Embedding Formulations, Complexity and Representability for Unions of Convex Sets
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1 , 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, Supported by NSF grant CMMI
2 Nonlinear Mixed 0-1 Integer Formulations Modeling Finite Alternatives = Unions of Convex Sets 1 / 15
3 Extended and Non-Extended Formulations for Extended Non-Extended Large, but strong (ideal * ) Small, but weak? * Integral y in extreme points of LP relaxation 2 / 15
4 Constructing Non-extended Ideal Formulations Pure Integer : Mixed Integer: 3 / 15
5 Embedding Formulation = Ideal non-extended (Cayley) Embedding 4 / 15
6 Alternative Encodings encodings guarantee validity x 2 x x x x x 1 0.0!0.5!0.5!0.5!1.0!1.0!1.0 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 x 2 x 1 For traditional or unary encoding: 6 / 15
8 Encoding Selection Matters Size of unary formulation is: (Lee and Wilson 01) f(x,y) y General Inequalities Variable Bounds Size of one binary formulation: (V. and Nemhauser 08) x Right embedding = significant computational advantage over alternatives (Extended, Big-M, etc.) 7 / 15
9 Complexity of Family of Polyhedra Embedding complexity = smallest ideal formulation 1 Relaxation complexity = smallest formulation / 15
10 Complexity Results 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 General Inequalities Total General Inequalities Total Still open questions (see V. 2015) 9 / 15
11 Example of Constant Sized Non-Ideal Formulation Polynomial sized coefficients: 80 fractional extreme points for n = / 15
12 Faces for Ideal Formulation with Unary Encoding Two types of facets (or faces): 1 0 Not all combinations of faces Which ones are valid? 11 / 15
13 Valid Combinations = Common Normals 12 / 15
14 Unary Embedding for Unions of Convex Sets x 2 x 1 x 2 Description of boundary of is easy if normals condition yields convex hull of 1 nonlinear constraint and point(s) x 1 13 / 15
15 Bad Example: Representability Issues x 2 x 1 x 2 Zariski closure of boundary x 1 Description with finite number of (quadratic) polynomial inequalities? can fail to be basic semi-algebraic 14 / 15
16 Summary = 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 ( and Complexity for Unions of Polyhedra, arxiv: ) for Convex Sets MINLP formulations Can have representability issues Open question: minimum number of auxiliary variables for fixing this
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