further advanced methods from mathematical optimization
DESCRIPTION
AACIMP 2009 Summer School lecture by Gerhard Wilhelm Weber. "Modern Operational Research and Its Mathematical Methods" course.TRANSCRIPT
Further Advanced Methods from Mathematical Optimization
4th International Summer SchoolAchievements and Applications of Contemporary Informatics, Mathematics and PhysicsNational University of Technology of the UkraineKiev, Ukraine, August 5-16, 2009
GerhardGerhardGerhardGerhardGerhardGerhardGerhardGerhard--------Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber ** and Baand Başşak Aktekeak Akteke--ÖztürkÖztürk
Institute of Applied Mathematics Institute of Applied Mathematics Middle East Technical University, Ankara, TurkeyMiddle East Technical University, Ankara, Turkey
** Faculty of Economics, Management and Law, Universi ty of Siegen, GermanyFaculty of Economics, Management and Law, Universi ty of Siegen, Germany
Center for Research on Optimization and Control, Univ ersity of Aveiro, Portugal
CE*OCEUEURORO CBBMCBBM EUEURORO ORDORD
Mathematical Optimization
On Foundations of Continuous Optimization
GerhardGerhardGerhardGerhardGerhardGerhardGerhardGerhard--------Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber *,*, Başak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak Akteke--------Öztürk Öztürk Öztürk Öztürk Öztürk Öztürk Öztürk Öztürk
Motivatio
4th International Summer SchoolAchievements and Applications of Contemporary Informatics, Mathematics and PhysicsNational University of Technology of the UkraineKiev, Ukraine, August 5-16, 2009
GerhardGerhardGerhardGerhardGerhardGerhardGerhardGerhard--------Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber *,*, Başak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak Akteke--------Öztürk Öztürk Öztürk Öztürk Öztürk Öztürk Öztürk Öztürk
Institute of Applied Mathematics Institute of Applied Mathematics Programs of Financial Mathematics, Actuarial Scienc es and Scientific ComputingPrograms of Financial Mathematics, Actuarial Scienc es and Scientific Computing
Department of Biomedical EngineeringDepartment of Biomedical EngineeringMiddle East Technical University, Ankara, TurkeyMiddle East Technical University, Ankara, Turkey
* * * * * * * * Faculty of Economics, Management and Law, Universi ty of Siegen, GermanyFaculty of Economics, Management and Law, Universi ty of Siegen, GermanyCenter for Research on Optimization and Control, Univ ersity of Aveiro, Portugal
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• differential topology, singularity theory, Morse theory, algebraic topology
• discrete math., discrete optimization, coding theory and cryptography,
complexity theory, theoretical informatics, logic
• probability theory, statistics, stochastic calculus,
Mathematics
financial math., actuarial sciences
• data mining and computational statistics
• calculus, nonsmooth analysis, theory of functions
• numerical mathematics
• dynamical systems, hybrid systems
• differential topology, singularity theory, Morse theory, algebraic topology
• discrete math., discrete optimization, coding theory and cryptography,
complexity theory, theoretical informatics, logic
• probability theory, statistics, stochastic calculus,
optimization
opt.
Mathematics
financial math., actuarial sciences
• data mining and computational statistics
• calculus, nonsmooth analysis, theory of functions
• numerical mathematics
• dynamical systems, hybrid systems
opt.
opt.
opt.
opt.
opt.
• linear optimization, conic opt., robust (& stochastic) opt.
• convex opt.
• nonsmooth opt., global opt.
• discontinuous opt. (in preparation)
Optimization
• discontinuous opt. (in preparation)
• nonlinear opt., semi-infinite opt., infinite programming
• calculus of variations, optimal control, opt. of PDEs
• variational inequalities
• discrete optimization, hybrid opt.
Networks and Optimization
min( ), ( ), ( )ij i im c d∗ ∗ ∗
l
( 1, . . . , )( , ) ( )n
j np m y yα∗ =≤∑
subject to
)2
1
0
l
M E C E D Eαα ακκ κ
α
∗ −∗ ∗ ∗
= ∞
+ + −∑ &
GSIP relaxation
Networks and Optimization
set of combined environmental effects combined environmental effects combined environmental effects combined environmental effects
( , ) :Y C D∗ ∗ =
1,..., 1,...,1,...,
( 0, ) ( 0, )i ii n i n
m
c d∗ ∗
= ==
× ∏ ∏l
l
1
1
1
, m in
( 1, . . . , )
( 1, . . . , )
( , ) ( )
( , ) ( )
( , ) ( )
&
i j i j ji
n
i ii
n
i ii
i i i
j n
m
p m y y
q c y y
d y y
m
α
β
ζ γ
δ
∗
=
∗
=
∗
=
=
=
≤
≤
≤
≥
∑
∑
∑
l l ll
o v e ra ll b o x c o n s t ra in ts
( ( , ))y Y C D∗ ∗∈
( 1, . . . , )i n=
Optimization
I, K, L finite
semi-infinite
Optimization
I, K, L finite
semi-infinitegeneralized
)(τψτ
ψ
Generalized Semi-Infinite Optimization
structurally stable
global local global
)(⋅ε
nIR
asymptotic
effect
homeom.
),( τϕ ⋅
nonsmooth problem:
clustering problem (for a finite set A)
Clustering
nonsmooth problem:
2
.
Clustering and Classification
2
max-min separability incremental approach
Incremental Approach to Classification
2
Adil Bagirov
Separable Functions
desirability functions (quality opt.)
max-, min-type
continuous selections
normal forms smoothening
stability
instabilitybilevelproblems
desirability functions (quality opt.)
max-, min-type
continuous selections
normal forms
Separable Functions
stability
instability modified subgradientalgorithm
• EURO
Examples: International
EURO WG on Complex Societal ProblemsEURO WG “OR in Comput. Biology, Bioinformatics & Me dicine”EURO WG “OR for Development” etc. 29 EURO WGs
• EUROPT
• SIAM
SIAG/OPTOptimization Society
9 Years 9 Years EUEUROPTROPT
Rhodes Ankara Reykjavik Prague Neringa BonnKiev Lisbon
The Pacific Optimization Research Activity Grou P
• Infrastructure planning is very important for many developing
countries such as China, Korea and Vietnam in Asian Pacific area.
Such planning and design were usually done by city designers,
industrial people, or politicians. Optimization problems arise naturally
in this kind of planning and may play an important role in saving costs
and improving performance.
Examples: Pacific Region
and improving performance.
• Traffic control and network optimization . Traffic has long been
a real problem for big cities in China and other Asian countries.
How to apply the network optimization and traffic control theory
to improve the traffic is a challenge for optimization community.
• High quality optimization journals of POP.
Xiaoling Sun (Fudan University, Shanghai)
• Journal of Industrial and Management Optimization
• Pacific Journal of Optimization
• Dynamics of Continuous, Discrete and Impulsive Systems (Series B)
Pacific Region
• Dynamics of Continuous, Discrete and Impulsive Systems (Series B)
• Global Journal of Technology and Optimization
(in preparation)
and , , etc.
On Foundations of Continuous Optimization
Parametric Unconstrained Optimization
Motivatio
4th International Summer SchoolAchievements and Applications of Contemporary Informatics, Mathematics and PhysicsNational University of Technology of the UkraineKiev, Ukraine, August 5-16, 2009
Parametric Unconstrained Optimization and Related Topics
GerhardGerhardGerhardGerhardGerhardGerhardGerhardGerhard--------Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber *,*, Başak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak Akteke--------Öztürk Öztürk Öztürk Öztürk Öztürk Öztürk Öztürk Öztürk
Institute of Applied Mathematics Institute of Applied Mathematics Programs of Financial Mathematics, Actuarial Scienc es and Scientific ComputingPrograms of Financial Mathematics, Actuarial Scienc es and Scientific Computing
Department of Biomedical EngineeringDepartment of Biomedical EngineeringMiddle East Technical University, Ankara, TurkeyMiddle East Technical University, Ankara, Turkey
* * * * * * * * Faculty of Economics, Management and Law, Universi ty of Siegen, GermanyFaculty of Economics, Management and Law, Universi ty of Siegen, GermanyCenter for Research on Optimization and Control, Univ ersity of Aveiro, Portugal
n