1. Automatic Control Laboratory, ETH Zürich www.control.ethz.ch Automatic dualization Johan Löfberg

2. Outline YALMIP? Recent developments Automatic dualization –Primal-dual conic problems –What is the problem? –Implementation Applications of automatic dualization Conclusions

3. What is YALMIP? Free MATLAB toolbox for rapid definition, manipulation and solution of optimization problems. Originally aimed towards linear semidefinite programming. Supports linear, quadratic, second-order cone, semidefinite, bilinear, geometric, parametric and mixed integer programming. Interfaces most state-of-the-art solvers.

4. Supported solvers

5. Recent developments 1. Strengthened modelling framework for non-convex problems Automatic derivation of LPAutomatic derivation of MILP 2. Improved integration with MPT for multi-parametric optimization 3. Performance improvements in automatic dualization

6. Automatic dualization

7. Primal-dual conic pairs Everything revolves around the primal-dual conic pair We work with mixed linear, quadratic and semidefinite cones

8. Primal-dual conic pairs Much more convenient in practice to allow free variables (equality constraints) But this is where the problems arise...

9. Primal or dual? Example: Silly LP Dimension of the problem? Primal: Dual:

10. Primal or dual? Example: MAXCUT Dimension of the problem? Primal: Dual: Ouch! variables YALMIP always interprets in dual form 

11. Problem: Data ( C, A, b, F, f ) not unique (including dimensions) Primal or dual? Problem: YALMIP always interprets in dual form Wanted: Solution: Automatic conversion from default dual form interpretation to a symbolic primal form interpretation Detect and extract primal form numerical model and return symbolic dual of this model.

12. Implementation

13. Where are we? External solver (SeDuMi, SDPA, CPLEX...) Solver interface YALMIP to solver format conversion YALMIP core Parsing, classification, conversion, solution treatment etc Symbolic modeling level User experience in MATLAB Solver specific format Internal numerical format Symbolic model Internal solvers (MIP, BMIBNB, CUTSDP, MPMIQP) Features SOS, Moment,Block-diagonalization, Dualization,...

14. Input: Symbolic YALMIP model (with constraints ) Implementation Output: Symbolic YALMIP model Notation: Primal cone: Variable X & a constraint of the type Translated primal cone: Constraint of the type Dual cone: Anything else (except equality constraints)

15. Implementation First step: Find simple primal cones Initialize: Detected cones and offsets X ={}, H = {}

16. Implementation Second step : Find translated primal cones

17. Implementation Third step: Introduce slacks for dual cones Status: All primal cone variables are detected, remaining variables are free and remaining constraints are equality constraints.

18. Implementation Fourth step: Extract numerical data from remaining constraints Intermediate result: Uniquely defined model Return: Symbolic dual form model of detected primal problem Original variables are duals to constraints in the resulting symbolic model. YALMIP automatically tracks this information and updates numerical data accordingly when the new model is solved.

19. Applications

20. MAXCUT (Computations using SDPT3 3.1)

21. Sum-of-squares Find s such that a polynomial p ( x, s ) is non-negative in x Can be addressed using sum-of-squares which leads to YALMIP formulates all SOS problems in this primal form, but solving them like this would be inefficient (since it interprets the model in dual form) Hence, dualization needed Stupid! Why not derive the dual form directly?

22. Sum-of-squares Generality! 1. Additional constraints on s does not complicate dualization 2. 3....or derive an image model (QR or sparse basis, one line of code) For non-convex (non-linear/integer/rank constrained) SOS problems, we can solve the problem as stated...

23. Sum-of-squares

24. Stability analysis Piecewise affine system (simplified) SDP for piecewise quadratic Lyapunov certificate Dualize if some H is very tall compared to state dimension

25. Outlook & conclusion Automatic decision to dualize? Dualize quadratic objective functions? Partial dualization? Performance for large-scale problems... Very simple but very useful Commands:

26. Download http://control.ee.ethz.ch/~joloef/yalmip.php