Polyhedral Computation Jayant Apte ASPITRG October 3, 2012 Jayant Apte. ASPITRG 1 Part I October 3, 2012 Jayant Apte. ASPITRG 2 The Three Problems of Polyhedral Computation ● Representation Conversion ● H-representation of polyhedron ● V-Representation of polyhedron ● Projection ● ● Redundancy Removal ● Compute the minimal representation of a polyhedron October 3, 2012 Jayant Apte. ASPITRG 3 Outline ● Preliminaries ● Representations of convex polyhedra, polyhedral cones ● Homogenization - Converting a general polyhedron in to a cone in ● Polar of a convex cone ● The three problems ● Representation Conversion – Review of LRS – Double Description Method ● Projection of polyhedral sets – Fourier-Motzkin Elimination – Block Elimination – Convex Hull Method (CHM) ● Redundancy removal – Redundancy removal using linear programming October 3, 2012 Jayant Apte. ASPITRG 4 Outline for Part 1 ● Preliminaries ● Representations of convex polyhedra, polyhedral cones ● Homogenization – Converting a general polyhedron in to a cone in ● Polar of a convex cone ● The first problem ● Representation Conversion – Review of LRS – Double Description Method October 3, 2012 Jayant Apte. ASPITRG 5 Preliminaries October 3, 2012 Jayant Apte. ASPITRG 6 Convex Polyhedron October 3, 2012 Jayant Apte. ASPITRG 7 Examples of polyhedra Bounded- Polytope Unbounded - polyhedron October 3, 2012 Jayant Apte. ASPITRG 8 H-Representation of a Polyhedron October 3, 2012 Jayant Apte. ASPITRG 9 V-Representation of a Polyhedron October 3, 2012 Jayant Apte. ASPITRG 10 Example (0.5,0.5,1.5) (1,1,1) (0,1,1) (0,0,1) (1,1,0) (0,1,0) (1,0,0) (0,0,0) October 3, 2012 Jayant Apte. ASPITRG 11 Switching between the two representations : Representation Conversion Problem ● H-representation V-representation: The Vertex Enumeration problem ● Methods: ● Reverse Search, Lexicographic Reverse Search ● Double-description method ● V-representation H-representation The Facet Enumeration Problem ● Facet enumeration can be accomplished by using polarity and doing Vertex Enumeration October 3, 2012 Jayant Apte. ASPITRG 12 Polyhedral Cone October 3, 2012 Jayant Apte. ASPITRG 13 A cone in October 3, 2012 Jayant Apte. ASPITRG 14 Homogenization October 3, 2012 Jayant Apte. ASPITRG 15 H-polyhedra October 3, 2012 Jayant Apte. ASPITRG 16 Example(d=2,d+1=3) October 3, 2012 Jayant Apte. ASPITRG 17 Example October 3, 2012 Jayant Apte. ASPITRG 18 V-polyhedra October 3, 2012 Jayant Apte. ASPITRG 19 Polar of a convex cone October 3, 2012 Jayant Apte. ASPITRG 20 Polar of a convex cone Looks like an H-representation!!! Our ticket back to the original cone!!! October 3, 2012 Jayant Apte. ASPITRG 21 Polar: Intuition October 3, 2012 Jayant Apte. ASPITRG 22 Polar: Intuition October 3, 2012 Jayant Apte. ASPITRG 23 Polar: Intuition October 3, 2012 Jayant Apte. ASPITRG 24 Polar: Intuition October 3, 2012 Jayant Apte. ASPITRG 25 Use of Polar ● Representation Conversion ● No need to have two different algorithms i.e. for and . ● Redundancy removal ● No need to have two different algorithms for removing redundancies from H-representation and V-representation ● Safely assume that input is always an H- representation for both these problems October 3, 2012 Jayant Apte. ASPITRG 26 Polarity: Intuition October 3, 2012 Jayant Apte. ASPITRG 27 Polar: Intuition Then take polar again to get facets of this cone Perform Vertex Enumeration on this cone. October 3, 2012 Jayant Apte. ASPITRG 28 Problem #1 Representation Conversion October 3, 2012 Jayant Apte. ASPITRG 29 Review of Lexicographic Reverse Search October 3, 2012 Jayant Apte. ASPITRG 30 Reverse Search: High Level Idea ● Start with dictionary corresponding to the optimal vertex ● Ask yourself 'What pivot would have landed me at this dictionary if i was running simplex?' ● Go to that dictionary by applying reverse pivot to current dictionary ● Ask the same question again ● Generate the so-called 'reverse search tree' October 3, 2012 Jayant Apte. ASPITRG 31 Reverse Search on a Cube Ref.David Avis, lrs: A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm October 3, 2012 Jayant Apte. ASPITRG 32 Double Description Method ● First introduced in: “ Motzkin, T. S.; Raiffa, H.; Thompson, G. L.; Thrall, R. M. (1953). "The double description method". Contributions to the theory of games. Annals of Mathematics Studies. Princeton, N. J.: Princeton University Press. pp. 51–73” ● The primitive algorithm in this paper is very inefficient. (How? We will see later) ● Several authors came up with their own efficient implementation(viz. Fukuda, Padberg) ● Fukuda's implementation is called cdd October 3, 2012 Jayant Apte. ASPITRG 33 Some Terminology October 3, 2012 Jayant Apte. ASPITRG 34 Double Description Method: The High Level Idea ● An Incremental Algorithm ● Starts with certain subset of rows of H-representation of a cone to form initial H-representation ● Adds rest of the inequalities one by one constructing the corresponding V-representation every iteration ● Thus, constructing the V-representation incrementally. October 3, 2012 Jayant Apte. ASPITRG 35 How it works? October 3, 2012 Jayant Apte. ASPITRG 36 How it works? October 3, 2012 Jayant Apte. ASPITRG 37 Initialization October 3, 2012 Jayant Apte. ASPITRG 38 Initialization Very trivial and inefficient October 3, 2012 Jayant Apte. ASPITRG 39 Example: Initialization October 3, 2012 Jayant Apte. ASPITRG 40 Example: Initialization October 3, 2012 Jayant Apte. ASPITRG 41 Example : Initialization -1 -1 18 1 -1 0 0 1 -4 A B C 4.0000 14.0000 9.0000 4.0000 4.0000 9.0000 1.0000 1.0000 1.0000 October 3, 2012 Jayant Apte. ASPITRG 42 How it works? October 3, 2012 Jayant Apte. ASPITRG 43 Iteration: Insert a new constraint October 3, 2012 Jayant Apte. ASPITRG 44 Iteration 1 A B C 4.0000 14.0000 9.0000 4.0000 4.0000 9.0000 1.0000 1.0000 1.0000 October 3, 2012 Jayant Apte. ASPITRG 45 Iteration 1: Insert a new constraint Constraint 2 October 3, 2012 Jayant Apte. ASPITRG 46 Iteration 1: Insert a new constraint October 3, 2012 Jayant Apte. ASPITRG 47 Iteration 1: Insert a new constraint October 3, 2012 Jayant Apte. ASPITRG 48 Main Lemma for DD Method October 3, 2012 Jayant Apte. ASPITRG 49 Iteration 1: Insert a new constraint October 3, 2012 Jayant Apte. ASPITRG 50 Iteration 1: Get the new DD pair -1 -1 18 1 -1 0 0 1 -4 -1 1 6 10.0000 12.0000 4.0000 9.0000 4.0000 6.0000 4.0000 9.0000 1.0000 1.0000 1.0000 1.0000 October 3, 2012 Jayant Apte. ASPITRG 51 Iteration 2 October 3, 2012 Jayant Apte. ASPITRG 52 Iteration 2: Insert new constraint October 3, 2012 Jayant Apte. ASPITRG 53 Iteration 2: Insert new constraint October 3, 2012 Jayant Apte. ASPITRG 54 Iteration 2: Insert new constraint October 3, 2012 Jayant Apte. ASPITRG 55 Get the new DD pair -1 -1 18 1 -1 0 0 1 -4 -1 1 6 0 -1 8 9.2000 10.0000 8.0000 10.0000 12.0000 4.0000 8.0000 8.0000 8.0000 4.0000 6.0000 4.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 October 3, 2012 Jayant Apte. ASPITRG 56 Iteration 3 October 3, 2012 Jayant Apte. ASPITRG 57 Get the new DD pair -1 -1 18 1 -1 0 0 1 -4 -1 1 6 0 -1 8 1 1 -12 A B C D E F G H I J 6.2609 6.4000 6.0000 8.0000 7.2000 9.2000 10.0000 8.0000 10.0000 12.0000 5.7391 5.6000 6.0000 4.0000 4.8000 8.0000 8.0000 8.0000 4.0000 6.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 October 3, 2012 Jayant Apte. ASPITRG 58 Termination October 3, 2012 Jayant Apte. ASPITRG 59 Efficiency Issues ● Is the implementation I just described good enough? October 3, 2012 Jayant Apte. ASPITRG 60 Efficiency Issues ● Is the implementation I just described good enough? ● Hell no! – The implementation just described suffers from profusion of redundancy October 3, 2012 Jayant Apte. ASPITRG 61 Efficiency Issues We can totally do without this one!! October 3, 2012 Jayant Apte. ASPITRG 62 Efficiency Issues And without all these!!! October 3, 2012 Jayant Apte. ASPITRG 63 Efficiency Issues October 3, 2012 Jayant Apte. ASPITRG 64 The Primitive DD method October 3, 2012 Jayant Apte. ASPITRG 65 What to do? ● Add some more structure ● Strengthen the Main Lemma October 3, 2012 Jayant Apte. ASPITRG 66 Add some more structure October 3, 2012 Jayant Apte. ASPITRG 67 Some definitions October 3, 2012 Jayant Apte. ASPITRG 68 October 3, 2012 Jayant Apte. ASPITRG 69 October 3, 2012 Jayant Apte. ASPITRG 70 Algebraic Characterization of adjacency October 3, 2012 Jayant Apte. ASPITRG 71 Combinatorial Characterization of adjacency (Combinatorial Oracle) October 3, 2012 Jayant Apte. ASPITRG 72 Example: Combinatorial Adjacency Oracle October 3, 2012 Jayant Apte. ASPITRG 73 Example: Combinatorial Adjacency Oracle Only the pairs and will be used to generate new rays October 3, 2012 Jayant Apte. ASPITRG 74 Strengthened Main Lemma for DD Method October 3, 2012 Jayant Apte. ASPITRG 75 Procedural Description October 3, 2012 Jayant Apte. ASPITRG 76 Part 2 October 3, 2012 Jayant Apte. ASPITRG 77 ● Projection of polyhedral sets – Fourier-Motzkin Elimination – Block Elimination – Convex Hull Method (CHM) ● Redundancy removal – Redundancy removal using linear programming October 3, 2012 Jayant Apte. ASPITRG 78 Projection of a Polyhedra October 3, 2012 Jayant Apte. ASPITRG 79 Fourier-Motzkin Elimination ● Named after Joseph Fourier and Theodore Motzkin October 3, 2012 Jayant Apte. ASPITRG 80 How it works? October 3, 2012 Jayant Apte. ASPITRG 81 How it works? Contd... October 3, 2012 Jayant Apte. ASPITRG 82 Efficiency of FM algorithm October 3, 2012 Jayant Apte. ASPITRG 83 Convex Hull Method(CHM) ● First appears in “C. Lassez and J.-L.