Wolfe’s Combinatorial Method is Exponential Jamie Haddock STOC June 26, 2018 UC Davis/UCLA Poster in Bradbury/Rose and Hershey/Crocker at 8 pm joint with Jes´ us A. De Loera and Luis Rademacher https://arxiv.org/abs/1710.02608 1
Minimum Norm Point in Polytope We are interested in solving the problem (MNP( P )): min x ∈ P � x � 2 where P is a polytope, and determining the minimum dimension face, F , which achieves distance � x � 2 . 2
Minimum Norm Point in Polytope We are interested in solving the problem (MNP( P )): min x ∈ P � x � 2 p 5 O p 1 P p 2 x p 4 p 3 2
Minimum Norm Point in Polytope We are interested in solving the problem (MNP( P )): min x ∈ P � x � 2 p 5 O p 1 P p 2 x p 4 p 3 ⊲ can be solved in polynomial time via interior-point methods 2
Minimum Norm Point in Polytope We are interested in solving the problem (MNP( P )): min x ∈ P � x � 2 p 5 O p 1 P p 2 x p 4 p 3 ⊲ can be solved in polynomial time via interior-point methods ⊲ no strongly-polynomial time algorithm known (even for simplex MNP) 2
Applications • nearest point problem for transportation polytopes 3
Applications • nearest point problem for transportation polytopes • colorful linear programming 3
Applications • nearest point problem for transportation polytopes • colorful linear programming • submodular function minimization 3
Applications • nearest point problem for transportation polytopes • colorful linear programming • submodular function minimization ⊲ subroutine in Fujishige-Wolfe method 3
Applications • nearest point problem for transportation polytopes • colorful linear programming • submodular function minimization ⊲ subroutine in Fujishige-Wolfe method ⊲ machine learning - vision, large-scale learning 3
Applications • nearest point problem for transportation polytopes • colorful linear programming • submodular function minimization ⊲ subroutine in Fujishige-Wolfe method ⊲ machine learning - vision, large-scale learning • linear programming 3
Our Results I Theorem (De Loera, H., Rademacher ’17) Linear programming reduces to distance to a V -simplex in strongly-polynomial time. 4
Our Results I Theorem (De Loera, H., Rademacher ’17) Linear programming reduces to distance to a V -simplex in strongly-polynomial time. • Step 1: LP reduces to “membership in V -polytope.” 4
Our Results I Theorem (De Loera, H., Rademacher ’17) Linear programming reduces to distance to a V -simplex in strongly-polynomial time. • Step 1: LP reduces to “membership in V -polytope.” • Step 2: “Membership in V -polytope” reduces to “distance to V -simplex.” 4
Our Results I Theorem (De Loera, H., Rademacher ’17) Linear programming reduces to distance to a V -simplex in strongly-polynomial time. • Step 1: LP reduces to “membership in V -polytope.” • Step 2: “Membership in V -polytope” reduces to “distance to V -simplex.” If a strongly-polynomial method for projection onto a polytope exists then this gives a strongly-polynomial method for LP. 4
Our Results I Theorem (De Loera, H., Rademacher ’17) Linear programming reduces to distance to a V -simplex in strongly-polynomial time. • Step 1: LP reduces to “membership in V -polytope.” • Step 2: “Membership in V -polytope” reduces to “distance to V -simplex.” If a strongly-polynomial method for projection onto a polytope exists then this gives a strongly-polynomial method for LP. It was previously known that linear programming reduces to MNP on a polytope in weakly-polynomial time [Fujishige, Hayashi, Isotani ’06]. 4
Background Lemma (Wolfe ’74) Let P = conv ( p 1 , p 2 , ..., p m ) . Then x ∈ P is MNP ( P ) if and only if x T p j ≥ � x � 2 2 for all j = 1 , 2 , ..., m . p 5 O p 1 P p 2 x p 4 p 3 { y : x T y = � x � 2 2 } 5
Background Lemma (Wolfe ’74) Let P = conv ( p 1 , p 2 , ..., p m ) . Then x ∈ P is MNP ( P ) if and only if x T p j ≥ � x � 2 2 for all j = 1 , 2 , ..., m . Def : An affinely independent set of input points Q = { q 1 , q 2 , ..., q k } is a corral if MNP(aff( Q )) ∈ relint(conv( Q )). 5
Background Lemma (Wolfe ’74) Let P = conv ( p 1 , p 2 , ..., p m ) . Then x ∈ P is MNP ( P ) if and only if x T p j ≥ � x � 2 2 for all j = 1 , 2 , ..., m . Def : An affinely independent set of input points Q = { q 1 , q 2 , ..., q k } is a corral if MNP(aff( Q )) ∈ relint(conv( Q )). q 1 q 2 O 5
Background Lemma (Wolfe ’74) Let P = conv ( p 1 , p 2 , ..., p m ) . Then x ∈ P is MNP ( P ) if and only if x T p j ≥ � x � 2 2 for all j = 1 , 2 , ..., m . Def : An affinely independent set of input points Q = { q 1 , q 2 , ..., q k } is a corral if MNP(aff( Q )) ∈ relint(conv( Q )). q 1 q 2 q 1 q 2 O q 3 O 5
Background Lemma (Wolfe ’74) Let P = conv ( p 1 , p 2 , ..., p m ) . Then x ∈ P is MNP ( P ) if and only if x T p j ≥ � x � 2 2 for all j = 1 , 2 , ..., m . Def : An affinely independent set of input points Q = { q 1 , q 2 , ..., q k } is a corral if MNP(aff( Q )) ∈ relint(conv( Q )). N q 1 q 2 N q 1 q 2 O q 3 O 5
Background Lemma (Wolfe ’74) Let P = conv ( p 1 , p 2 , ..., p m ) . Then x ∈ P is MNP ( P ) if and only if x T p j ≥ � x � 2 2 for all j = 1 , 2 , ..., m . Def : An affinely independent set of input points Q = { q 1 , q 2 , ..., q k } is a corral if MNP(aff( Q )) ∈ relint(conv( Q )). N q 1 N q 2 q 1 q 2 O q 3 O N q 1 q 2 X O 5
Background Lemma (Wolfe ’74) Let P = conv ( p 1 , p 2 , ..., p m ) . Then x ∈ P is MNP ( P ) if and only if x T p j ≥ � x � 2 2 for all j = 1 , 2 , ..., m . Def : An affinely independent set of input points Q = { q 1 , q 2 , ..., q k } is a corral if MNP(aff( Q )) ∈ relint(conv( Q )). N q 1 q 2 N q 1 q 2 O q 3 O Note : Singletons are corrals. 5
Background Lemma (Wolfe ’74) Let P = conv ( p 1 , p 2 , ..., p m ) . Then x ∈ P is MNP ( P ) if and only if x T p j ≥ � x � 2 2 for all j = 1 , 2 , ..., m . Def : An affinely independent set of input points Q = { q 1 , q 2 , ..., q k } is a corral if MNP(aff( Q )) ∈ relint(conv( Q )). N q 1 q 2 N q 1 q 2 O q 3 O Note : Singletons are corrals. Note : There is a corral in P whose convex hull contains MNP( P ). 5
Idea of Wolfe’s Method • combinatorial method for computing MNP on a vertex-representation polytope 6
Idea of Wolfe’s Method • combinatorial method for computing MNP on a vertex-representation polytope • searches through sequence of corrals whose MNPs have strictly decreasing norm until finding optimal 6
Idea of Wolfe’s Method • combinatorial method for computing MNP on a vertex-representation polytope • searches through sequence of corrals whose MNPs have strictly decreasing norm until finding optimal • maintains set C defining the current corral and the MNP in C , x 6
Idea of Wolfe’s Method • combinatorial method for computing MNP on a vertex-representation polytope • searches through sequence of corrals whose MNPs have strictly decreasing norm until finding optimal • maintains set C defining the current corral and the MNP in C , x Sketch: ⊲ Start with any point in P as C and x . 6
Idea of Wolfe’s Method • combinatorial method for computing MNP on a vertex-representation polytope • searches through sequence of corrals whose MNPs have strictly decreasing norm until finding optimal • maintains set C defining the current corral and the MNP in C , x Sketch: ⊲ Start with any point in P as C and x . ⊲ If C not the optimal corral (checked via optimality criterion): 6
Idea of Wolfe’s Method • combinatorial method for computing MNP on a vertex-representation polytope • searches through sequence of corrals whose MNPs have strictly decreasing norm until finding optimal • maintains set C defining the current corral and the MNP in C , x Sketch: ⊲ Start with any point in P as C and x . ⊲ If C not the optimal corral (checked via optimality criterion): - Find improving point p (given via optimality criterion) to add to C . 6
Idea of Wolfe’s Method • combinatorial method for computing MNP on a vertex-representation polytope • searches through sequence of corrals whose MNPs have strictly decreasing norm until finding optimal • maintains set C defining the current corral and the MNP in C , x Sketch: ⊲ Start with any point in P as C and x . ⊲ If C not the optimal corral (checked via optimality criterion): - Find improving point p (given via optimality criterion) to add to C . - Let x “follow gravity” within conv( C ) while dimension of current face decreases to a face which is a corral and update C . 6
Related Methods ⊲ related to von Neumann’s algorithm for linear programming ⊲ related to Frank-Wolfe method for convex programming ⊲ related to Gilbert’s procedure for quadratic programming 7
Related Methods ⊲ related to von Neumann’s algorithm for linear programming ⊲ related to Frank-Wolfe method for convex programming ⊲ related to Gilbert’s procedure for quadratic programming ⊲ generalized by Hanson-Lawson procedure for non-negative least-squares 7
Our Results II Theorem (De Loera, H., Rademacher ’17) There exists a set of points P ( d ) ⊂ R d for d = 2 k − 1 where Wolfe’s method with the minnorm insertion rule visits a sequence of corrals C ( d ) of length 5 · 2 k − 1 − 4 . 8
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