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

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

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 Introduction

Financial Portfolio Example Problem of choosing an efficient portfolio of assets

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): Financial Portfolio Example Original formulationParametric formulation

Financial Portfolio Example

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 Primal Quadratic Optimization problem: Quadratic Optimization Dual Quadratic Optimization problem: KKT conditions:

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

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 Parametric Quadratic Programming Primal and dual perturbed problems: Properties: Domain of is a closed interval Optimal partition plays a key role

Parametric Quadratic Programming

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

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

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

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

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

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:

Algorithm and Numerical Illustration Illustrative problem:

Algorithm and Numerical Illustration Illustrative problem:

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

The End Thank You