i
k
tradition/ inertia
learning social behaviour
𝐩swarmbest,k
𝐩ik
𝐩ibest,k
∆𝐩ik+1
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M15
Karush–Kuhn–Tucker Conditions
If 𝐩∗ is a regular point and a local minimizer of the optimization problem
min𝐩∈P
f(𝐩) with P = {𝐩 ∈ Rh | 𝐠(𝐩) = 𝟎, 𝐡(𝐩) ≤ 0, 𝐠: Rh → Rl, 𝐡: Rh → Rm} ,
then Lagrange multipliers 𝛌∗ and 𝛍∗ exist, for which 𝐩∗, 𝛌∗, 𝛍∗ fulfill the following conditions
∂f
∂𝐩− ∑ λi
∂gi∂𝐩
− ∑ μj∂hj
∂𝐩= 𝟎
m
j=1
l
i=1
𝐠(𝐩) = 𝟎
𝐡(𝐩) ≤ 𝟎
𝛍 ≤ 𝟎
μjhj(𝐩) = 0 , j = 1(1)m
If we introduce the Lagrange function
L(𝐩, 𝛌, 𝛍) ≔ f(𝐩) − ∑ μigi(𝐩)
l
i=1
− ∑ μjhj(𝐩)
m
j=1
,
we can write the Karush–Kuhn–Tucker conditions as follows
∂L
∂𝐩= 𝟎 ,
∂L
∂𝛌= 𝟎 ,
∂L
∂𝛍≥ 𝟎 ,
𝛍 ≤ 𝟎 , μjhj(𝐩) = 𝟎 , j = 1(1)m .
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M16.1
Lagrange-Newton-Method = Sequential Quadratic Programming (SQP)
= Recursive Quadratic Programming (RQP)
= Variable Metric Method
Here simplifying assumption: only equality constraints
min𝐩∈𝑃
f(𝐩) with P = {𝐩 ∈ ℝh|𝐠(𝐩) = 𝟎}
Karush-Kuhn-Tucker Condition (KKT) for minimizer 𝐩
𝐚(𝐩∗, 𝛌) =
[ ∂L
∂𝐩∂L
∂𝛌]
= [∇f(𝐩∗) − ∑∇gi(𝐩
∗)λi𝐠(𝐩∗)
] = [𝟎𝟎]
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M16.2
[ ∇2f − ∑∇2giλi −
∂𝐠T
∂𝐩∂𝐠
∂𝐩𝟎
] (v)
∙ [𝐩(v+1) − 𝐩(v)
𝛌(v+1) − 𝛌(v)] = − [∇f −
∂𝐠T
∂𝐩∙ 𝛌
𝐠]
(v)
[ 𝐖(v)
∂𝐠(v)T
∂𝐩
∂𝐠(v)
∂𝐩𝟎
]
∙ [δ𝐩(v+1)
−𝛌(v+1)] = − [
∇f (v)
𝐠(v)]
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M16.3
In the case that the performance function and constraint equations are general nonlinear
functions the parameter variation δ𝐩 is not necessarily the best possible parameter varia-
tion for the original optimization problem. In order to achieve a higher flexibility the method
can be combined with a line search, δ𝐩 = α𝐬.
[ 𝐖(v)
∂𝐠(v)T
∂𝐩
∂𝐠(v)
∂𝐩𝟎
]
∙ [𝐬
−𝛌] = − [
∇f (v)
𝐠(v)]
equivalent to
min𝐬∈S
1
2𝐬 ∙ 𝐖(v) ∙ 𝐬 with S = {𝐬 ∈ ℝh|
∂𝐠∂𝐩
∙ 𝐬 + 𝐠 = 𝟎}
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M17
Comparison of Various Deterministic Optimization
Algorithms for Nonlinear
Constrained Optimization Problems
name algo-
rithm
author availability mean number of function calls percent-
age of
failure
f gi, hj ∇f ∇gi, ∇hj [%]
SUMT SUMT McCormick
et al.
2335 24046 99 1053 69.9
NLP SUMT Rufer 1043 8635 111 957 15.6
VF02AD SQP Powell Harwell
Subroutine
Library
16 179 16 179 6.2
NLPQL
(NCONF,
NCONG)
SQP Schittkowski IMSL–
Library
18 181 16 64 3.3
The results are based on 240 test runs with 80 test problems, using
3 different initial parameters for each problem.
see
Schittkowski, K.: Nonlinear Programming Codes. Information, Tests, Performance.
Berlin: Springer, 1980.
Schittkowski, K.: NLPQL: A FORTRAN Subroutine Solving Constrained Nonlinear Pro-
gramming Problems. Annals of Operations Res. 5 (1985/86) 485-500.
Institute of Eng. and Comp. Mechanics Optimization of Mechanical Systems Prof. Dr.-Ing. Prof. E.h. P. Eberhard WT 20/21 M18
Principles of Reduction in Multicriteria Optimization