Applications of Parametric Quadratic Optimization

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

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

3. Introduction: Parametric Optimization 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

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

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

6. PQO:NumericalIllustration Solution output type lu B N T () ----------------------------------------------------------------------------------------------------------------- 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 invariancy interval +3.33333 Inf 3 4 5 1 2 0.002 - 0.00 + 0.00

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

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

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

10. Financial Portfolio Example

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

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 • perturbations in the propagation environment due to excessive heat on the line or neighboring bundles may lead this assumption not to hold varying we can investigate the robustness of the power allocation under the effect of uncertainty in the noise power

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 • if the actual noise is lower than the nominal, the user may decide to decrease the transmitted power the parameter is now used to express the uncertainty in noise power as well as power increment to reduce the effect of noise

17. Multiparametric Quadratic Optimization

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. 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.