BiParametric Convex Quadratic Optimization Tams Terlaky Lehigh University

Bi-Parametric Convex Quadratic Optimization Tamás Terlaky Lehigh University Joint work with Alireza Ghaffari-Hadigheh and Oleksandr Romanko RUTCOR 2009: Dedicated to the 80 th Birthday of Professor András Prékopa

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Outline Introduction n Quadratic optimization, optimal partition n Uni-Parametric quadratic optimization n Bi-Parametric quadratic optimization n Numerical illustration n Fundamental properties n Algorithm n Conclusions and future work n 3

Introduction: Parametric Optimization General framework of parametric optimization n Multidimensional parameter is introduced into objective function and/or constraints n The goal is to find • • n n – optimal solution – optimal value function Generalization of sensitivity analysis Applications: -objective multi optimization 4

Introduction: Multi-Objective Optimization as Parametric Problem n Multi-objective optimization: OBJECTIVE SPACE n f 2 Multi-objective optimization with weighting method: identify Pareto frontier (all nondominated solutions) f* 2 n Parametric formulation: f 1 f* 1 5

Introduction: Quadratic Optimization and Its Parametric Counterpart n Convex Quadratic Optimization (QO) problem: n Bi-Parametric Convex Quadratic Optimization (PQO) problem: n Bi-parametric QO generalizes three models: uni-parametric QO 6

Sensity Analysis: Just be careful! 7

Optimal Partition for QO Convex Quadratic Optimization problems: n Primal n Optimality conditions: n Maximally complementary solution: • • LO: QO: IPMs !!! Dual and - strictly complementary solution , but may not hold maximally complementary solution maximizes the number of non-zero coordinates in and 8

Optimal Partition for QO n The optimal partition of the index set {1, 2, …, n} is n The optimal partition is unique!!! An optimal solution is maximally complementary iff: n Example: for maximally complementary solution with: 9

Uni-Parametric Quadratic Optimization n Primal and dual perturbed problems: n For some solution partition n The left and right extreme points of the invariancy interval: we are given the maximally complementary optimal of and with the optimal. - invariancy interval - transition points 10

Gengyang Uni-Parametric QO: Optimal Partition in the Neighboring Invariancy Interval Solve two auxiliary problems n How to proceed from the current invariancy interval to the next one? (1) (2) z z z 11

Uni-param QO: Numerical Illustration Solver output type l u B N T ( ) --------------------------------------------------------transition point -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 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 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 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 4 5 1 2 0. 00 invariancy interval +3. 33333 Inf 3 4 5 1 2 0. 00 2 - 0. 00 + 0. 00 12

Bi-Parametric Quadratic Optimization n Primal and dual perturbed problems: n Invariancy regions instead of invariancy intervals n Illustrative example: 13

Bi-Parametric Quadratic Optimization n Illustrative example Invariancy regions 14

Bi-Parametric Quadratic Optimization n Illustrative example: Optimal value function 15

Bi-Parametric Quadratic Optimization n The optimalregions value function is aare bivariate quadratic Invariancy that transition lines or Invariancy region is a convex set and its closure The optimal boundary value of a function non-trivial is continuous invariancy region and piecewise consists ofregions. a is Optimal partition is constant: on invariancy function on invariancy region singletons are called trivial regions. Otherwise, they are called a polyhedron that might be unbounded. bivariate finite number quadratic of line segments. non-trivial invariancy regions. 16

Bi-Parametric QO: Algorithm n Idea: reduce bi-parametric QO problem to a series of uni-paramteric QO problems with where 17

Bi-Parametric QO: Algorithm n Start from Choose n Solve n , determine the optimal partition , and where n n n Solve where Now, two points and of the invariancy region are known Consider cases and on the boundary 18

Bi-Parametric QO: Algorithm n Case 19

Bi-Parametric QO: Algorithm n Case Ø : and 20

Bi-Parametric QO: Algorithm n Case Ø : and 21

Bi-Parametric QO: Algorithm n Case Ø : and Ø : back to the first or the second case 22

Bi-Parametric QO: Algorithm n Invariancy region exploration 23

Bi-Parametric QO: Algorithm n Enumerating all invariancy regions To-be-processed queue Completed queue vertex cell edge 24

Conclusions and Future Work n § § n n Developed an IPM-based technique for solving bi-parametric problems that • extends the results of the uni-parametric case • allows solving both bi-parametric linear and bi-parametric quadratic optimization problems • systematically explores the optimal value surface Polynomial-time algorithm in the output size Applications in finance, IMRT, data mining Improving the implementation Extending methodology to • Parametric Second Order Conic Optimization • Multi-Parametric Quadratic Optimization 25

References n 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. n A. Ghaffari-Hadigheh, O. Romanko, and T. Terlaky. Sensitivity Analysis in Convex Quadratic Optimization: Simultaneous Perturbation of the Objective and Right-Hand-Side Vectors. Algorithmic Operations Research, Vol. 2(2), 2007. n A. Ghaffari-Hadigheh, O. Romanko, and T. Terlaky. Bi. Parametric Convex Quadratic Optimization. To appear in Optimization Methods and Software, 2009. n A. Ghaffari-Hadigheh, O. Romanko, and T. Terlaky. On Bi. Parametric Programming in Quadratic Optimization. Proceedings of Eur. OPT-2008, 2008. 26