6th International Summer School<br />National University of Technology of the Ukraine<br />Kiev, Ukraine,  August 8-20, 20...
Outline<br />Bio- and Financial Systems<br />Genetic ,Gene-Environment and Eco-Finance Networks <br />Time-Continuous and ...
Bio-Systems <br />environment<br />medicine<br />food<br />bio materials<br />bio energy<br />development<br />education<b...
Stock Markets<br />
Regulatory Networks:  Examples<br />Further examples:<br />Socio-econo-networks, stock markets, portfolio optimization, im...
Bio-Systems <br /> Medicine<br />Environment<br />...    Finance<br /> Health Care<br />prediction of gene patterns  based...
DNA experiments<br />Ex.:yeastdata<br />http://genome-www5.stanford.edu/<br />
Analysis of DNA experiments<br />
E0 : metabolic state of a cell at t0 (:=gene expression pattern),ith element of the vector E0 :=expression level of gene i...
Modeling & Prediction<br />data<br />prediction,   anticipation<br />      least squares  –  max likelihood<br />Expressio...
Modeling & Prediction<br />Ex.:<br />Ex.:   Euler,   Runge-Kutta<br />M<br />We analyze the influence of em-parameters  <b...
Stability<br /><ul><li>For which parameters, i.e., for which setM</li></ul>(hence,dynamics),  isstabilityguaranteed ?<br /...
Stability<br />combinatorial  algorithm<br /><ul><li>For which parameters, i.e., for which setM</li></ul>(hence,dynamics),...
Genetic Network<br />Ex. :<br />
Genetic Network<br />0.4E1<br />gene2<br />gene1<br />0.2 E2<br />1 E1<br />gene3<br />gene4<br />
Gene-Environment Networks<br />             if  gene j regulates gene i<br />             otherwise<br />
Model Class<br />:    time-autonomous form, where<br />:     d-vector  of concentration levels of proteins and<br />      ...
Model Class<br />(i):  a constant (nxn)-matrix<br />                                                             :  an (nx...
Model Class<br />In general, in the d-dimensional extended space,<br />                                                   ...
Time-Discretized Model<br />-  Euler’s method, <br />-  Runge-Kutta methods, e.g., 2nd-order Heun's method<br />3rd-order ...
Time-Discretized Model  <br />(**)<br />:  in the extended spacedenotes the DNA microarray experimental data<br />and the ...
Matrix Algebra<br />:     (nxn)- and  (nxm)-matrices, respectively<br />:        (n+m)x(n+m) -matrix<br />:    (n+m)-vecto...
Matrix Algebra<br />Final canonical block form of :                     =  .<br />
Optimization Problem<br />mixed-integer least-squares optimization problem:<br />Boolean variables<br />subject to<br />Ug...
Mixed-Integer Problem<br />:  constant (nxn)-matrix with entries            representing the effect<br />    which the exp...
Numerical Example<br />MINLP for data: <br />Gebert et al. (2004a)<br />Apply 3rd-order Heun method:<br />Take<br />using ...
Numerical Example<br />Apply 3rd-order Heun’s time discretization :<br />
____    gene A<br />........   gene B<br />_ . _ .   gene C<br />- - - -    gene D<br />Results of Euler Method for all ge...
____    gene A<br />........   gene B<br />_ . _ .   gene C<br />- - - -    gene D<br />Results of 3rd-order Heun Method f...
Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />
Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />
Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />
Model Class under Interval Uncertainty<br />
Model Class under Interval Uncertainty<br />θ2,2<br />hybrid<br />local model<br />θ2,1<br />θ1,1<br />θ1,2<br />
Model Class under Interval Uncertainty<br />min<br />subject to   <br />
Generalized Semi-Infinite Programming<br />I, K, L   finite<br />
Generalized Semi-Infinite Programming<br />Jongen, Weber, Guddat et al.<br />homeom.<br />        asymptotic<br />effect<b...
Generalized Semi-Infinite Programming<br />Thm.    (W. 1999/2003, 2006):<br />Fulfilled!<br />
Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />
Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />Coalitions under uncertainty<br />
Regulatory Networks:   Interactions<br />Determine the degree of connectivity.<br />
Time-Discrete Model<br />Clusters and Ellipsoids:<br />Target clusters: 	             C1,C2,…,CREnvironmental clusters:	 D...
Time-Discrete Model<br />Time-Discrete Model:<br />Target  Target<br />Environment  Target<br />(<br />R<br />)<br />(<b...
Time-Discrete Model<br />Ellipsoidal Calculus:<br /><ul><li>  Affine-linear transformations
  Sums of ellipsoids
  Intersections / fusions of ellipsoids</li></ul>AE + b<br />E1 + E2<br />inner / outer approximations<br />E1∩ E2<br />R...
Set-Theoretic Regression Problem<br />Ellipsoidal Calculus<br />The Regression Problem:<br />Maximize(overlap of ellipsoid...
Set-Theoretic Regression Problem<br />Measures for the size of intersection:<br /><ul><li>Volume->ellipsoid matrix determi...
Sum of squares of semiaxes->  trace of covariance matrix
Length of largest semiaxes->  eigenvalues of covariance matrix</li></ul>E<br />semidefinite programming                   ...
Curse of Dimensionality<br />Ci<br />1<br />1<br />0<br />χij <br />1<br />=<br />Cj<br />Tj<br />0<br />
Curse of Dimensionality<br />Mixed-Integer Regression Problem:<br />R<br />S<br />T<br />Σ<br />Σ<br />Σ<br />−<br />−<br ...
Curse of Dimensionality<br />Scale free networks(metabolic networks, world wide web,…)<br /><ul><li>High error tolerance
High attack vulnerability(removal of important nodes)</li></li></ul><li>Curse of Dimensionality<br />Continuous Regression...
Curse of Dimensionality<br />Continuous Regression Problem:<br />R<br />S<br />T<br />Σ<br />Σ<br />Σ<br />−<br />−<br />^...
Cost Games<br />Cost games are very important in the practice of OR.<br />Ex.: <br /><ul><li>  airport game,
  unanimity game,
  production economy with landowners and peasants,
  bankrupcy game, etc..</li></ul>There is also a cost game in environmental protection (TEM model):<br />The aim is to rea...
Cost Games<br />The central problem in cooperative game theory is how to allocate the gain<br />    among the individual p...
TEM Model      <br />
TEM Model      <br />Influence of memory parameter on the emissions reduced and financial means expended<br />
TEM Model      <br />
Games<br />cooperative <br />
IntervalGames<br />cooperative<br />.<br />.<br />.<br />.<br />
Ellipsoid Games                                                        Interval Games<br />cooperative<br />.<br />.<br />...
Ellipsoid Games                                                        Interval Games<br />cooperative<br />.<br />.<br />...
Ellipsoid Games                                                        Interval Games<br />cooperative<br />.<br />.<br />...
Ellipsoid Games                                                        Interval Games<br />cooperative<br />.<br />.<br />...
Ellipsoid Games                                                        Interval Games<br />cooperative<br />.<br />.<br />...
IntervalGames<br />cooperative<br />
IntervalGames<br />cooperative<br />Interval Glove Game<br />
               Ellipsoid Games                                                        <br />cooperative<br />
               Ellipsoid Games                                                        <br />cooperative<br />Ellipsoid  Gl...
               Ellipsoid Games                                                        <br />cooperative<br />Ellipsoid Kyo...
               Ellipsoid Games                                                        <br />cooperative<br />Ellipsoid Mal...
               Ellipsoid Games                                                        <br />cooperative<br />re<br />r<br ...
               Ellipsoid Games                                                        <br />cooperative<br />re<br />r<br ...
Finance Networks<br />.<br />
Finance Networks with Bubbles<br />.<br />
Finance Networks with Bubbles<br />.<br />hybrid<br />
Financial Dynamics<br />drift    diffusion       <br />Ex.:       price,          wealth,        interest rate,        vol...
Financial Dynamics<br />Milstein Scheme:<br />and, based on finitely many data:<br />
Financial Dynamics Identified<br />Tikhonovregularization<br />conic quadratic programming<br />Interior Point Methods<br />
Financial Dynamics Identified<br />Özmen, Weber, Batmaz<br />Important  new class of (Generalized) Partial Linear Models: ...
Financial Dynamics Identified<br />Özmen, Weber, Batmaz<br />Robust  CMARS:              <br />confidence interval<br />.<...
Financial Dynamics Identified<br />Özmen, Weber, Batmaz<br />Robust  CMARS:              <br />confidence interval<br />.<...
Portfolio Optimization Identified<br />            max utility !     or<br />mincosts!or<br />min risk!<br />martingale me...
Portfolio Optimization Identified<br />            max utility !     or<br />mincosts!or<br />min risk!<br />martingale me...
Portfolio Optimization Identified<br />            max utility !     or<br />mincosts!or<br />min risk! <br />martingale m...
Portfolio Optimization Identified<br />            max utility !     or<br />mincosts!or<br />min risk! <br />martingale m...
HybridStochastic Control<br />Control of Stochastic Hybrid Systems, R.Raffard<br /><ul><li>standard Brownian motion
continuous stateSolves an SDE whose jumps are governed by the discrete state.
discrete stateContinuous time Markov chain.
control</li></li></ul><li>Applications<br />hybrid<br /><ul><li>Engineering:Maintain dynamical system in safe domain for m...
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New Insights and Applications of Eco-Finance Networks and Collaborative Games

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AACIMP 2011 Summer School. Operational Research stream. Lecture by Gerhard-Wilhelm Weber.

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New Insights and Applications of Eco-Finance Networks and Collaborative Games

  1. 1. 6th International Summer School<br />National University of Technology of the Ukraine<br />Kiev, Ukraine, August 8-20, 2011<br />New Insights and Applications of Eco-Finance Networks<br />and Collaborative Games<br />Gerhard-Wilhelm Weber 1*<br />SırmaZeynepAlparslanGök2, Erik Kropat3, ÖzlemDefterli4,<br /> Fatma Yelikaya-Özkurt1,Armin Fügenschuh5<br />1 Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey<br /> 2 Department of Mathematics, SüleymanDemirel University, Isparta, Turkey 3 Department of Computer Science, Universität der Bundeswehr München, Munich, Germany 4 Department of Mathematics and Computer Science, Cankaya University, Ankara, Turkey 5 Optimierung, Zuse Institut Berlin, Germany<br />* Faculty of Economics, Management and Law, University of Siegen, Germany<br />Center for Research on Optimization and Control, University of Aveiro, Portugal<br />Universiti Teknologi Malaysia, Skudai, Malaysia<br />
  2. 2. Outline<br />Bio- and Financial Systems<br />Genetic ,Gene-Environment and Eco-Finance Networks <br />Time-Continuous and Time-Discrete Models<br />Optimization Problems<br />Numerical Example and Results<br />Networks under Uncertainty<br />Ellipsoidal Model<br />Optimization of the Ellipsoidal Model<br />Kyoto Game<br />Ellipsoidal Game Theory<br />Related Aspects from Finance<br />Hybrid Stochastic Control<br />Conclusion<br />
  3. 3. Bio-Systems <br />environment<br />medicine<br />food<br />bio materials<br />bio energy<br />development<br />education<br />health care<br />sustainability<br />
  4. 4. Stock Markets<br />
  5. 5. Regulatory Networks: Examples<br />Further examples:<br />Socio-econo-networks, stock markets, portfolio optimization, immune system, epidemiological processes …<br />
  6. 6. Bio-Systems <br /> Medicine<br />Environment<br />... Finance<br /> Health Care<br />prediction of gene patterns based on<br />DNA microarraychip experiments<br />with<br />M.U. Akhmet, H. Öktem <br />S.W. Pickl, E. Quek Ming Poh<br />T. Ergenç, B. Karasözen <br />J. Gebert, N. Radde <br />Ö. Uğur, R. Wünschiers<br />M. Taştan, A. Tezel, P. Taylan <br />F.B. Yilmaz, B. Akteke-Öztürk<br />S. Özöğür, Z. Alparslan-Gök <br /> A. Soyler, B. Soyler, M. Çetin<br />S. Özöğür-Akyüz, Ö. Defterli<br /> N. Gökgöz, E. Kropat<br />
  7. 7. DNA experiments<br />Ex.:yeastdata<br />http://genome-www5.stanford.edu/<br />
  8. 8. Analysis of DNA experiments<br />
  9. 9. E0 : metabolic state of a cell at t0 (:=gene expression pattern),ith element of the vector E0 :=expression level of gene i,Mk := I + hkM(Ek) , Ek (k є IN0) is recursively defined as Ek+1 := MkEk.<br />Metabolic Shift<br />Gebert et al. (2006)<br />
  10. 10. Modeling & Prediction<br />data<br />prediction, anticipation<br /> least squares – max likelihood<br />Expression<br />expression data<br />matrix-valued function – metabolic reaction<br />
  11. 11. Modeling & Prediction<br />Ex.:<br />Ex.: Euler, Runge-Kutta<br />M<br />We analyze the influence of em-parameters <br />on the dynamics (expression-metabolic).<br />
  12. 12. Stability<br /><ul><li>For which parameters, i.e., for which setM</li></ul>(hence,dynamics), isstabilityguaranteed ?<br />stable<br />feasible<br />M<br />metabolic reaction<br />unstable <br />unfeasible<br />goodness-of-fit (model) test<br />Def.:Mis stable :<br />B : (complex) bounded neighbourhood of<br />M :<br />
  13. 13. Stability<br />combinatorial algorithm<br /><ul><li>For which parameters, i.e., for which setM</li></ul>(hence,dynamics), isstabilityguaranteed ?<br />stable<br />feasible<br />M<br />metabolic reaction<br />unstable <br />unfeasible<br /> Akhmet, Gebert, Öktem, Pickl, Weber (2005),<br /> Gebert, Laetsch, Pickl, Weber, Wünschiers (2006),<br /> Weber, Ugur, Taylan, Tezel (2009),<br /> Ugur, Pickl, Weber, Wünschiers (2009)<br />
  14. 14. Genetic Network<br />Ex. :<br />
  15. 15. Genetic Network<br />0.4E1<br />gene2<br />gene1<br />0.2 E2<br />1 E1<br />gene3<br />gene4<br />
  16. 16. Gene-Environment Networks<br /> if gene j regulates gene i<br /> otherwise<br />
  17. 17. Model Class<br />: time-autonomous form, where<br />: d-vector of concentration levels of proteins and<br /> of certain levels of environmental factors <br />: change in the gene-expression data in time<br />: initial values of the gene-expression levels<br />: experimental data vectors obtained from microarray experiments <br />and environmental measurements <br />: the gene-expression level (concentration rate) of the i th gene at time t<br />denotes anyone of the first n coordinates in the<br />d-vector of genetic and environmental states.<br />Weber et al. (2008c), Chen et al. (1999), <br />Gebert et al. (2004a),<br />Gebert et al. (2006), Gebert et al. (2007), <br />Tastan (2005), Yilmaz (2004), Yilmaz et al. (2005),<br />Sakamoto and Iba (2001), Tastan et al. (2005)<br />: the set of genes.<br />
  18. 18. Model Class<br />(i): a constant (nxn)-matrix<br /> : an (nx1)-vector of gene-expression levels<br />represents and t the dynamical system of the n genes <br /> and their interaction alone.<br /> : : (nxn)-matrix with entries as functions of polynomials, exponential, trigonometric, <br /> splines or wavelets, containing some parameters to be optimized.<br />(iii)<br />Weber et al. (2008c), Tastan (2005), <br />Tastan et al. (2006),<br />Ugur et al. (2009), Tastan et al. (2005), <br />Yilmaz (2004), Yilmaz et al. (2005),<br />Weber et al. (2008b), Weber et al. (2009b)<br />environmental effects<br />(*)<br />n genes , m environmental effects<br />: (n+m)-vector and <br /> (n+m)x(n+m)-matrix, respectively.<br />
  19. 19. Model Class<br />In general, in the d-dimensional extended space,<br /> with <br /> : : (dxd)-matrix,<br /> : (dx1)-vectors.<br />Ugur and Weber (2007), <br />Weber et al. (2008c),<br />Weber et al. (2008b), <br />Weber et al. (2009b)<br />
  20. 20. Time-Discretized Model<br />- Euler’s method, <br />- Runge-Kutta methods, e.g., 2nd-order Heun's method<br />3rd-order Heun's method is introduced byDefterli et al. (2009)<br />we rewrite it as <br />where<br />Ergenc and Weber (2004), <br />Tastan (2005), Tastan et al. (2006), Tastan et al. 2005)<br />
  21. 21. Time-Discretized Model <br />(**)<br />: in the extended spacedenotes the DNA microarray experimental data<br />and the data of environmental items <br /> obtained at the time-level<br />: approximationsobtained<br /> by the iterative formula above<br />: initial values<br />kthapproximation (prediction):<br />
  22. 22. Matrix Algebra<br />: (nxn)- and (nxm)-matrices, respectively<br />: (n+m)x(n+m) -matrix<br />: (n+m)-vectors<br />Applying the 3rd-order Heun’s method to (*) gives the iterative formula (**), where <br />
  23. 23. Matrix Algebra<br />Final canonical block form of : = .<br />
  24. 24. Optimization Problem<br />mixed-integer least-squares optimization problem:<br />Boolean variables<br />subject to<br />Ugur and Weber (2007),<br />Weber et al.(2008c),<br />Weber et al. (2008b),<br />Weber et al. (2009b), <br />Gebert et al. (2004a), <br />Gebert et al. (2006), <br />Gebert et al. (2007)<br />, , : th : the numbers of genes regulated by gene (its outdegree), <br /> by environmental item , or by the cumulative environment, resp..<br />
  25. 25. Mixed-Integer Problem<br />: constant (nxn)-matrix with entries representing the effect<br /> which the expression level of gene has on the change of expression of gene<br />genetic regulation network <br />mixed-integer nonlinear optimization problem (MINLP):<br />subject to <br />: constant vectorrepresenting the lower bounds <br /> for the decrease of the transcript concentration.<br />Binary variables : <br />
  26. 26. Numerical Example<br />MINLP for data: <br />Gebert et al. (2004a)<br />Apply 3rd-order Heun method:<br />Take<br />using modeling language Zimpl 3.0, we solve<br />by SCIP 1.2 as a branch-and-cutframework, together with SOPLEX 1.4.1 as LP solver<br />
  27. 27. Numerical Example<br />Apply 3rd-order Heun’s time discretization :<br />
  28. 28. ____ gene A<br />........ gene B<br />_ . _ . gene C<br />- - - - gene D<br />Results of Euler Method for all genes:<br />
  29. 29. ____ gene A<br />........ gene B<br />_ . _ . gene C<br />- - - - gene D<br />Results of 3rd-order Heun Method for all genes:<br />
  30. 30. Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />
  31. 31. Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />
  32. 32. Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />
  33. 33. Model Class under Interval Uncertainty<br />
  34. 34. Model Class under Interval Uncertainty<br />θ2,2<br />hybrid<br />local model<br />θ2,1<br />θ1,1<br />θ1,2<br />
  35. 35. Model Class under Interval Uncertainty<br />min<br />subject to <br />
  36. 36. Generalized Semi-Infinite Programming<br />I, K, L finite<br />
  37. 37. Generalized Semi-Infinite Programming<br />Jongen, Weber, Guddat et al.<br />homeom.<br /> asymptotic<br />effect<br />: structurally stable<br />global local global<br />
  38. 38. Generalized Semi-Infinite Programming<br />Thm. (W. 1999/2003, 2006):<br />Fulfilled!<br />
  39. 39. Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />
  40. 40. Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />Coalitions under uncertainty<br />
  41. 41. Regulatory Networks: Interactions<br />Determine the degree of connectivity.<br />
  42. 42. Time-Discrete Model<br />Clusters and Ellipsoids:<br />Target clusters: C1,C2,…,CREnvironmental clusters: D1,D2,…,DS<br />Target ellipsoids: X1,X2,…,XRXi = E(μi , Σi) Environmental ellipsoids: E1,E2,…,ES Ej = E(ρj ,Πj) <br />Center<br />Covariance matrix<br />
  43. 43. Time-Discrete Model<br />Time-Discrete Model:<br />Target  Target<br />Environment  Target<br />(<br />R<br />)<br />(<br />S<br />)<br />Targetcluster<br />TT<br />(k)<br />(k+1)<br />ET<br />(k)<br />X<br />ξ<br />X<br />A<br />+<br />+<br />=<br />E<br />A<br />j r<br /> r<br /> j0<br />j s<br /> j<br /> s<br />r =1<br />s =1<br />(<br />R<br />)<br />(<br />S<br />)<br />Environmental cluster<br />TE<br />(k)<br />(k+1)<br />EE<br />(k)<br />X<br />ζ<br />E<br />A<br />+<br />+<br />=<br />E<br />A<br />i r<br /> r<br /> i0<br />is<br /> i<br /> s<br />r =1<br />s =1<br />Target  Environment<br />Environment  Environment<br /> Determine system matrices and intercepts.<br />
  44. 44. Time-Discrete Model<br />Ellipsoidal Calculus:<br /><ul><li> Affine-linear transformations
  45. 45. Sums of ellipsoids
  46. 46. Intersections / fusions of ellipsoids</li></ul>AE + b<br />E1 + E2<br />inner / outer approximations<br />E1∩ E2<br />Ros et al. (2002)<br />Parameterized family of ellipsoidal approximations<br />Kurzhanski, Varaiya (2008)<br />
  47. 47. Set-Theoretic Regression Problem<br />Ellipsoidal Calculus<br />The Regression Problem:<br />Maximize(overlap of ellipsoids) <br />Determine<br />EE<br />TT<br />ET<br />TE<br />, A<br />A<br />, A<br />, A<br />matrices<br />and <br />is<br />j r<br />j s<br />i r<br />,ζ<br />vectors<br />ξ<br /> i0<br /> j0<br /><br /><br /><br /><br /><br /><br />measurement<br />R<br />S<br />T<br />Σ<br />Σ<br />Σ<br />−<br />−<br />^<br />(k)<br />(k)<br />(k)<br /><br />(k)<br />+<br /><br />^<br />E<br />E<br />X<br />X<br />∩<br />∩<br /> s<br /> r<br /> r<br /> s<br />r = 1<br />s = 1<br />k= 1<br />prediction<br />
  48. 48. Set-Theoretic Regression Problem<br />Measures for the size of intersection:<br /><ul><li>Volume->ellipsoid matrix determinant
  49. 49. Sum of squares of semiaxes-> trace of covariance matrix
  50. 50. Length of largest semiaxes-> eigenvalues of covariance matrix</li></ul>E<br />semidefinite programming interior point methods<br />
  51. 51. Curse of Dimensionality<br />Ci<br />1<br />1<br />0<br />χij <br />1<br />=<br />Cj<br />Tj<br />0<br />
  52. 52. Curse of Dimensionality<br />Mixed-Integer Regression Problem:<br />R<br />S<br />T<br />Σ<br />Σ<br />Σ<br />−<br />−<br />^<br />(k)<br />(k)<br />(k)<br /><br />(k)<br />+<br /><br />^<br />E<br />E<br />X<br />maximize<br />X<br />∩<br />∩<br /> s<br /> r<br /> r<br /> s<br />r = 1<br />s = 1<br />k= 1<br />α<br />TT<br />≤<br />deg(C )TT <br />bounds on outdegrees<br />such that<br /><br /><br /><br /><br /><br /><br />j<br />j<br />α<br />TE<br />≤<br />deg(C )TE <br />j<br />j<br />α<br />ET<br />≤<br />deg(D )ET <br />i<br />i<br />α<br />EE<br />≤<br />deg(D )EE <br />i<br />i<br />
  53. 53. Curse of Dimensionality<br />Scale free networks(metabolic networks, world wide web,…)<br /><ul><li>High error tolerance
  54. 54. High attack vulnerability(removal of important nodes)</li></li></ul><li>Curse of Dimensionality<br />Continuous Regression Problem:<br />R<br />S<br />T<br />Σ<br />Σ<br />Σ<br />−<br />−<br />^<br />(k)<br />(k)<br />(k)<br /><br />(k)<br />+<br /><br />^<br />E<br />E<br />X<br />X<br />∩<br />maximize<br />∩<br /> s<br /> r<br /> r<br /> s<br />r = 1<br />s = 1<br />k= 1<br />R<br />Σ<br />α<br />TT<br />TT<br />≤<br />PTT ( TT <br />)<br />such that<br />, ξ<br />A<br /><br /><br /><br /><br /><br /><br />j<br />j r<br />jr<br />j0<br />r =1<br />R<br />α<br />TE<br />Σ<br />≤<br />TE<br />PTE ( TE <br />)<br />,ξ<br />A<br />j<br />j r<br /> j0<br />jr<br />r =1<br />bounds on outdegrees<br />R<br />ET<br />Σ<br />α<br />ET<br />PET ( ET <br />≤<br />, ζ<br />A<br />)<br />i<br />i s<br /> i0<br />is<br />Continuous Constraints /Probabilities <br />s =1<br />R<br />Σ<br />EE<br />α<br />EE<br />PEE ( EE <br />)<br />≤<br />, ζ<br />A<br />i<br />i s<br /> i0<br />is<br />s =1<br />
  55. 55. Curse of Dimensionality<br />Continuous Regression Problem:<br />R<br />S<br />T<br />Σ<br />Σ<br />Σ<br />−<br />−<br />^<br />(k)<br />(k)<br />(k)<br /><br />(k)<br />+<br /><br />^<br />E<br />E<br />X<br />X<br />∩<br />maximize<br />∩<br /> s<br /> r<br /> r<br /> s<br />r = 1<br />s = 1<br />k= 1<br />R<br />Σ<br />α<br />TT<br />TT<br />≤<br />PTT ( TT <br />)<br />such that<br />, ξ<br />A<br /><br /><br /><br /><br /><br /><br />j<br />j r<br />jr<br />j0<br />r =1<br />R<br />α<br />TE<br />Σ<br />≤<br />TE<br />PTE ( TE <br />)<br />,ξ<br />A<br />j<br />j r<br /> j0<br />jr<br />r =1<br />R<br />ET<br />Σ<br />α<br />ET<br />PET ( ET <br />≤<br />, ζ<br />A<br />)<br />i<br />i s<br /> i0<br />is<br />s =1<br />R<br />Ex.:<br />Robust Optimization<br />Σ<br />EE<br />α<br />EE<br />PEE ( EE <br />)<br />≤<br />, ζ<br />A<br />i<br />i s<br /> i0<br />is<br />s =1<br />
  56. 56. Cost Games<br />Cost games are very important in the practice of OR.<br />Ex.: <br /><ul><li> airport game,
  57. 57. unanimity game,
  58. 58. production economy with landowners and peasants,
  59. 59. bankrupcy game, etc..</li></ul>There is also a cost game in environmental protection (TEM model):<br />The aim is to reach a state which is mentioned in Kyoto Protocol by choosing control parameters such that the emissions of each player become minimized.<br />For example, the value is taken as a control parameter.<br />
  60. 60. Cost Games<br />The central problem in cooperative game theory is how to allocate the gain<br /> among the individual players in a “fair” way. <br />There are various notions of fairness and corresponding allocation rules <br />(solution concepts).<br />Any with is an allocation.<br />So, a core allocation <br />guarantees each coalition to be satisfied <br />in the sense that it gets at least what it could get on its own. <br />
  61. 61. TEM Model <br />
  62. 62. TEM Model <br />Influence of memory parameter on the emissions reduced and financial means expended<br />
  63. 63. TEM Model <br />
  64. 64. Games<br />cooperative <br />
  65. 65. IntervalGames<br />cooperative<br />.<br />.<br />.<br />.<br />
  66. 66. Ellipsoid Games Interval Games<br />cooperative<br />.<br />.<br />.<br />.<br />
  67. 67. Ellipsoid Games Interval Games<br />cooperative<br />.<br />.<br />.<br />.<br />
  68. 68. Ellipsoid Games Interval Games<br />cooperative<br />.<br />.<br />.<br />.<br />
  69. 69. Ellipsoid Games Interval Games<br />cooperative<br />.<br />.<br />.<br />.<br />
  70. 70. Ellipsoid Games Interval Games<br />cooperative<br />.<br />.<br />.<br />.<br />Robust Optimization<br />
  71. 71. IntervalGames<br />cooperative<br />
  72. 72. IntervalGames<br />cooperative<br />Interval Glove Game<br />
  73. 73. Ellipsoid Games <br />cooperative<br />
  74. 74. Ellipsoid Games <br />cooperative<br />Ellipsoid Glove Game<br />
  75. 75. Ellipsoid Games <br />cooperative<br />Ellipsoid Kyoto Game<br />Ellipsoid Glove Game<br /> , : (individual roles in TEM Model)<br />: (individual role in TEM Model)<br />
  76. 76. Ellipsoid Games <br />cooperative<br />Ellipsoid Malacca Police Game<br />R<br />
  77. 77. Ellipsoid Games <br />cooperative<br />re<br />r<br />r<br />r<br />r<br />
  78. 78. Ellipsoid Games <br />cooperative<br />re<br />r<br />Farkas Lemma<br />r<br />r<br />r<br />
  79. 79. Finance Networks<br />.<br />
  80. 80. Finance Networks with Bubbles<br />.<br />
  81. 81. Finance Networks with Bubbles<br />.<br />hybrid<br />
  82. 82. Financial Dynamics<br />drift diffusion <br />Ex.: price, wealth, interest rate, volatility <br /> processes<br />
  83. 83. Financial Dynamics<br />Milstein Scheme:<br />and, based on finitely many data:<br />
  84. 84. Financial Dynamics Identified<br />Tikhonovregularization<br />conic quadratic programming<br />Interior Point Methods<br />
  85. 85. Financial Dynamics Identified<br />Özmen, Weber, Batmaz<br />Important new class of (Generalized) Partial Linear Models: <br />Important new class of (Generalized) Partial Linear Models: <br />
  86. 86. Financial Dynamics Identified<br />Özmen, Weber, Batmaz<br />Robust CMARS: <br />confidence interval<br />.<br />.<br />.<br />.<br />. <br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />. <br />. <br />. <br />. <br />.<br />.<br />outlier<br />outlier<br />semi-length of confidence interval<br />
  87. 87. Financial Dynamics Identified<br />Özmen, Weber, Batmaz<br />Robust CMARS: <br />confidence interval<br />.<br />.<br />.<br />.<br />. <br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />.<br />. <br />. <br />. <br />. <br />.<br />.<br />outlier<br />outlier<br />semi-length of confidence interval<br />
  88. 88. Portfolio Optimization Identified<br /> max utility ! or<br />mincosts!or<br />min risk!<br />martingale method: <br /> Optimization Problem<br /> Representation Problem<br />or stochastic control<br />
  89. 89. Portfolio Optimization Identified<br /> max utility ! or<br />mincosts!or<br />min risk!<br />martingale method: <br />Optimization Problem<br />Representation Problem<br />or stochastic control<br />Parameter Estimation<br />
  90. 90. Portfolio Optimization Identified<br /> max utility ! or<br />mincosts!or<br />min risk! <br />martingale method: <br /> Optimization Problem<br /> Representation Problem<br />or stochastic control<br />Parameter Estimation<br />
  91. 91. Portfolio Optimization Identified<br /> max utility ! or<br />mincosts!or<br />min risk! <br />martingale method: <br />Optimization Problem<br />Representation Problem<br />or stochastic control<br />Parameter Estimation<br />
  92. 92. HybridStochastic Control<br />Control of Stochastic Hybrid Systems, R.Raffard<br /><ul><li>standard Brownian motion
  93. 93. continuous stateSolves an SDE whose jumps are governed by the discrete state.
  94. 94. discrete stateContinuous time Markov chain.
  95. 95. control</li></li></ul><li>Applications<br />hybrid<br /><ul><li>Engineering:Maintain dynamical system in safe domain for maximum time.
  96. 96. Systems biology: Parameter identification.
  97. 97. Finance: Optimal portfolio selection.</li></li></ul><li>Method:1st step<br />hybrid<br />Derive a PDE satisfied by the objective function in terms of the generator:<br /><ul><li>Example 1: </li></ul> If <br /> then<br /><ul><li>Example 2:</li></ul> If<br /> then<br />
  98. 98. Method:2ndand 3rd step<br />hybrid<br />Rewrite original problem as deterministic PDE optimization program:<br />Solve PDE optimization program using adjoint method.<br /> Simple and robust…<br />
  99. 99. References<br />http://www3.iam.metu.edu.tr/iam/images/7/73/Willi-CV.pdf<br />Thank you very much for your attention!<br />gweber@metu.edu.tr<br />
  100. 100. References Part 1<br />Achterberg, T., Constraint integer programming, PhD. Thesis, Technische Universitat Berlin, Berlin, 2007.<br />Aster, A., Borchers, B., and Thurber, C., Parameter Estimation and Inverse Problems. Academic Press, San Diego; 2004.<br />Chen, T., He, H.L., and Church, G.M., Modeling gene expression with differential equations, Proceedings of Pacific Symposium on Biocomputing 1999, 29-40.<br />Ergenc, T,. and Weber, G.-W., Modeling and prediction of gene-expression patterns reconsidered with Runge-Kutta discretization, Journal of Computational Technologies 9, 6 (2004) 40-48.<br />Gebert, J., Laetsch, M., Pickl, S.W., Weber, G.-W., and Wünschiers ,R., Genetic networks and anticipation of gene expression patterns, Computing Anticipatory Systems: CASYS(92)03 - Sixth International Conference,AIP Conference Proceedings 718 (2004) 474-485.<br />Hoon, M.D., Imoto, S., Kobayashi, K., Ogasawara, N ., andMiyano, S., Inferring gene regulatory networks from time-ordered gene expression data of Bacillus subtilis using dierential equations, Proceedings of Pacific Symposium on Biocomputing (2003) 17-28.<br />Pickl, S.W., and Weber, G.-W., Optimization of a time-discrete nonlinear dynamical system from a problem of ecology - an analytical and numerical approach, Journal of Computational Technologies 6, 1 (2001) 43-52.<br />Sakamoto, E., and Iba, H., Inferring a system of differential equations for a gene regulatory network by using genetic programming, Proc. Congress on Evolutionary Computation 2001, 720-726.<br />Tastan, M., Analysis and Prediction of Gene Expression Patterns by Dynamical Systems, and by a Combinatorial Algorithm, MSc Thesis, Institute of Applied Mathematics, METU, Turkey, 2005.<br />
  101. 101. References Part 1<br />Tastan , M., Pickl, S.W., and Weber, G.-W., Mathematical modeling and stability analysis of gene-expression patterns in an extended space and with Runge-Kutta discretization, Proceedings of Operations Research, Bremen, 2006, 443-450.<br />Wunderling, R., Paralleler und objektorientierter Simplex Algorithmus, PhD Thesis. Technical Report ZIB-TR 96-09. Technische Universitat Berlin, Berlin, 1996.<br />Weber, G.-W., Alparslan -Gök, S.Z ., and Dikmen, N.. Environmental and life sciences: Gene-environment networks-optimization, games and control - a survey on recent achievements, deTombe, D. (guest ed.), special issue of Journal of Organizational Transformation and Social Change 5, 3 (2008) 197-233.<br />Weber, G.-W., Taylan, P., Alparslan-Gök, S.Z., Özögur, S., and Akteke-Öztürk, B., Optimization of gene-environment networks in the presence of errors and uncertainty with Chebychev approximation, TOP 16, 2 (2008) 284-318.<br />Weber, G.-W., Alparslan-Gök, S.Z ., and Söyler, B., A new mathematical approach in environmental and life sciences: gene-environment networks and their dynamics,Environmental Modeling & Assessment 14, 2 (2009) 267-288.<br />Weber, G.-W., and Ugur, O., Optimizing gene-environment networks: generalized semi-infinite programming approach with intervals,Proceedings of International Symposium on Health Informatics and Bioinformatics Turkey '07, HIBIT, Antalya, Turkey, April 30 - May 2 (2007).<br />Yılmaz, F.B., A Mathematical Modeling and Approximation of Gene Expression Patterns by Linear and Quadratic Regulatory Relations and Analysis of Gene Networks, MSc Thesis, Institute of Applied Mathematics, METU, Turkey, 2004.<br />
  102. 102. References Part 2<br />Aster, A., Borchers, B., and Thurber, C., Parameter Estimation and Inverse Problems, Academic Press, 2004.<br />Boyd, S., and Vandenberghe, L., Convex Optimization, Cambridge University Press, 2004.<br />Buja, A., Hastie, T., and Tibshirani, R., Linear smoothers and additive models, The Ann. Stat. 17,2(1989) 453-510.<br />Fox, J., Nonparametric regression, Appendix to an R and S-Plus Companion to Applied Regression, Sage Publications, 2002. <br />Friedman, J.H., Multivariate adaptive regression splines, Annals of Statistics 19, 1 (1991) 1-141.<br />Hastie, T., and Tibshirani, R., Generalized additive models, Statist. Science 1, 3 (1986) 297-310.<br />Hastie, T., and Tibshirani, R., Generalized additive models: some applications, J. Amer. Statist. Assoc. 82, 398 (1987) 371-386.<br />Hastie, T., Tibshirani, R., and Friedman, J.H., The Element of Statistical Learning, Springer, 2001.<br />Hastie, T.J., and Tibshirani, R.J., Generalized Additive Models, New York, Chapman and Hall, 1990.<br />Kloeden, P.E, Platen, E., and Schurz, H., Numerical Solution of SDE ThroughComputer Experiments, Springer, 1994.<br />Korn, R., and Korn, E., Options Pricing and Portfolio Optimization: Modern Methods ofFinancial Mathematics, Oxford University Press, 2001.<br />Nash, G., and Sofer, A., Linear and Nonlinear Programming, McGraw-Hill, New York, 1996. <br />Nemirovski, A., Lectures on modern convex optimization, Israel Institute of Technology (2002).<br />
  103. 103. References Part 2<br />Nemirovski, A., Modern Convex Optimization, lecture notes, Israel Institute of Technology (2005).<br />Nesterov, Y.E , and Nemirovskii,A.S., Interior Point Methods in Convex Programming, SIAM, 1993.<br />Önalan, Ö., Martingale measures for NIG Lévyprocesses with applications to mathematicalfinance, presentation at Advanced Mathematical Methods for Finance, Side, Antalya, Turkey, April 26-29, 2006.<br />Taylan, P., Weber, G.-W.,and Kropat, E.,Approximation of stochastic differential equationsby additive modelsusing splines and conic programming, International Journal of Computing Anticipatory Systems 21(2008) 341-352.<br />Taylan, P., Weber, G.-W., and Beck, A.,New approaches to regression by generalized additive modelsand continuous optimization for modernapplications in finance, science and techology, Optimization 56, 5-6 (2007) 1-24.<br />Taylan, P., Weber, G.-W.,andYerlikaya, F., A new approach to multivariate adaptive regression splineby using Tikhonov regularization and continuous optimization, TOP 18, 2 (December 2010) 377-395.<br />Seydel, R., Tools for ComputationalFinance, Springer, Universitext, 2004.<br />Weber, G.-W., Taylan, P., Akteke-Öztürk, B., and Uğur, Ö., Mathematical and datamining contributions<br />dynamics and optimization of gene-environment networks,inthe special issue Organization in Matter<br />fromQuarks to Proteins of Electronic Journalof Theoretical Physics.<br />Weber, G.-W.,Taylan, P., Yıldırak, K.,and Görgülü, Z.K., Financial regression and organization, DCDIS-B (Dynamics of Continuous, Discrete andImpulsive Systems (Series B)) 17, 1b (2010) 149-174. <br />
  104. 104. Appendix<br />DNA experiments<br />Control Material<br />Test Material<br />Laser Scan of the Array<br />mRNA<br />-Isolation<br />Sequence Data<br />(cDNA, Genome,<br />cDNA<br />-Synthesis<br />Genbank, etc.)<br />and <br />Labeling<br />Selection or Design and<br />Synthesis of the Probes<br />Hybridization<br />Picture Analysis<br />Array Production<br />Array Preparation<br />Sample Preparation<br />Data Analysis<br />
  105. 105. Identifying Stochastic Differential Equations<br />Appendix<br />Application<br />F. Yerlikaya Özkurt, G.-W. Weber, P. Taylan<br /> Evaluation of the models based on performance values:<br /><ul><li>CMARS performs better than Tikhonov regularization with respect to all themeasures for both data sets.
  106. 106. On the other hand, GLM with CMARS (GPLM) performs better than both Tikhonov regularization and CMARS with respect to all the measures for both data sets.</li>

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