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Robert Weismantel
Bridging the gap between fixed and variable dimension
ETH Zurich
Lecture 6
Robert Weismantel Lecture 6 1 / 27
Table of contents
Part 1
Affine TU decomposition of matrices(based on joint work with J. Bader, R. Hildebrand and R. Zenklusen)
Part 2
A polyhedral Frobenius problem(based on joint work with T. Oertel and D. Adjiashvili)
Robert Weismantel Lecture 6 2 / 27
Return to ILP
goal: solve Integer LinearProgramming problem (IP)
max cTxs.t. Ax ≤ b
x ∈ Zn
x ∈ Rn, W x ∈ Zk
2x1+x2∈Z
c
••
•
NP-hard [Karp, 1971], polynomial if n is constant [Lenstra, 1983]
MILP relaxation is stronger than LP relaxation
MILP relaxation is polynomial time solvable if k = O( log nlog log n ) [Kannan,
1986], based on [Lenstra, Lenstra and Lovasz, 1982]
Robert Weismantel Lecture 6 3 / 27
Return to ILP
goal: solve Integer LinearProgramming problem (IP)
max cTxs.t. Ax ≤ b
����x ∈ Zn x ∈ Rn
, W x ∈ Zk
2x1+x2∈Z
c
•
•
•
NP-hard [Karp, 1971], polynomial if n is constant [Lenstra, 1983]
MILP relaxation is stronger than LP relaxation
MILP relaxation is polynomial time solvable if k = O( log nlog log n ) [Kannan,
1986], based on [Lenstra, Lenstra and Lovasz, 1982]
Robert Weismantel Lecture 6 3 / 27
Return to ILP
goal: solve Integer LinearProgramming problem (IP)
max cTxs.t. Ax ≤ b
����x ∈ Zn x ∈ Rn, W x ∈ Zk
2x1+x2∈Zc
•••
NP-hard [Karp, 1971], polynomial if n is constant [Lenstra, 1983]
MILP relaxation is stronger than LP relaxation
MILP relaxation is polynomial time solvable if k = O( log nlog log n ) [Kannan,
1986], based on [Lenstra, Lenstra and Lovasz, 1982]
Robert Weismantel Lecture 6 3 / 27
Return to ILP
goal: solve Integer LinearProgramming problem (IP)
max cTxs.t. Ax ≤ b
����x ∈ Zn x ∈ Rn, W x ∈ Zk
2x1+x2∈Zc
•••
NP-hard [Karp, 1971], polynomial if n is constant [Lenstra, 1983]
MILP relaxation is stronger than LP relaxation
MILP relaxation is polynomial time solvable if k = O( log nlog log n ) [Kannan,
1986], based on [Lenstra, Lenstra and Lovasz, 1982]
Robert Weismantel Lecture 6 3 / 27
W x ∈ Zk reformulation
goal: describe the integer hull of a polyhedron P
Desired property
For W ∈ Zk×n,
conv(P ∩ Zn) ⊆ conv({x ∈ P |W x ∈ Zk})
number of integrality constraints reduced from n to k
MILPs with few integer variables are solved efficiently, e.g. usingcutting planes and branch-and-bound techniques [Lodi, 2010]
Robert Weismantel Lecture 6 4 / 27
W x ∈ Zk reformulation
goal: describe the integer hull of a polyhedron P
Desired property
For W ∈ Zk×n,
conv(P ∩ Zn)= conv({x ∈ P |W x ∈ Zk})
number of integrality constraints reduced from n to k
MILPs with few integer variables are solved efficiently, e.g. usingcutting planes and branch-and-bound techniques [Lodi, 2010]
Robert Weismantel Lecture 6 4 / 27
W x ∈ Zk reformulation
goal: describe the integer hull of a polyhedron P
Desired property
For W ∈ Zk×n,
conv(P ∩ Zn)= conv({x ∈ P |W x ∈ Zk})
number of integrality constraints reduced from n to k
MILPs with few integer variables are solved efficiently, e.g. usingcutting planes and branch-and-bound techniques [Lodi, 2010]
Robert Weismantel Lecture 6 4 / 27
A 0-1 knapsack polytope
conv({
x ∈ {0, 1}8 | x1 + x2 + x3 + x4 + 4x5 + 4x6 + 5x7 + 5x8 ≤ 11})
polyhedral description: 39 inequalities
replace x ∈ Z8 by x5 + x6 + x7 + x8 ∈ Z
for all knapsack polytopes possible: aggregate variables withneighboring weights
Robert Weismantel Lecture 6 5 / 27
A 0-1 knapsack polytope
conv({
x ∈ {0, 1}8 | x1 + x2 + x3 + x4 + 4x5 + 4x6 + 5x7 + 5x8 ≤ 11})
polyhedral description: 39 inequalities
replace x ∈ Z8 by x5 + x6 + x7 + x8 ∈ Z
for all knapsack polytopes possible: aggregate variables withneighboring weights
Robert Weismantel Lecture 6 5 / 27
The parity polytope
convex hull of all {0, 1}n vectors that have an even cardinality support
polyhedral description: 2n−1 inequalities [Jeroslow, 1975]
extended formulation with 4n − 1 inequalities [Carr and Konjevod, 2004]
described by
conv({x ∈ [0, 1]n | 1
2
n∑i=1
xi integral}),
projection of
conv({(x, z) ∈ [0, 1]n×R |n∑
i=1
xi−2z = 0, z ∈ Z})
(1, 1, 0)
(0, 0, 0)
(1, 0, 1)
(0, 1, 1)
Robert Weismantel Lecture 6 6 / 27
The parity polytope
convex hull of all {0, 1}n vectors that have an even cardinality support
polyhedral description: 2n−1 inequalities [Jeroslow, 1975]
extended formulation with 4n − 1 inequalities [Carr and Konjevod, 2004]
described by
conv({x ∈ [0, 1]n | 1
2
n∑i=1
xi integral}),
projection of
conv({(x, z) ∈ [0, 1]n×R |n∑
i=1
xi−2z = 0, z ∈ Z})
(1, 1, 0)
(0, 0, 0)
(1, 0, 1)
(0, 1, 1)
Robert Weismantel Lecture 6 6 / 27
The parity polytope
convex hull of all {0, 1}n vectors that have an even cardinality support
polyhedral description: 2n−1 inequalities [Jeroslow, 1975]
extended formulation with 4n − 1 inequalities [Carr and Konjevod, 2004]
described by
conv({x ∈ [0, 1]n | 1
2
n∑i=1
xi integral}),
projection of
conv({(x, z) ∈ [0, 1]n×R |n∑
i=1
xi−2z = 0, z ∈ Z})
(1, 1, 0)
(0, 0, 0)
(1, 0, 1)
(0, 1, 1)
Robert Weismantel Lecture 6 6 / 27
The parity polytope
convex hull of all {0, 1}n vectors that have an even cardinality support
polyhedral description: 2n−1 inequalities [Jeroslow, 1975]
extended formulation with 4n − 1 inequalities [Carr and Konjevod, 2004]
described by
conv({x ∈ [0, 1]n | 1
2
n∑i=1
xi integral}),
projection of
conv({(x, z) ∈ [0, 1]n×R |n∑
i=1
xi−2z = 0, z ∈ Z})
(1, 1, 0)
(0, 0, 0)
(1, 0, 1)
(0, 1, 1)
Robert Weismantel Lecture 6 6 / 27
Knapsack polytopes
Let n ≥ 4, a ∈ Zn, b ∈ Z.
Theorem.
For
P ={
x ∈ [0, 1]n | aTx ≤ b}
there exists W ∈ Z(n−2)×n s.t.
conv(P ∩ Zn) = conv({x ∈ P |W x ∈ Zn−2}).
Robert Weismantel Lecture 6 7 / 27
Knapsack polytopes
Let n ≥ 4, a ∈ Zn, b ∈ Z.
Theorem.
For P ={
x ∈ [0, 1]n | aTx ≤ b}
there exists W ∈ Z(n−2)×n s.t.
conv(P ∩ Zn) = conv({x ∈ P |W x ∈ Zn−2}).
Robert Weismantel Lecture 6 7 / 27
Knapsack polytopes: k ≤ n − 2 sufficient
Proof idea.
consider conv(P ∩ {0, 1}n) and conv({0, 1}n \ P)
, there existsbn2c-dimensional face of either one not intersecting the respectiveother polytope
any 2-dimensional plane through three 0-1 points can be described bya system W x = d with TU matrix W ∈ Z(n−2)×n
conv(P ∩ Zn) = conv({x ∈ P |W x ∈ Zn−2})
Robert Weismantel Lecture 6 8 / 27
Knapsack polytopes: k ≤ n − 2 sufficient
Proof idea.
consider conv(P ∩ {0, 1}n) and conv({0, 1}n \ P) , there existsbn2c-dimensional face of either one not intersecting the respectiveother polytope
any 2-dimensional plane through three 0-1 points can be described bya system W x = d with TU matrix W ∈ Z(n−2)×n
conv(P ∩ Zn) = conv({x ∈ P |W x ∈ Zn−2})
Robert Weismantel Lecture 6 8 / 27
Knapsack polytopes: k ≤ n − 2 sufficient
Proof idea.
consider conv(P ∩ {0, 1}n) and conv({0, 1}n \ P) , there existsbn2c-dimensional face of either one not intersecting the respectiveother polytope
any 2-dimensional plane through three 0-1 points can be described bya system W x = d with TU matrix W ∈ Z(n−2)×n
conv(P ∩ Zn) = conv({x ∈ P |W x ∈ Zn−2})
Robert Weismantel Lecture 6 8 / 27
Knapsack polytopes: k ≤ n − 2 sufficient
Proof idea.
consider conv(P ∩ {0, 1}n) and conv({0, 1}n \ P) , there existsbn2c-dimensional face of either one not intersecting the respectiveother polytope
any 2-dimensional plane through three 0-1 points can be described bya system W x = d with TU matrix W ∈ Z(n−2)×n
conv(P ∩ Zn) = conv({x ∈ P |W x ∈ Zn−2})
Robert Weismantel Lecture 6 8 / 27
Definition.
Affine TU decomposition: A = A + UW with U integral, [A;W ] TUAffine TU-dimension of A: minimal number of rows of W
matrices with affine TU-dimension 0: totally unimodular
Definition [Padberg, 1988].
A ∈ Zn×n is almost totally unimodular if A is not TUbut every proper submatrix of A is TU
1 0 11 1 00 1 1
building block for k-balanced matrices [Conforti, 2006]
almost TU matrices have affine TU-dimension 1:[a A] = [0n A] + a(1, 0, . . . , 0)
affine TU-dimension 1=nearly totally unimodular matrices [Gijswijt,
2005] (e.g. incidence matrix of nearly bipartite graphs)
Robert Weismantel Lecture 6 9 / 27
Definition.
Affine TU decomposition: A = A + UW with U integral, [A;W ] TUAffine TU-dimension of A: minimal number of rows of W
matrices with affine TU-dimension 0: totally unimodular
Definition [Padberg, 1988].
A ∈ Zn×n is almost totally unimodular if A is not TUbut every proper submatrix of A is TU
1 0 11 1 00 1 − 1
building block for k-balanced matrices [Conforti, 2006]
almost TU matrices have affine TU-dimension 1:[a A] = [0n A] + a(1, 0, . . . , 0)
affine TU-dimension 1=nearly totally unimodular matrices [Gijswijt,
2005] (e.g. incidence matrix of nearly bipartite graphs)
Robert Weismantel Lecture 6 9 / 27
Definition.
Affine TU decomposition: A = A + UW with U integral, [A;W ] TUAffine TU-dimension of A: minimal number of rows of W
matrices with affine TU-dimension 0: totally unimodular
Definition [Padberg, 1988].
A ∈ Zn×n is almost totally unimodular if A is not TUbut every proper submatrix of A is TU
1 0 11 1 00 1 1
building block for k-balanced matrices [Conforti, 2006]
almost TU matrices have affine TU-dimension 1:[a A] = [0n A] + a(1, 0, . . . , 0)
affine TU-dimension 1=nearly totally unimodular matrices [Gijswijt,
2005] (e.g. incidence matrix of nearly bipartite graphs)
Robert Weismantel Lecture 6 9 / 27
Definition.
Affine TU decomposition: A = A + UW with U integral, [A;W ] TUAffine TU-dimension of A: minimal number of rows of W
matrices with affine TU-dimension 0: totally unimodular
Definition [Padberg, 1988].
A ∈ Zn×n is almost totally unimodular if A is not TUbut every proper submatrix of A is TU
1 0 11 1 00 1 1
building block for k-balanced matrices [Conforti, 2006]
almost TU matrices have affine TU-dimension 1:[a A]
= [0n A] + a(1, 0, . . . , 0)
affine TU-dimension 1=nearly totally unimodular matrices [Gijswijt,
2005] (e.g. incidence matrix of nearly bipartite graphs)
Robert Weismantel Lecture 6 9 / 27
Definition.
Affine TU decomposition: A = A + UW with U integral, [A;W ] TUAffine TU-dimension of A: minimal number of rows of W
matrices with affine TU-dimension 0: totally unimodular
Definition [Padberg, 1988].
A ∈ Zn×n is almost totally unimodular if A is not TUbut every proper submatrix of A is TU
1 0 11 1 00 1 1
building block for k-balanced matrices [Conforti, 2006]
almost TU matrices have affine TU-dimension 1:[a A] = [0n A] + a(1, 0, . . . , 0)
affine TU-dimension 1=nearly totally unimodular matrices [Gijswijt,
2005] (e.g. incidence matrix of nearly bipartite graphs)
Robert Weismantel Lecture 6 9 / 27
Definition.
Affine TU decomposition: A = A + UW with U integral, [A;W ] TUAffine TU-dimension of A: minimal number of rows of W
matrices with affine TU-dimension 0: totally unimodular
Definition [Padberg, 1988].
A ∈ Zn×n is almost totally unimodular if A is not TUbut every proper submatrix of A is TU
1 0 11 1 00 1 1
building block for k-balanced matrices [Conforti, 2006]
almost TU matrices have affine TU-dimension 1:[a A] = [0n A] + a(1, 0, . . . , 0)
affine TU-dimension 1=nearly totally unimodular matrices [Gijswijt,
2005] (e.g. incidence matrix of nearly bipartite graphs)
Robert Weismantel Lecture 6 9 / 27
Theorem.
Affine TU decomposition: A = A + UW , U integral, [A;W ] TU.
Forall integral b, the polyhedron P = {x ∈ Rn | Ax ≤ b} satisfies
conv(P ∩ Zn) = conv({x ∈ P |W x ∈ Zk}).
Proof.
Pr ={
x ∈ Rn | Ax ≤ b− Ur,W x = r}
is an integral polyhedron for each r ∈ Zk ,
conv({
x ∈ Rn | Ax ≤ b,W x ∈ Zk})
= conv
⋃r∈Zk
Pr
is an integral set
still holds with additional bounds l ≤ x ≤ u (l,u ∈ Zn)
Robert Weismantel Lecture 6 10 / 27
Theorem.
Affine TU decomposition: A = A + UW , U integral, [A;W ] TU. Forall integral b, the polyhedron P = {x ∈ Rn | Ax ≤ b} satisfies
conv(P ∩ Zn) = conv({x ∈ P |W x ∈ Zk}).
Proof.
Pr ={
x ∈ Rn | Ax ≤ b− Ur,W x = r}
is an integral polyhedron for each r ∈ Zk ,
conv({
x ∈ Rn | Ax ≤ b,W x ∈ Zk})
= conv
⋃r∈Zk
Pr
is an integral set
still holds with additional bounds l ≤ x ≤ u (l,u ∈ Zn)
Robert Weismantel Lecture 6 10 / 27
Theorem.
Affine TU decomposition: A = A + UW , U integral, [A;W ] TU. Forall integral b, the polyhedron P = {x ∈ Rn | Ax ≤ b} satisfies
conv(P ∩ Zn) = conv({x ∈ P |W x ∈ Zk}).
Proof.
Pr ={
x ∈ Rn | Ax ≤ b− Ur,W x = r}
is an integral polyhedron for each r ∈ Zk
,
conv({
x ∈ Rn | Ax ≤ b,W x ∈ Zk})
= conv
⋃r∈Zk
Pr
is an integral set
still holds with additional bounds l ≤ x ≤ u (l,u ∈ Zn)
Robert Weismantel Lecture 6 10 / 27
Theorem.
Affine TU decomposition: A = A + UW , U integral, [A;W ] TU. Forall integral b, the polyhedron P = {x ∈ Rn | Ax ≤ b} satisfies
conv(P ∩ Zn) = conv({x ∈ P |W x ∈ Zk}).
Proof.
Pr ={
x ∈ Rn | Ax ≤ b− Ur,W x = r}
is an integral polyhedron for each r ∈ Zk ,
conv({
x ∈ Rn | Ax ≤ b,W x ∈ Zk})
= conv
⋃r∈Zk
Pr
is an integral set
still holds with additional bounds l ≤ x ≤ u (l,u ∈ Zn)
Robert Weismantel Lecture 6 10 / 27
Theorem.
Affine TU decomposition: A = A + UW , U integral, [A;W ] TU. Forall integral b, the polyhedron P = {x ∈ Rn | Ax ≤ b} satisfies
conv(P ∩ Zn) = conv({x ∈ P |W x ∈ Zk}).
Proof.
Pr ={
x ∈ Rn | Ax ≤ b− Ur,W x = r}
is an integral polyhedron for each r ∈ Zk ,
conv({
x ∈ Rn | Ax ≤ b,W x ∈ Zk})
= conv
⋃r∈Zk
Pr
is an integral set
still holds with additional bounds l ≤ x ≤ u (l,u ∈ Zn)
Robert Weismantel Lecture 6 10 / 27
Master knapsack polytopeP = conv({x ∈ {0, 1}n |
∑ni=1 ixi ≤ b})
use fewer than n integrality constraints by affine TU decomposition
aggregate neighboring coefficients:
(1, 2, . . . , n − 1, n) = (0, 1, 0, 1, . . . , 0, 1) + (1, 3, . . . , n − 1)
1 1
1 1. . .
1 1
affine TU decomposition with dn2e rows
there exists affine TU decomposition with 2d√ne − 2 rows
Robert Weismantel Lecture 6 11 / 27
Master knapsack polytopeP = conv({x ∈ {0, 1}n |
∑ni=1 ixi ≤ b})
use fewer than n integrality constraints by affine TU decomposition
aggregate neighboring coefficients:
(1, 2, . . . , n − 1, n) = (0, 1, 0, 1, . . . , 0, 1) + (1, 3, . . . , n − 1)
1 1
1 1. . .
1 1
affine TU decomposition with dn2e rows
there exists affine TU decomposition with 2d√ne − 2 rows
Robert Weismantel Lecture 6 11 / 27
Master knapsack polytopeP = conv({x ∈ {0, 1}n |
∑ni=1 ixi ≤ b})
use fewer than n integrality constraints by affine TU decomposition
aggregate neighboring coefficients:
(1, 2, . . . , n − 1, n) = (0, 1, 0, 1, . . . , 0, 1) + (1, 3, . . . , n − 1)
1 1
1 1. . .
1 1
affine TU decomposition with dn2e rows
there exists affine TU decomposition with 2d√ne − 2 rows
Robert Weismantel Lecture 6 11 / 27
Excursion: A generalization of total dual integrality
Affine TU decompositions A = A + UW ∈ Zm×n:conv({x ∈ Rn | Ax ≤ b,W x ∈ Zk}) is integral for all integral b
Now we ask integrality for specific fixed b ∈ Rm
Observation
Let W ∈ Zk×n. If the system Ax ≤ b,W x = d is total dual integral foreach integral vector d, then
conv({x ∈ Rn | Ax ≤ b,W x ∈ Zk})
is integral.
Robert Weismantel Lecture 6 12 / 27
Excursion: A generalization of total dual integrality
Affine TU decompositions A = A + UW ∈ Zm×n:conv({x ∈ Rn | Ax ≤ b,W x ∈ Zk}) is integral for all integral b
Now we ask integrality for specific fixed b ∈ Rm
Observation
Let W ∈ Zk×n. If the system Ax ≤ b,W x = d is total dual integral foreach integral vector d, then
conv({x ∈ Rn | Ax ≤ b,W x ∈ Zk})
is integral.
Robert Weismantel Lecture 6 12 / 27
A Knapsack Example
Let b, k ∈ Z+, k ≥ 3. Assume k−1k+1b ≤ ai ≤ b for i ∈ B, and
bk+1 < ai ≤ b
k for i ∈ S .
Theorem
The system∑i ′∈S:ai′>ai
xi ′ +∑
j∈B:aj>b−ai
xj + (k − 1) ·∑
j∈B xj ≤ k for all i ∈ S ,
xi ≤ 1 for all i ∈ S ,xi ≥ 0 for all i ∈ S ∪ B∑
j∈B xj ≤ 1∑j∈B xj = d
is integral for all d ∈ Z.
Robert Weismantel Lecture 6 13 / 27
A Knapsack Example
Let b, k ∈ Z+, k ≥ 3. Assume k−1k+1b ≤ ai ≤ b for i ∈ B, and
bk+1 < ai ≤ b
k for i ∈ S .
Theorem
The system∑i ′∈S:ai′>ai
xi ′ +∑
j∈B:aj>b−ai
xj + (k − 1) ·∑
j∈B xj ≤ k for all i ∈ S ,
xi ≤ 1 for all i ∈ S ,xi ≥ 0 for all i ∈ S ∪ B∑
j∈B xj ≤ 1∑j∈B xj = d
is integral for all d ∈ Z.
Robert Weismantel Lecture 6 13 / 27
Determining the affine TU-dimension
affine TU-dimension 0: decomposition procedure to decide if a matrixis TU [Seymour, 1980] in polynomial time [Truemper, 1990]
Theorem.
Given A ∈ Zm×n, one can decide in polynomial time if A has affineTU-dimension k , provided that m and k are fixed.
Proof.
W.l.o.g., A has distinct columns.Then the (unknown) TU matrix [A;W ] has distinct columns, there arepolynomially many such matrices.For each of these [A;W ], decide in polynomial time if there is an integralU such that A = A + UW (every row of U needs to be an integral solutionvector to a system of m linear equations).
Robert Weismantel Lecture 6 14 / 27
Determining the affine TU-dimension
affine TU-dimension 0: decomposition procedure to decide if a matrixis TU [Seymour, 1980] in polynomial time [Truemper, 1990]
Theorem.
Given A ∈ Zm×n, one can decide in polynomial time if A has affineTU-dimension k , provided that m and k are fixed.
Proof.
W.l.o.g., A has distinct columns.Then the (unknown) TU matrix [A;W ] has distinct columns, there arepolynomially many such matrices.For each of these [A;W ], decide in polynomial time if there is an integralU such that A = A + UW (every row of U needs to be an integral solutionvector to a system of m linear equations).
Robert Weismantel Lecture 6 14 / 27
Determining the affine TU-dimension
affine TU-dimension 0: decomposition procedure to decide if a matrixis TU [Seymour, 1980] in polynomial time [Truemper, 1990]
Theorem.
Given A ∈ Zm×n, one can decide in polynomial time if A has affineTU-dimension k , provided that m and k are fixed.
Proof.
W.l.o.g., A has distinct columns.Then the (unknown) TU matrix [A;W ] has distinct columns, there arepolynomially many such matrices.
For each of these [A;W ], decide in polynomial time if there is an integralU such that A = A + UW (every row of U needs to be an integral solutionvector to a system of m linear equations).
Robert Weismantel Lecture 6 14 / 27
Determining the affine TU-dimension
affine TU-dimension 0: decomposition procedure to decide if a matrixis TU [Seymour, 1980] in polynomial time [Truemper, 1990]
Theorem.
Given A ∈ Zm×n, one can decide in polynomial time if A has affineTU-dimension k , provided that m and k are fixed.
Proof.
W.l.o.g., A has distinct columns.Then the (unknown) TU matrix [A;W ] has distinct columns, there arepolynomially many such matrices.For each of these [A;W ], decide in polynomial time if there is an integralU such that A = A + UW (every row of U needs to be an integral solutionvector to a system of m linear equations).
Robert Weismantel Lecture 6 14 / 27
Theorem.
One can decide in polynomial time if A ∈ Zm×n has affine TU-dimensionk , provided that m and k are fixed.
Theorem.
To decide if A ∈ Zm×n has affine TU-dimension n is an NP-completeproblem.
Conjecture.
One can decide in polynomial time if A ∈ Zm×n has affine TU-dimensionk , provided that k is fixed.
Robert Weismantel Lecture 6 15 / 27
Theorem.
One can decide in polynomial time if A ∈ Zm×n has affine TU-dimensionk , provided that m and k are fixed.
Theorem.
To decide if A ∈ Zm×n has affine TU-dimension n is an NP-completeproblem.
Conjecture.
One can decide in polynomial time if A ∈ Zm×n has affine TU-dimensionk , provided that k is fixed.
Robert Weismantel Lecture 6 15 / 27
Theorem.
One can decide in polynomial time if A ∈ Zm×n has affine TU-dimensionk , provided that m and k are fixed.
Theorem.
To decide if A ∈ Zm×n has affine TU-dimension n is an NP-completeproblem.
Conjecture.
One can decide in polynomial time if A ∈ Zm×n has affine TU-dimensionk , provided that k is fixed.
Robert Weismantel Lecture 6 15 / 27
Table of contents
Part 2
A polyhedral Frobenius problem
Application of parametric integer optimization to nonlinear discretesystems. (based on joint work with T. Oertel and D. Adjiashvili)
Robert Weismantel Lecture 6 16 / 27
Motivation: bridge the gap
Nonlinear optimization in mixed integer variables
min{f (x , y) : Ax + By ≤ b, y ∈ Rn, x ∈ Zd}
Many results in fixed dimension d
d fixed, f convex (Grotschel, Lovasz, Schrijver ’1988)
d fixed, f and constraints quasi-convex polynomials (Khachiyan,Porkolab ’2000)
d fixed, f concave (Cook, Hartmann, Kannan, McDiarmid ’1992)
d = 2 and f polynomial of degree two (Del Pia, Weismantel ’2014)
Few results in variable dimension d
Separable-convex IP (Hochbaum, Shanthikumar ’1990)
N-fold IP (De Loera, Hemmecke, Onn, Weismantel ’2008)
Dynamic programming
Robert Weismantel Lecture 6 17 / 27
Motivation: bridge the gap
Nonlinear optimization in mixed integer variables
min{f (x , y) : Ax + By ≤ b, y ∈ Rn, x ∈ Zd}
Many results in fixed dimension d
d fixed, f convex (Grotschel, Lovasz, Schrijver ’1988)
d fixed, f and constraints quasi-convex polynomials (Khachiyan,Porkolab ’2000)
d fixed, f concave (Cook, Hartmann, Kannan, McDiarmid ’1992)
d = 2 and f polynomial of degree two (Del Pia, Weismantel ’2014)
Few results in variable dimension d
Separable-convex IP (Hochbaum, Shanthikumar ’1990)
N-fold IP (De Loera, Hemmecke, Onn, Weismantel ’2008)
Dynamic programming
Robert Weismantel Lecture 6 17 / 27
Motivation: bridge the gap
Nonlinear optimization in mixed integer variables
min{f (x , y) : Ax + By ≤ b, y ∈ Rn, x ∈ Zd}
Many results in fixed dimension d
d fixed, f convex (Grotschel, Lovasz, Schrijver ’1988)
d fixed, f and constraints quasi-convex polynomials (Khachiyan,Porkolab ’2000)
d fixed, f concave (Cook, Hartmann, Kannan, McDiarmid ’1992)
d = 2 and f polynomial of degree two (Del Pia, Weismantel ’2014)
Few results in variable dimension d
Separable-convex IP (Hochbaum, Shanthikumar ’1990)
N-fold IP (De Loera, Hemmecke, Onn, Weismantel ’2008)
Dynamic programming
Robert Weismantel Lecture 6 17 / 27
Under which conditions on the input is the followingproblem tractable?
(Parametric integer optimization problem)
min {f (Wx) : Ax ≤ b, x ∈ Zn}
The input
Matrices A ∈ Zm×n and W ∈ Zd×n, a vector b ∈ Zm
We assume to have access to a fiber oracle.
Given a y ∈ Zd returns x ∈ {z ∈ Zn : Az ≤ b},such that Wx = y , or states that no such x exists.
A function f : Qd → Q presented by an integer minimization oracle.
(Query: y∗ ← arg min{f (y) : By ≤ c , y ∈ Λ})
Robert Weismantel Lecture 6 18 / 27
Under which conditions on the input is the followingproblem tractable?
(Parametric integer optimization problem)
min {f (Wx) : Ax ≤ b, x ∈ Zn}
The input
Matrices A ∈ Zm×n and W ∈ Zd×n, a vector b ∈ Zm
We assume to have access to a fiber oracle.
Given a y ∈ Zd returns x ∈ {z ∈ Zn : Az ≤ b},such that Wx = y , or states that no such x exists.
A function f : Qd → Q presented by an integer minimization oracle.
(Query: y∗ ← arg min{f (y) : By ≤ c , y ∈ Λ})
Robert Weismantel Lecture 6 18 / 27
Under which conditions on the input is the followingproblem tractable?
(Parametric integer optimization problem)
min {f (Wx) : Ax ≤ b, x ∈ Zn}
The input
Matrices A ∈ Zm×n and W ∈ Zd×n, a vector b ∈ Zm
We assume to have access to a fiber oracle.
Given a y ∈ Zd returns x ∈ {z ∈ Zn : Az ≤ b},such that Wx = y , or states that no such x exists.
A function f : Qd → Q presented by an integer minimization oracle.
(Query: y∗ ← arg min{f (y) : By ≤ c , y ∈ Λ})
Robert Weismantel Lecture 6 18 / 27
Under which conditions on the input is the followingproblem tractable?
(Parametric integer optimization problem)
min {f (Wx) : Ax ≤ b, x ∈ Zn}
The input
Matrices A ∈ Zm×n and W ∈ Zd×n, a vector b ∈ Zm
We assume to have access to a fiber oracle.
Given a y ∈ Zd returns x ∈ {z ∈ Zn : Az ≤ b},such that Wx = y , or states that no such x exists.
A function f : Qd → Q presented by an integer minimization oracle.
(Query: y∗ ← arg min{f (y) : By ≤ c , y ∈ Λ})
Robert Weismantel Lecture 6 18 / 27
Under which conditions on the input is the followingproblem tractable?
(Parametric integer optimization problem)
min {f (Wx) : Ax ≤ b, x ∈ Zn}
The input
Matrices A ∈ Zm×n and W ∈ Zd×n, a vector b ∈ Zm
We assume to have access to a fiber oracle.
Given a y ∈ Zd returns x ∈ {z ∈ Zn : Az ≤ b},such that Wx = y , or states that no such x exists.
A function f : Qd → Q presented by an integer minimization oracle.
(Query: y∗ ← arg min{f (y) : By ≤ c , y ∈ Λ})
Robert Weismantel Lecture 6 18 / 27
Some Clear Boundaries of Tractability
min {f (Wx) : Ax ≤ b, x ∈ Zn}
We assume W to be given in unary representation.
Partition problem. w1, · · · ,wn ∈ Z+, D = 12
∑ni=1 wi .
Solvemin{(wT x − D)2 : x ∈ {0, 1}n}.
With fiber oracle - solvable with a single oracle call.
We assume d to be fixed
Leverage algorithms for minimization in fixed d .
Solvemin{f (y) : y ∈ Zd}
where f is convex.
Robert Weismantel Lecture 6 19 / 27
Some Clear Boundaries of Tractability
min {f (Wx) : Ax ≤ b, x ∈ Zn}We assume W to be given in unary representation.
Partition problem. w1, · · · ,wn ∈ Z+, D = 12
∑ni=1 wi .
Solvemin{(wT x − D)2 : x ∈ {0, 1}n}.
With fiber oracle - solvable with a single oracle call.
We assume d to be fixed
Leverage algorithms for minimization in fixed d .
Solvemin{f (y) : y ∈ Zd}
where f is convex.
Robert Weismantel Lecture 6 19 / 27
Some Clear Boundaries of Tractability
min {f (Wx) : Ax ≤ b, x ∈ Zn}We assume W to be given in unary representation.
Partition problem. w1, · · · ,wn ∈ Z+, D = 12
∑ni=1 wi .
Solvemin{(wT x − D)2 : x ∈ {0, 1}n}.
With fiber oracle - solvable with a single oracle call.
We assume d to be fixed
Leverage algorithms for minimization in fixed d .
Solvemin{f (y) : y ∈ Zd}
where f is convex.
Robert Weismantel Lecture 6 19 / 27
Some Clear Boundaries of Tractability
min {f (Wx) : Ax ≤ b, x ∈ Zn}We assume W to be given in unary representation.
Partition problem. w1, · · · ,wn ∈ Z+, D = 12
∑ni=1 wi .
Solvemin{(wT x − D)2 : x ∈ {0, 1}n}.
With fiber oracle - solvable with a single oracle call.
We assume d to be fixed
Leverage algorithms for minimization in fixed d .
Solvemin{f (y) : y ∈ Zd}
where f is convex.
Robert Weismantel Lecture 6 19 / 27
Some Clear Boundaries of Tractability
min {f (Wx) : Ax ≤ b, x ∈ Zn}We assume W to be given in unary representation.
Partition problem. w1, · · · ,wn ∈ Z+, D = 12
∑ni=1 wi .
Solvemin{(wT x − D)2 : x ∈ {0, 1}n}.
With fiber oracle - solvable with a single oracle call.
We assume d to be fixed
Leverage algorithms for minimization in fixed d .
Solvemin{f (y) : y ∈ Zd}
where f is convex.
Robert Weismantel Lecture 6 19 / 27
Some Clear Boundaries of Tractability
min {f (Wx) : Ax ≤ b, x ∈ Zn}We assume W to be given in unary representation.
Partition problem. w1, · · · ,wn ∈ Z+, D = 12
∑ni=1 wi .
Solvemin{(wT x − D)2 : x ∈ {0, 1}n}.
With fiber oracle - solvable with a single oracle call.
We assume d to be fixed
Leverage algorithms for minimization in fixed d .
Solvemin{f (y) : y ∈ Zd}
where f is convex.Robert Weismantel Lecture 6 19 / 27
Our Main Result (from a optimization point of view)
The assumptions
Let A ∈ Zm×n, W ∈ Zd×n, b ∈ Zm and f : Rd → R.
Let d be a fixed constant.
Let ∆ denote the maximum sub-determinant of A and W.
Theorem.
There is an algorithm that solves the non-linear optimization problem
min {f (Wx) : Ax ≤ b, x ∈ Zn}.
The number of oracle calls it performs (to the optimization and fiberoracles) is polynomial in n and ∆.
Robert Weismantel Lecture 6 20 / 27
Our Main Result (from a optimization point of view)
The assumptions
Let A ∈ Zm×n, W ∈ Zd×n, b ∈ Zm and f : Rd → R.
Let d be a fixed constant.
Let ∆ denote the maximum sub-determinant of A and W.
Theorem.
There is an algorithm that solves the non-linear optimization problem
min {f (Wx) : Ax ≤ b, x ∈ Zn}.
The number of oracle calls it performs (to the optimization and fiberoracles) is polynomial in n and ∆.
Robert Weismantel Lecture 6 20 / 27
Our Approach
Understand the set
R = {Wx : x ∈ Zn, Ax ≤ b} ⊂ Zd
Typically,R ⊂ Q = {Wx : Ax ≤ b} ∩ Zd but R 6= Q.
[0, 3]3 ∩ Z3
W =
(1 2 1−2 0 1
)
Robert Weismantel Lecture 6 21 / 27
The Main Tool - (Diagonal) Frobenius Number
Definition (Frobenius Number)
Given integers a1, · · · , an with gcd(a1, · · · , an) = 1, the Frobenius numberF (a1, · · · , an) is the largest integer k that can not be expressed as apositive integer combination of a1, · · · , an.
Some important facts about the Frobenius number
Finding F (a1, · · · , an) is also known as the coin problem.
F (a1, a2) = a1a2 − (a1 + a2) (Sylvester ’1884).
NP-hard to compute F (a1, · · · , an) (Kannan ’1992).
F (a1, · · · , an) ≤ cn‖(a1, · · · , an)‖22 (e.g. Brauer ’1942).
Robert Weismantel Lecture 6 22 / 27
The Main Tool - (Diagonal) Frobenius Number
Definition (Frobenius Number)
Given integers a1, · · · , an with gcd(a1, · · · , an) = 1, the Frobenius numberF (a1, · · · , an) is the largest integer k that can not be expressed as apositive integer combination of a1, · · · , an.
Some important facts about the Frobenius number
Finding F (a1, · · · , an) is also known as the coin problem.
F (a1, a2) = a1a2 − (a1 + a2) (Sylvester ’1884).
NP-hard to compute F (a1, · · · , an) (Kannan ’1992).
F (a1, · · · , an) ≤ cn‖(a1, · · · , an)‖22 (e.g. Brauer ’1942).
Robert Weismantel Lecture 6 22 / 27
The Main Tool - (Diagonal) Frobenius Number
Definition (Frobenius Number)
Given integers a1, · · · , an with gcd(a1, · · · , an) = 1, the Frobenius numberF (a1, · · · , an) is the largest integer k that can not be expressed as apositive integer combination of a1, · · · , an.
Some important facts about the Frobenius number
Finding F (a1, · · · , an) is also known as the coin problem.
F (a1, a2) = a1a2 − (a1 + a2) (Sylvester ’1884).
NP-hard to compute F (a1, · · · , an) (Kannan ’1992).
F (a1, · · · , an) ≤ cn‖(a1, · · · , an)‖22 (e.g. Brauer ’1942).
Robert Weismantel Lecture 6 22 / 27
The Main Tool - (Diagonal) Frobenius Number
Definition (Frobenius Number)
Given integers a1, · · · , an with gcd(a1, · · · , an) = 1, the Frobenius numberF (a1, · · · , an) is the largest integer k that can not be expressed as apositive integer combination of a1, · · · , an.
Some important facts about the Frobenius number
Finding F (a1, · · · , an) is also known as the coin problem.
F (a1, a2) = a1a2 − (a1 + a2) (Sylvester ’1884).
NP-hard to compute F (a1, · · · , an) (Kannan ’1992).
F (a1, · · · , an) ≤ cn‖(a1, · · · , an)‖22 (e.g. Brauer ’1942).
Robert Weismantel Lecture 6 22 / 27
The Main Tool - (Diagonal) Frobenius Number
Definition (Frobenius Number)
Given integers a1, · · · , an with gcd(a1, · · · , an) = 1, the Frobenius numberF (a1, · · · , an) is the largest integer k that can not be expressed as apositive integer combination of a1, · · · , an.
Some important facts about the Frobenius number
Finding F (a1, · · · , an) is also known as the coin problem.
F (a1, a2) = a1a2 − (a1 + a2) (Sylvester ’1884).
NP-hard to compute F (a1, · · · , an) (Kannan ’1992).
F (a1, · · · , an) ≤ cn‖(a1, · · · , an)‖22 (e.g. Brauer ’1942).
Robert Weismantel Lecture 6 22 / 27
The Main Tool - (Diagonal) Frobenius Number
Definition (Diagonal Frobenius Number)
Let M ∈ Zd×m (d ≤ m) and v = M1 such that
M has HNF Identity, and
C (M) = {λ : Mλ ≥ 0} is a pointed cone.
F (M) is the smallest integer t st (tv + C (M)) ∩ Zd ⊂ {Mx : x ∈ Zm+}.
M =
(2 3 40 1 −1
)
Robert Weismantel Lecture 6 23 / 27
The Main Tool - (Diagonal) Frobenius Number
Definition (Diagonal Frobenius Number)
Let M ∈ Zd×m (d ≤ m) and v = M1 such that
M has HNF Identity, and
C (M) = {λ : Mλ ≥ 0} is a pointed cone.
F (M) is the smallest integer t st (tv + C (M)) ∩ Zd ⊂ {Mx : x ∈ Zm+}.
M =
(2 3 40 1 −1
)
Robert Weismantel Lecture 6 23 / 27
The Main Tool - (Diagonal) Frobenius Number
Definition (Diagonal Frobenius Number)
Let M ∈ Zd×m (d ≤ m) and v = M1 such that
M has HNF Identity, and
C (M) = {λ : Mλ ≥ 0} is a pointed cone.
F (M) is the smallest integer t st (tv + C (M)) ∩ Zd ⊂ {Mx : x ∈ Zm+}.
M =
(2 3 40 1 −1
)
v =
(90
)
Robert Weismantel Lecture 6 23 / 27
The Main Tool - (Diagonal) Frobenius Number
Definition (Diagonal Frobenius Number)
Let M ∈ Zd×m (d ≤ m) and v = M1 such that
M has HNF Identity, and
C (M) = {λ : Mλ ≥ 0} is a pointed cone.
F (M) is the smallest integer t st (tv + C (M)) ∩ Zd ⊂ {Mx : x ∈ Zm+}.
M =
(2 3 40 1 −1
)
v =
(90
)F (M) = 1
Robert Weismantel Lecture 6 23 / 27
The Main Tool - (Diagonal) Frobenius Number
Definition (Diagonal Frobenius Number)
Let M ∈ Zd×m (d ≤ m) and v = M1 such that
M has HNF Identity, and
C (M) = {λ : Mλ ≥ 0} is a pointed cone.
The diagonal Frobenius number F (M) is the smallest integer t such that
(tv + C (M)) ∩ Zd ⊂ {Mx : x ∈ Zm+}.
Theorem (Aliev, Henk 2010).
F (M) ≤ (m − d)√m
2
√det(MMT ).
For fixed d , the bound is polynomial in the unary encoding of M.
Robert Weismantel Lecture 6 24 / 27
{Wx : x ∈ P ∩ Zn} vs. {Wx : x ∈ P}∩Zd
P = Rn+ w ∈ Zn
P = Rn+ W ∈ Zd×n
P = {x ∈ Rn : Ax ≤ b}
W ∈ Zd×n
Robert Weismantel Lecture 6 25 / 27
{Wx : x ∈ P ∩ Zn} vs. {Wx : x ∈ P}∩Zd
P = Rn+ w ∈ Zn
P = Rn+ W ∈ Zd×n
P = {x ∈ Rn : Ax ≤ b}
W ∈ Zd×n
Robert Weismantel Lecture 6 25 / 27
{Wx : x ∈ P ∩ Zn} vs. {Wx : x ∈ P}∩Zd
P = Rn+ w ∈ Zn
P = Rn+ W ∈ Zd×n
P = {x ∈ Rn : Ax ≤ b}
W ∈ Zd×n
Robert Weismantel Lecture 6 25 / 27
{Wx : x ∈ P ∩ Zn} vs. {Wx : x ∈ P}∩Zd
P = Rn+ w ∈ Zn
P = Rn+ W ∈ Zd×n
P = {x ∈ Rn : Ax ≤ b}
W ∈ Zd×n
Robert Weismantel Lecture 6 25 / 27
δ-Regular Sets
Definition (δ-regular set).
We call a set S ⊂ Zd δ-regular, with respect to a region B ⊂ Rd , if thereexists a family of full-dimensional affine sub-lattices Λ1, · · · ,Λk of Zd withdeterminants det(Λi ) ≤δ such that S ∩ B =
⋃i Λi ∩ B.
Robert Weismantel Lecture 6 26 / 27
δ-Regular Sets
Definition (δ-regular set).
We call a set S ⊂ Zd δ-regular, with respect to a region B ⊂ Rd , if thereexists a family of full-dimensional affine sub-lattices Λ1, · · · ,Λk of Zd withdeterminants det(Λi ) ≤δ such that S ∩ B =
⋃i Λi ∩ B.
Robert Weismantel Lecture 6 26 / 27
δ-Regular Sets
Definition (δ-regular set).
We call a set S ⊂ Zd δ-regular, with respect to a region B ⊂ Rd , if thereexists a family of full-dimensional affine sub-lattices Λ1, · · · ,Λk of Zd withdeterminants det(Λi ) ≤δ such that S ∩ B =
⋃i Λi ∩ B.
Robert Weismantel Lecture 6 26 / 27
Main Result (from a geometric point of view)
The notation
Let P = {x ∈ Rn : Ax ≤ b}, with A ∈ Zm×n and b ∈ Zm.
Let ∆ denote the maximum absolute sub-determinant of A.
Let Q = WP and let R = W (P ∩ Zn) with W ∈ Zd×n.
Define Qγ := {x ∈ Rd : x + Bγ ⊂ Q}.
Qγ
Theorem.
The set R is δ-regular with respect to the polyhedron Qγ , where γ and δare bounded polynomially in ∆, ‖W ‖max and n.
Robert Weismantel Lecture 6 27 / 27
Main Result (from a geometric point of view)
The notation
Let P = {x ∈ Rn : Ax ≤ b}, with A ∈ Zm×n and b ∈ Zm.
Let ∆ denote the maximum absolute sub-determinant of A.
Let Q = WP and let R = W (P ∩ Zn) with W ∈ Zd×n.
Define Qγ := {x ∈ Rd : x + Bγ ⊂ Q}.
Qγ
Theorem.
The set R is δ-regular with respect to the polyhedron Qγ , where γ and δare bounded polynomially in ∆, ‖W ‖max and n.
Robert Weismantel Lecture 6 27 / 27
Main Result (from a geometric point of view)
The notation
Let P = {x ∈ Rn : Ax ≤ b}, with A ∈ Zm×n and b ∈ Zm.
Let ∆ denote the maximum absolute sub-determinant of A.
Let Q = WP and let R = W (P ∩ Zn) with W ∈ Zd×n.
Define Qγ := {x ∈ Rd : x + Bγ ⊂ Q}.
Qγ
Theorem.
The set R is δ-regular with respect to the polyhedron Qγ , where γ and δare bounded polynomially in ∆, ‖W ‖max and n.
Robert Weismantel Lecture 6 27 / 27
Main Result (from a geometric point of view)
The notation
Let P = {x ∈ Rn : Ax ≤ b}, with A ∈ Zm×n and b ∈ Zm.
Let ∆ denote the maximum absolute sub-determinant of A.
Let Q = WP and let R = W (P ∩ Zn) with W ∈ Zd×n.
Define Qγ := {x ∈ Rd : x + Bγ ⊂ Q}.
Qγ
Theorem.
The set R is δ-regular with respect to the polyhedron Qγ , where γ and δare bounded polynomially in ∆, ‖W ‖max and n.
Robert Weismantel Lecture 6 27 / 27
Main Result (from a geometric point of view)
The notation
Let P = {x ∈ Rn : Ax ≤ b}, with A ∈ Zm×n and b ∈ Zm.
Let ∆ denote the maximum absolute sub-determinant of A.
Let Q = WP and let R = W (P ∩ Zn) with W ∈ Zd×n.
Define Qγ := {x ∈ Rd : x + Bγ ⊂ Q}.
Qγ
Theorem.
The set R is δ-regular with respect to the polyhedron Qγ , where γ and δare bounded polynomially in ∆, ‖W ‖max and n.
Robert Weismantel Lecture 6 27 / 27
The Algorithm
For all affine lattices Λ ⊂ Zd with det(Λ) ≤ δ, solvemin{f (y) : y ∈ Qγ ∩ Λ} ⇒ yΛ.
Determinex∗ = argmin{f (Wx) : x ∈ P ∩ Zn such that Wx = yΛ for some Λ}.For each affine subspace L, sufficiently close to boundary
Recursively find a solution to min{f (Wx) : x ∈W−1L ∩ P} ⇒ x ′.Replace x∗ with x ′ if its objective value is smaller.
Return x∗.
Robert Weismantel Lecture 6 28 / 27
The Algorithm
For all affine lattices Λ ⊂ Zd with det(Λ) ≤ δ, solvemin{f (y) : y ∈ Qγ ∩ Λ} ⇒ yΛ.Determinex∗ = argmin{f (Wx) : x ∈ P ∩ Zn such that Wx = yΛ for some Λ}.
For each affine subspace L, sufficiently close to boundaryRecursively find a solution to min{f (Wx) : x ∈W−1L ∩ P} ⇒ x ′.Replace x∗ with x ′ if its objective value is smaller.
Return x∗.
Robert Weismantel Lecture 6 28 / 27
The Algorithm
For all affine lattices Λ ⊂ Zd with det(Λ) ≤ δ, solvemin{f (y) : y ∈ Qγ ∩ Λ} ⇒ yΛ.Determinex∗ = argmin{f (Wx) : x ∈ P ∩ Zn such that Wx = yΛ for some Λ}.For each affine subspace L, sufficiently close to boundary
Recursively find a solution to min{f (Wx) : x ∈W−1L ∩ P} ⇒ x ′.
Replace x∗ with x ′ if its objective value is smaller.
Return x∗.
Robert Weismantel Lecture 6 28 / 27
The Algorithm
For all affine lattices Λ ⊂ Zd with det(Λ) ≤ δ, solvemin{f (y) : y ∈ Qγ ∩ Λ} ⇒ yΛ.Determinex∗ = argmin{f (Wx) : x ∈ P ∩ Zn such that Wx = yΛ for some Λ}.For each affine subspace L, sufficiently close to boundary
Recursively find a solution to min{f (Wx) : x ∈W−1L ∩ P} ⇒ x ′.Replace x∗ with x ′ if its objective value is smaller.
Return x∗.Robert Weismantel Lecture 6 28 / 27
The Algorithm
For all affine lattices Λ ⊂ Zd with det(Λ) ≤ δ, solvemin{f (y) : y ∈ Qγ ∩ Λ} ⇒ yΛ.Determinex∗ = argmin{f (Wx) : x ∈ P ∩ Zn such that Wx = yΛ for some Λ}.For each affine subspace L, sufficiently close to boundary
Recursively find a solution to min{f (Wx) : x ∈W−1L ∩ P} ⇒ x ′.Replace x∗ with x ′ if its objective value is smaller.
Return x∗.Robert Weismantel Lecture 6 28 / 27