nonlinear integer programming - technionie.technion.ac.il/~onn/talks/paris.pdf · 2015-06-08 ·...
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Shmuel OnnTechnion – Israel Institute of Technology
Nonlinear Integer Programmingis
Fixed-Parameter Tractable
(with Berstein, De Loera, Hemmecke, Lee, Romanchuk, Rothblum, Weismantel)
Outline
Overview our theory of Graver bases for integer programming
Shmuel Onn
(with Berstein, De Loera, Hemmecke, Lee, Romanchuk, Rothblum, Weismantel)
Outline
Overview our theory of Graver bases for integer programming
(with Hemmecke and my student Romanchuk, Mathematical Programming, 2013)
Explain the fixed-parameter tractability of integer programming
Shmuel Onn
(with Berstein, De Loera, Hemmecke, Lee, Romanchuk, Rothblum, Weismantel)
Outline
Overview our theory of Graver bases for integer programming
Explain the fixed-parameter tractability of integer programming
Describe an approximation hierarchy for integer programming
Shmuel Onn
(with Hemmecke and my student Romanchuk, Mathematical Programming, 2013)
(Non)-Linear Integer Programming
l,u: lower/upper bounds in Zn f: function from Zn to R
with data: A: integer m x n matrix b: right-hand side in Zm
Shmuel Onn
The problem is: min/max { f(x) : Ax ≤ b, l ≤ x ≤ u, x in Zn }
(Non)-Linear Integer Programming
Shmuel Onn
Our theory enables polynomial time solution of broad naturaluniversal (non)-linear integer programs in variable dimension
The problem is: min/max { f(x) : Ax ≤ b, l ≤ x ≤ u, x in Zn }
(Non)-Linear Integer Programming
Shmuel Onn
under which integer programming is fixed-parameter tractable.
The problem is: min/max { f(x) : Ax ≤ b, l ≤ x ≤ u, x in Zn }
It gives variable dimension parameterization of integer programming,
Our theory enables polynomial time solution of broad naturaluniversal (non)-linear integer programs in variable dimension
Graver Basesand
Nonlinear Integer Programming
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Graver Bases
The Graver basis of an integer matrix A is the finite set G(A) of conformal-minimal nonzero integer vectors x satisfying Ax = 0.
x is conformal-minimal if no other y in same orthant has all |yi| ≤ |xi|
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circuits: ±(2 -1 0), ±(1 0 -1), ±(0 1 -2)
Example: Consider A=(1 2 1). Then G(A) consists of
non-circuits: ±(1 -1 1)
Graver Bases
The Graver basis of an integer matrix A is the finite set G(A) of conformal-minimal nonzero integer vectors x satisfying Ax = 0.
Reference: N-fold integer programming (De Loera, Hemmecke, Onn, Weismantel)Discrete Optimization (Volume in memory of George Dantzig), 2008
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Theorem 1: linear optimization in polytime with G(A):
Some Theorems on(Non)-Linear Integer Programming
max { wx : Ax = b, l ≤ x ≤ u, x in Zn }
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Reference: A polynomial oracle-time algorithm for convex integer minimization (Hemmecke, Onn, Weismantel) Mathematical Programming, 2011
Theorem 2: separable convex minimization in polytime with G(A):
Some Theorems on(Non)-Linear Integer Programming
min {Σfi(xi) : Ax = b, l ≤ x ≤ u, x in Zn }
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Theorem 3: integer point lp-nearest to x in polytime with G(A):
Reference: A polynomial oracle-time algorithm for convex integer minimization (Hemmecke, Onn, Weismantel) Mathematical Programming, 2011
Some Theorems on(Non)-Linear Integer Programming
min {|x - x|p : Ax = b, l ≤ x ≤ u, x in Zn}
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Reference: Convex integer maximization via Graver bases (De Loera, Hemmecke, Onn, Rothblum, Weismantel) Journal of Pure and Applied Algebra, 2009
Theorem 4: weighted convex maximization in polytime with G(A):
where W is d x n matrix and f convex function on Zd
(balancing d linear criteria or player utilities Wix)
Some Theorems on(Non)-Linear Integer Programming
max {f(Wx) : Ax = b, l ≤ x ≤ u, x in Zn }
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Reference: The quadratic Graver cone, quadratic integer minimization & extensions(Lee, Onn, Romanchuk, Weismantel), Mathematical Programming, 2012
Theorem 5: quadratic minimization in polytime with G(A):
where V lies in cone K2(A) of possibly indefinite matrices, enabling minimization of some convex and some non-convex quadratics.
Some Theorems on(Non)-Linear Integer Programming
min {xTVx : Ax = b, l ≤ x ≤ u, x in Zn }
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Theorem 6: polynomial minimization in polytime with G(A):
Reference: The quadratic Graver cone, quadratic integer minimization & extensions(Lee, Onn, Romanchuk, Weismantel), Mathematical Programming, 2012
where p is possibly indefinite polynomial of degree d in cone Kd(A),enabling minimization of some (non)-convex degree d polynomials.
Some Theorems on(Non)-Linear Integer Programming
min {p(x) : Ax = b, l ≤ x ≤ u, x in Zn }
The Main Iterative Algorithm
Rn
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Proof of Theorems 1 and 2(linear and separable convex minimization)
R
f
with the Graver basis G(A)
To solve min {Σ fi(xi) : Ax = b, l ≤ x ≤ u, x in Zn }
Do:
Rn
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R
f1. Find initial point by auxiliary program
Proof of Theorems 1 and 2(linear and separable convex minimization)
with the Graver basis G(A)
To solve min {Σ fi(xi) : Ax = b, l ≤ x ≤ u, x in Zn }
Do:
Rn
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R
ff1. Find initial point by auxiliary program
Proof of Theorems 1 and 2(linear and separable convex minimization)
2. Iteratively improve by Graver-best steps,that is, by best cz with c in Z and z in G(A).
with the Graver basis G(A)
To solve min {Σ fi(xi) : Ax = b, l ≤ x ≤ u, x in Zn }
Do:
Rn
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1. Find initial point by auxiliary program
R
ff
Using supermodality of f and integer Caratheodorytheorem (Cook-Fonlupt-Schrijver, Sebo) we can show polytime convergence to some optimal solution
Proof of Theorems 1 and 2(linear and separable convex minimization)
2. Iteratively improve by Graver-best steps,that is, by best cz with c in Z and z in G(A).
with the Graver basis G(A)
To solve min {Σ fi(xi) : Ax = b, l ≤ x ≤ u, x in Zn }
Do:
N-Fold Integer Programming
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N-Fold Products
A(n) =
n
The n-fold product of an (r,s) x t bimatrix A =is the (r+ns) x nt matrix
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N-Fold Products
The n-fold product of an (r,s) x t bimatrix A =is the (r+ns) x nt matrix
A(n) =
n
Lemma 1: For fixed A we can compute the Graver basis G(A(n)) in polynomial time O(ng(A)) with g(A) the Graver complexity of A.
(Non)-Linear N-Fold Integer Programming
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A(n) =
n
References: see Nonlinear Discrete Optimization (Onn), Zurich Lectures in Advanced Mathematics, European Mathematical Society, 2010
min{f(x) : A(n)x = b, l ≤ x ≤ u, x in Znt}
Theorem 7: for various f can solve in polynomial time O(ng(A) L):
Some Applications
1
1
0
33
0
2 0
14
50
86
9
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The multi-index transportation problem of Motzkin (1952) concerns minimization over m1 X . . . X mk X mk+1 tables with given margins:
1. Multiway Tables
1
1
0
33
0
2 0
14
50
86
9
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The multi-index transportation problem of Motzkin (1952) concerns minimization over m1 X . . . X mk X mk+1 tables with given margins.
If all mi fixed then solvable by fixed dimension theory.
1. Multiway Tables
1
1
0
33
0
2 0
14
50
86
9
n
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The multi-index transportation problem of Motzkin (1952) concerns minimization over m1 X . . . X mk X n tables with given margins.
1. Multiway Tables
1
1
0
33
0
2 0
14
50
86
9
n
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The multi-index transportation problem of Motzkin (1952) concerns minimization over m1 X . . . X mk X n tables with given margins.
Corollary 1: (Non)-linear optimization over m1 X . . . X mk X n tables with given margins is doable in polynomial time O(ng(m1,…,mk) L) .
If one side n is variable then it is an n-fold program.
1. Multiway Tables
1
1
0
33
0
2 0
14
50
86
9
n
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However, if two sides are variable then have universality:Every integer program is one over 3 X m X n tables with given line-sums.
The multi-index transportation problem of Motzkin (1952) concerns minimization over m1 X . . . X mk X n tables with given margins.
1. Multiway Tables
Corollary 1: (Non)-linear optimization over m1 X . . . X mk X n tables with given margins is doable in polynomial time O(ng(m1,…,mk) L) .
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suppliers
Km,n
consumers
s1
sm
c1
cn
cost: Σijcij(Σk xijk) aij
capacities: uij
2. Multicommodity Flows
flow
Corollary 2: For any fixed l commodities and m suppliers, can find min-congestion multicommodity-flow for n consumers in polytime.
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Better Way of Finding Graver-Best Steps
Reference: N-fold integer programming in cubic time(Hemmecke, Onn, Romanchuk) Mathematical Programming, 2013
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min {wx : A(n)x = b, l ≤ x ≤ u, x in Znt}.
Better Way of Finding Graver-Best StepsLet x be feasible at some iteration of solving the n-fold program
A(n) =
n
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S0={0} Si-1=V(A) Si=V(A) Sn={v inV(A) : A1v=0}
vi-1 vi
hi := vi–vi-1 inV(A)li ≤ xi+chi ≤ ui
length wihi
V(A) := {v inZt : v is the sum of at most g(A) elements of G(A2) }
Lemma 2: A Graver-best step for x can be computed in O(n2) time, solvingO(|G(A2)|
g(A)n) dynamic programs each in time O(|G(A2)|2g(A)n).
Better Way of Finding Graver-Best Steps
A =
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Present Complexity of N-fold Integer Programming
Theorem 9: For any fixed bimatrix A, the following linear n-fold integer program is solvable in time O(n3 log(w,b,l,u)):
max{wx : A(n)x = b, l ≤ x ≤ u, x in Znt}
Reference: N-fold integer programming in cubic time(Hemmecke, Onn, Romanchuk) Mathematical Programming, 2013
Combining Lemma 2 with our bound on number of iterations gives:
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Present Complexity of N-fold Integer Programming
Theorem 9: For any fixed bimatrix A, the following linear n-fold integer program is solvable in time O(n3 log(w,b,l,u)):
Reference: N-fold integer programming in cubic time(Hemmecke, Onn, Romanchuk) Mathematical Programming, 2013
Combining Lemma 2 with our bound on number of iterations gives:
Instead of O(ng(A) log(w,b,l,u))
max{wx : A(n)x = b, l ≤ x ≤ u, x in Znt}
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With more work we can obtain the following:
Theorem 10: For any fixed r,s,t and p, the following p-piecewise affine separable convex n-fold integer program with variable (r,s) x t bimatrixA is solvable in time cubic in n, linear in log(f,b,l,u), and polynomial in A:
min{Σ fi(xi) : A(n)x = b, l ≤ x ≤ u, x in Znt}
Reference: N-fold integer programming in cubic time(Hemmecke, Onn, Romanchuk) Mathematical Programming, 2013
Present Complexity of N-fold Integer Programming
Fixed-Parameter Tractabilityand
Graver Approximation Hierarchy
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Integer Programming is Fixed-Parameter Tractable
Corollary 3: (Non)-linear optimization over m1 X . . . X mk X n tables with given margins is doable in fixed-parameter cubic time O(n3 L).
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Integer Programming is Fixed-Parameter Tractable
Corollary 3: (Non)-linear optimization over m1 X . . . X mk X n tables with given margins is doable in fixed-parameter cubic time O(n3 L).
Instead of O(ng(m1,…,mk) L)
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Integer Programming is Fixed-Parameter Tractable
Now, have parameterization of all integer programming by universality:Every integer program is one over 3 X m X n tables with given line-sums.
Corollary 3: (Non)-linear optimization over m1 X . . . X mk X n tables with given margins is doable in fixed-parameter cubic time O(n3 L).
Instead of O(ng(m1,…,mk) L)
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Integer Programming is Fixed-Parameter Tractable
Now, have parameterization of all integer programming by universality:Every integer program is one over 3 X m X n tables with given line-sums.
Hence (nonlinear) integer programming is fixed-parameter tractable:for each fixed m it is solvable in time O(n3 L) instead of O(ng(m) L).
Corollary 3: (Non)-linear optimization over m1 X . . . X mk X n tables with given margins is doable in fixed-parameter cubic time O(n3 L).
Instead of O(ng(m1,…,mk) L)
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Our algorithm naturally enables a hierarchy of approximations of the universal integer program over 3 X m X n tables with variable m,n.
Graver Approximation Hierarchy
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Graver Approximation Hierarchy
Vd(m) := { v in Z3m : v is the sum of at most d circuits of K3,m }
Our algorithm naturally enables a hierarchy of approximations of the universal integer program over 3 X m X n tables with variable m,n.
At level d of this hierarchy, we find approximated Graver-best stepsin time O(m9d n2), approximating V(A) in the dynamic programs by
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Graver Approximation Hierarchy
Vd(m) := { v in Z3m : v is the sum of at most d circuits of K3,m }
At level d of this hierarchy, we find approximated Graver-best stepsin time O(m9d n2), approximating V(A) in the dynamic programs by
We apply Graver-best steps while possible. If the approximation is satisfactory then we stop, else we proceed to the next level.
Our algorithm naturally enables a hierarchy of approximations of the universal integer program over 3 X m X n tables with variable m,n.
Some Bibliography (available at http://ie.technion.ac.il/~onn)
- The complexity of 3-way tables (SIAM J. Comp.)- Convex combinatorial optimization (Disc. Comp. Geom.)- Markov bases of 3-way tables (J. Symb. Comp.)- All linear and integer programs are slim 3-way programs (SIAM J. Opt.)- Graver complexity of integer programming (Annals Combin.)- N-fold integer programming (Disc. Opt. in memory of Dantzig) - Convex integer maximization via Graver bases (J. Pure App. Algebra)- Polynomial oracle-time convex integer minimization (Math. Prog.)- The quadratic Graver cone, quadratic integer minimization & extensions (Math Prog.)- N-fold integer programming in cubic time (Math. Prog.)
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Comprehensive development is in my monograph
available electronically from my homepage
(with kind permission of EMS)
The cubic time algorithm and thefixed-parameter tractability are in
N-fold integer programming in cubic timeMathematical Programming, 2013