Parametric Quadratic Optimization

1. Parametric Quadratic Optimization Oleksandr Romanko Joint work with Alireza Ghaffari Hadigheh and Tamás Terlaky McMaster University January 19, 2004

2. Outline • Introduction • Origins - financial portfolio example • Quadratic optimization, optimal partition • Parametric quadratic optimization • Invariancy intervals and transition points • Differentiability • Algorithm and numerical illustration • Conclusions and future work

3. Introduction 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 • Many applications

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

5. 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):

6. Financial Portfolio Example

7. Quadratic Optimization • Primal Quadratic Optimization problem: • Dual Quadratic Optimization problem: • KKT conditions: • Maximally complementary solution: • LO: and - strictly complementary solution • QO: , but and can be both zero maximally complementary solution maximizes the number of non-zero coordinates in and

8. Optimal Partition • The optimalpartition of the index set {1, 2,…, n} is The optimalpartition is unique!!! • An optimal solution (x,y,s) is maximally complementary iff: • The support set of a vector v is: • For any maximally complementary solution :

9. Parametric Quadratic Programming • Primal and dual perturbed problems: • Notation: • - feasible sets of the problems • - optimal solution sets of • We are interested in: • Studying properties of the functions and . • Designing an algorithm for computing and without discretizing the space of • Properties: • Domain of is a closed interval • Optimal partition plays a key role

10. Parametric Quadratic Programming

11. Invariancy Intervals • For some we are given the maximally complementary optimal solution of and with the optimal partition . • The left and right extreme points of the invariancy interval: - invariancy interval - transition points • On an invariancy interval a convex combination of the maximally complementary optimal solutions for and is a maximally complementary optimal solution for the corresponding .

12. Optimal Value Function The optimal value function is: • quadratic on the invariancy intervals and: • strictly convex if • linear if • strictly concave if • continuous and piecewise quadratic on its domain

13. Transition Points • Equivalent statements: • is a transition point • or is discontinuous at • invariancy interval = (singleton) • How to proceed from the current invariancy interval to the next one? • Derivatives In a non-transition point the first order derivative of the optimal value function is

14. Derivatives • Derivatives in transition points: The left and right derivatives of the optimal value function at :

15. Derivatives • Derivatives in transition points: The right derivative of the optimal value function at :

16. Optimal Partition in the Neighboring Invariancy Interval • Solving an auxiliary self-dual quadratic problem we can obtain the optimal partition in the neighboring invariancy interval:

17. Algorithm and Numerical Illustration Illustrative problem:

18. Algorithm and Numerical Illustration Illustrative problem:

19. Conclusions and Future Work • The methodology allows: • solving both parametric linear and parametric quadratic optimization problems • doing simultaneous perturbation sensitivity analysis • All auxiliary problems can be solved in polynomial time • Future work: • extending methodology to the Parametric Second Order Conic Optimization (robust optimization, financial and engineering applications) • completing the Matlab/C implementation of the algorithm

20. The EndThank You