1. Applications of Parametric Quadratic Optimization Oleksandr Romanko Joint work with Alireza Ghaffari Hadigheh and Tamás Terlaky November 1, 2004

2. 2 Outline Introduction Parametric QO Numerical illustration Simultaneous perturbation Financial portfolio example DSL example Multiparametric QO Conclusions and future work

3. 3 Parametric optimization Parameter is introduced into objective function and/or constraints The goal is to find – optimal solution – optimal value function Allows to do sensitivity analysis Applications Introduction: Parametric Optimization

4. 4 Convex Quadratic Optimization (QO) problem: Quadratic Optimization and Its Parametric Counterpart Parametric Convex Quadratic Optimization (PQO) problem:

5. 5 Optimal Partition and Invariancy Intervals The optimal partition of the index set {1, 2,…, n} is The optimal partition is unique!!! Invariancy intervals: Covering all invariancy intervals:

6. 6 PQO: Numerical Illustration type l u B N T  ( ) ----------------------------------------------------------------------------------------------------------------- invariancy interval +3.33333 Inf 3 4 5 1 2 0.00 2 - 0.00 + 0.00 transition point -8.00000 -8.00000 3 5 1 4 2 -0.00 invariancy interval -8.00000 -5.00000 2 3 5 1 4 8.50 2 + 68.00 + 0.00 transition point -5.00000 -5.00000 2 1 3 4 5 -127.50 invariancy interval -5.00000 +0.00000 1 2 3 4 5 4.00 2 + 35.50 - 50.00 transition point +0.00000 +0.00000 1 2 3 4 5 -50.00 invariancy interval +0.00000 +1.73913 1 2 3 4 5 -6.91 2 + 35.50 - 50.00 transition point +1.73913 +1.73913 2 3 4 5 1 -9.15 invariancy interval +1.73913 +3.33333 2 3 4 5 1 -3.60 2 + 24.00 - 40.00 transition point +3.33333 +3.33333 3 4 5 1 2 0.00 Solution output

7. 7 Simultaneous perturbation parametric QO can be extended to multiparametric QO: Simultaneous Perturbation Simultaneous perturbation parametric QO generalizes two models:

8. 8 Financial Portfolio Example Problem of choosing an efficient portfolio of assets

9. 9 Mean-variance formulation: Minimize portfolio risk subject to predetermined level of portfolio expected return.  x i, i=1,…,n asset holdings,  portfolio expected return,  portfolio variance. Portfolio optimization problem (Markowitz, 1956): Original formulationParametric formulation Financial Portfolio Example

10. 10 Financial Portfolio Example

11. 11 Mean-variance formulation: extensions Financial Portfolio Example the investor's risk aversion parameter influences not only risk-return preferences (in the objective function), but also  budget constraints  transaction volumes  upper bounds on asset holdings  etc.

12. 12 DSL Example Optimal multi-user spectrum management for Digital Subscriber Lines (DSL) M users are connected to one service provider via telephone line (DSL) the total bandwidth of the channel is divided into N subcarriers (frequency tones) that are shared by all users

13. 13 DSL Example Optimal multi-user spectrum management for Digital Subscriber Lines (DSL) Each user i tries to allocate his total transmission power to subcarriers to maximize his data transfer rate

14. 14 Allocating each users' total transmission power among the subcarriers "intelligently" may result in higher overall data rates DSL Example Optimal multi-user spectrum management for Digital Subscriber Lines (DSL) Current DSL systems use fixed power levels Noncooperative game – each user behaves selfishly Nash equilibrium points of the noncooperative rate maximization game correspond to optimal solutions of quadratic optim. problem

15. 15 DSL Example Optimal multi-user spectrum management for Digital Subscriber Lines (DSL) the quadratic formulation assumes that the noise power on each subcarrier k is perfectly known apriori varying we can investigate the robustness of the power allocation under the effect of uncertainty in the noise power perturbations in the propagation environment due to excessive heat on the line or neighboring bundles may lead this assumption not to hold

16. 16 DSL Example Optimal multi-user spectrum management for Digital Subscriber Lines (DSL) to mitigate the adverse effect of excess noise, the i -th user may decide to increase the transmitted power in steps of size the parameter is now used to express the uncertainty in noise power as well as power increment to reduce the effect of noise if the actual noise is lower than the nominal, the user may decide to decrease the transmitted power

17. 17 Multiparametric Quadratic Optimization

18. 18 Conclusions and Future Work Background and applications of solving parametric convex QO problems Simultaneous parameterization Extending methodology to Multiparametric QO Parametric Second Order Conic Optimization (robust optimization, financial and engineering applications)

19. 19 References A. Ghaffari Hadigheh, O. Romanko, and T. Terlaky. Sensitivity Analysis in Convex Quadratic Optimization: Simultaneous Perturbation of the Objective and Right-Hand-Side Vectors. Submitted to Optimization, 2003. A. B. Berkelaar, C. Roos, and T. Terlaky. The optimal set and optimal partition approach to linear and quadratic programming. In Advances in Sensitivity Analysis and Parametric Programming, T. Gal and H. J. Greenberg, eds., Kluwer, Boston, USA, 1997. T. Luo. Optimal Multi-user Spectrum Management for Digital Subscriber Lines. Presentation at the ICCOPT conference, Troy, USA, 2004.