1. Embedding Formulations, Complexity and Representability for Unions of Convex Sets Juan Pablo Vielma Massachusetts Institute of Technology CMO-BIRS Workshop: Modern Techniques in Discrete Optimization: Mathematics, Algorithms and Applications, Oaxaca, Mexico. November, 2015. Supported by NSF grant CMMI-1351619 Minkowski Sums: Good or Evil?

2. Nonlinear Mixed 0-1 Integer Formulations Modeling Finite Alternatives = Unions of Convex Sets Embedding Formulations1 / 15 -

3. Extended and Non-Extended Formulations for Large, but strong (ideal * ) Extended Non-Extended Small, but weak? Embedding Formulations * Integral y in extreme points of LP relaxation 2 / 15

4. Constructing Non-extended Ideal Formulations Embedding Formulations Pure Integer : Mixed Integer: 3 / 15

5. Embedding Formulation = Ideal non-Extended Embedding Formulations (Cayley) Embedding 4 / 15

6. Alternative Encodings Embedding Formulations 0-1 encodings guarantee validity Options for 0-1 encodings: – Traditional or Unary encoding – Binary encodings: – Others (e.g. incremental encoding unary) 5 / 15

7. Unary Encoding, Minkowski Sum and Cayley Trick Embedding Formulations For traditional or unary encoding: 6 / 15

8. Encoding Selection Matters Embedding Formulations Size of unary formulation is: (Lee and Wilson ’01) Variable Bounds General Inequalities Size of one binary formulation: (V. and Nemhauser ’08) Right embedding = significant computational advantage over alternatives (Extended, Big-M, etc.) 7 / 15

9. Complexity of Family of Polyhedra Embedding Formulations Embedding complexity = smallest ideal formulation Relaxation complexity = smallest formulation 8 / 15

10. Lower and Upper bounds for special structures: – e.g. for Special Order Sets of Type 2 (SOS2) on n variables Embedding complexity (ideal) Relaxation complexity (non-ideal) Relation to other complexity measures Still open questions (see V. 2015) Complexity Results Embedding Formulations General Inequalities Total General Inequalities Total 9 / 15

11. Example of Constant Sized Non-Ideal Formulation Embedding Formulations Polynomial sized coefficients: – 80 fractional extreme points for n = 5 3 4 10 / 15

12. Faces for Ideal Formulation with Unary Encoding Embedding Formulations Two types of facets (or faces): – – – Not all combinations of faces – Which ones are valid? 11 / 15

13. Valid Combinations = Common Normals Embedding Formulations12 / 15

14. Description of boundary of is easy if “normals condition” yields convex hull of 1 nonlinear constraint and point(s) Unary Embedding for Unions of Convex Sets Embedding Formulations13 / 15

15. Bad Example: Representability Issues Embedding Formulations can fail to be basic semi-algebraic Description with finite number of (quadratic) polynomial inequalities? Zariski closure of boundary 14 / 15

16. Summary Embedding Formulations = Systematic procedure for strong (ideal) non-extended formulations – Encoding can significantly affect size Complexity of Union of Polyhedra beyond convex hull – Embedding Complexity (non-extended ideal formulation) – Relaxation Complexity (any non-extended formulation) – Still open questions on relations between complexity ( Embedding Formulations and Complexity for Unions of Polyhedra, arXiv:1506.01417 ) Embedding Formulations for Convex Sets – MINLP formulations – Can have representability issues Open question: minimum number of auxiliary variables for fixing this Embedding Formulations

17. Example: Pizza Slices Embedding Formulations16 / 24 = 4 conic + 4 linear inequalities

18. Final Positive Results Embedding Formulations17 / 24 Unions of Homothetic Convex Bodies (all extreme points exposed) Generalizes polyhedral results from Balas ‘85, Jeroslow ’88 and Blair ‘90

19. Easy to Recover and Generalize Existing Results Embedding Formulations18 / 24 Isotone function results from Hijazi et al. ‘12 and Bonami et al. ’15 (n=1, 2): – Can generalize to n ≥ 3 and two functions per set: Other special cases (previous slide)

20. Right Embedding = Significant Improvements Results from Nemhauser, Ahmed and V. ’10 using CPLEX 11 Embedding Formulations19 / 24 Non-extended and ideal formulations provide a significant computational advantage