Further Advanced Methods from Mathematical Optimization

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AACIMP 2009 Summer School lecture by Gerhard Wilhelm Weber. "Modern Operational Research and Its Mathematical Methods" course.

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Further Advanced Methods from Mathematical Optimization

  1. 1. 4th International Summer School Achievements and Applications of Contemporary Informatics, Mathematics and Physics National University of Technology of the Ukraine Kiev, Ukraine, August 5-16, 2009 Further Advanced Methods from Mathematical Optimization Gerhard-Wilhelm Weber * and Başak Akteke-Öztürk Gerhard- Akteke- Institute of Applied Mathematics Middle East Technical University, Ankara, Turkey * Faculty of Economics, Management and Law, University of Siegen, Germany Center for Research on Optimization and Control, University of Aveiro, Portugal EURO CBBM EURO EURO ORD EURO CE*OC
  2. 2. 4th International Summer School Achievements and Applications of Contemporary Informatics, Mathematics and Physics National University of Technology of the Ukraine Kiev, Ukraine, August 5-16, 2009 Motivatio On Foundations of Continuous Optimization Gerhard- Gerhard-Wilhelm Weber *, Başak Akteke-Öztürk Akteke- Institute of Applied Mathematics n Programs of Financial Mathematics, Actuarial Sciences and Scientific Computing Department of Biomedical Engineering Middle East Technical University, Ankara, Turkey * Faculty of Economics, Management and Law, University of Siegen, Germany Center for Research on Optimization and Control, University of Aveiro, Portugal
  3. 3. Mathematics • 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, financial math., actuarial sciences • data mining and computational statistics • calculus, nonsmooth analysis, theory of functions • numerical mathematics • dynamical systems, hybrid systems
  4. 4. Mathematics • differential topology, singularity theory, Morse theory, algebraic topology optimization • discrete math., discrete optimization, coding theory and cryptography, complexity theory, theoretical informatics, logic opt. • probability theory, statistics, stochastic calculus, financial math., actuarial sciences opt. • data mining and computational statistics opt. • calculus, nonsmooth analysis, theory of functions opt. • numerical mathematics opt. • dynamical systems, hybrid systems opt.
  5. 5. Optimization • linear optimization, conic opt., robust (& stochastic) opt. • convex opt. • nonsmooth opt., global opt. • 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.
  6. 6. Networks and Optimization
  7. 7. Networks and Optimization GSIP relaxation 2 l ∗ −1 ) min ∑ α =0 ∗ & M Eκα + C E κα + D∗ − Eκα ∗ (mij ∗ ), (cil∗ ), (d i ∗ ) ∞ subject to n ∑ i =1 p ij ( m ij ∗ , y ) ≤ α j ( y ) ( j = 1, ..., n ) n ∑ q il ( c il ∗ , y ) ≤ β l ( y ) ( l = 1, ..., m ) ( y ∈ Y (C ∗ , D∗ )) i =1 n ∑ i =1 ζ i ( d i∗ , y ) ≤ γ ( y ) set of combined environmental effects m ii ≥ δ i , m in ( i = 1, . . . , n ) Y (C ∗ , D∗ ) := & o v e r a ll b o x c o n s t r a in t s ( ∏ i =1,..., n 0, ci∗l  ) × (   ∏ i =1,..., n 0, d i∗  )   l =1,..., m
  8. 8. Optimization semi-infinite I, K, L finite
  9. 9. Optimization generalized semi-infinite I, K, L finite
  10. 10. Generalized Semi-Infinite Optimization ψ (τ ) τ ψ ϕ (⋅,τ ) homeom. structurally stable asymptotic effect ε (⋅) IR n global local global
  11. 11. Clustering clustering problem (for a finite set A) nonsmooth problem: . 2
  12. 12. Clustering and Classification 2 max-min separability incremental approach
  13. 13. Incremental Approach to Classification 2 Adil Bagirov
  14. 14. Separable Functions desirability functions (quality opt.) max-, min-type continuous selections normal forms smoothening stability instability bilevel problems
  15. 15. Separable Functions desirability functions (quality opt.) max-, min-type continuous selections normal forms stability instability modified subgradient algorithm
  16. 16. Examples: International • EURO EURO WG on Complex Societal Problems EURO WG “OR in Comput. Biology, Bioinformatics & Medicine” EURO WG “OR for Development” etc. 29 EURO WGs • EUROPT 9 Years EUROPT EUROPT Rhodes Ankara Reykjavik Prague Neringa Bonn • SIAM Kiev Lisbon SIAG/OPT The Pacific Optimization Optimization Society Research Activity GrouP
  17. 17. Examples: Pacific Region • 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. • 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)
  18. 18. Pacific Region • Journal of Industrial and Management Optimization • Pacific Journal of Optimization • Dynamics of Continuous, Discrete and Impulsive Systems (Series B) • Global Journal of Technology and Optimization (in preparation) and , , etc.
  19. 19. 4th International Summer School Achievements and Applications of Contemporary Informatics, Mathematics and Physics National University of Technology of the Ukraine Kiev, Ukraine, August 5-16, 2009 Motivatio On Foundations of Continuous Optimization Parametric Unconstrained Optimization and Related Topics n Gerhard- Gerhard-Wilhelm Weber *, Başak Akteke-Öztürk Akteke- Institute of Applied Mathematics Programs of Financial Mathematics, Actuarial Sciences and Scientific Computing Department of Biomedical Engineering Middle East Technical University, Ankara, Turkey * Faculty of Economics, Management and Law, University of Siegen, Germany Center for Research on Optimization and Control, University of Aveiro, Portugal

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