Embedding Formulations, Complexity and Representability for Unions of Convex Sets

Juan Pablo VielmaMassachusetts 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?

Embedding Formulations

Nonlinear Mixed 0-1 Integer Formulations

• Modeling Finite Alternatives = Unions of Convex Sets

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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 3 / 15

Embedding Formulations

Constructing Non-extended Ideal Formulations

• Pure Integer : • Mixed Integer:

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Embedding Formulations

Embedding Formulation = Ideal non-Extended

(Cayley) Embedding

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Embedding Formulations

Alternative Encodings

• 0-1 encodings guarantee validity

• Options for 0-1 encodings:– Traditional or Unary encoding

– Binary encodings:– Others (e.g. incremental encoding unary)

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Embedding Formulations

Unary Encoding, Minkowski Sum and Cayley Trick

For traditional or unary encoding:

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Embedding Formulations

Encoding Selection Matters

• 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.)

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Embedding Formulations

Complexity of Family of Polyhedra

• Embedding complexity = smallest ideal formulation

• Relaxation complexity = smallest formulation

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Embedding Formulations

• 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

General InequalitiesTotal

General InequalitiesTotal

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Embedding Formulations

Example of Constant Sized Non-Ideal Formulation

• Polynomial sized coefficients:–

• 80 fractional extreme points for n = 5

3 4

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Embedding Formulations

Faces for Ideal Formulation with Unary Encoding

• Two types of facets (or faces):–

–

– Not all combinations of faces

– Which ones are valid?

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Embedding Formulations

Valid Combinations = Common Normals

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Embedding Formulations

• 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

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Embedding Formulations

Bad Example: Representability Issues

can fail to be basic semi-algebraic

Description with finite number of (quadratic) polynomial inequalities?

Zariski closure of boundary

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Embedding Formulations

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

Example: Pizza Slices

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= 4 conic + 4 linear inequalities

Embedding Formulations

Final Positive Results

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• Unions of Homothetic Convex Bodies

(all extreme points exposed)

• Generalizes polyhedral results from Balas ‘85, Jeroslow ’88 and Blair ‘90

Embedding Formulations

Easy to Recover and Generalize Existing Results

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• 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)

Embedding Formulations

Right Embedding = Significant Improvements

4 81 s

10 s

100 s

1000 s

10000 s

ExtendedGood "Big-M"Binary Embedding

• Results from Nemhauser, Ahmed and V. ’10 using CPLEX 11

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• Non-extended and ideal formulations provide a significant computational advantage