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

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