Post on 02-Oct-2020
Completely Positive Programs
John E. Mitchell
Department of Mathematical Sciences
RPI, Troy, NY 12180 USA
May 2018
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Convex vs nonconvex problems
Optimization problems are often divided into convex and nonconvex
problems.
Integer optimization problems are examples of nonconvex optimization
problems.
The perception is that convex problems are “easy” to solve to global
optimality and nonconvex problems are “hard”.
Completely positive programs are convex conic optimization problems
which typically cannot be solved in polynomial time, so they are convex
problems that are “hard”.
They provide exact reformulations of quadratic integer programs as
convex optimization problems.
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A quadratic integer program
Consider the quadratic integer program
minx2IRn cT x + 1
2xT Qx
subject to Ax = bx � 0
xj 2 {0, 1}, j 2 B ✓ {1, . . . , n}
(1)
where A 2 IRm⇥n, b 2 IRm, c 2 IRn, and Q is a symmetric n ⇥ n matrix;
Q need not be positive semidefinite.
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Constructing a relaxation
Let ai represent the i th row of A (written as a column vector in IRn).
We follow a similar sequence of steps to those used to construct the
SDP relaxation of MAXCUT.
First, we lift, introducing a matrix X 2 IRn⇥n of variables equal to xxT :
minx2IRn,X2IRn⇥n cT x + 1
2Q • X
subject to Ax = baT
i Xai = b2
i , i = 1, . . . ,mx � 0
xj = Xjj , j 2 B ✓ {1, . . . , n}xj 2 {0, 1}, j 2 B ✓ {1, . . . , n}X = xxT
(2)
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xTQx = t r a c e (xTQ×)= trace(Qxxt)= t race
(ax)=QoX
Axebait(÷÷/×
= b
"÷÷iN÷÷.":#i
[!!!)x xT[a, a , . . . an]e b b '
[{II,]x x ' [a, a . . . . am]= b bT
[!÷÷)x [a , a . . . . am] = bb'
-m x m m e t r i x
Diagonal en t r i e s :ajtka; = bit
Constructing a relaxation
We can relax this problem as:
minx2IRn,X2IRn⇥n1
2
0 cT
c Q
�•
1 xT
x X
�
subject to Ax = baT
i Xai = b2
i , i = 1, . . . ,mx � 0
xj = Xjj , j 2 B ✓ {1, . . . , n}1 xT
x X
�2 K
(3)
for a convex cone K . If we take K = Sn+1
+ , the cone of symmetric
positive semidefinite matrices, we obtain the SDP relaxation.
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E x tI Q .X = t a x + 'exit'sg .x
⇒ LeoEff i ¥1
f.)axis-1: ¥11 : ; ]w w(net)x (ntl) i n a c o n e .
Eg: i n SSI'
Completely positive matrices
We can impose stronger restrictions on K . For example, we can
choose K to be the cone of completely positive matrices.
Definition
The cone of q ⇥ q completely positive matrices is
CPq := {X 2 IRq⇥q : X = ZZ T for some matrix Z
with every entry of Z nonnegative}.
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N o limit o n number of column, i n 2 .
Examples of completely positive matrices
It is easy to see that CPn✓ Sn
+, so taking K = CPn+1
gives a relaxation
that is at least as tight as the SDP relaxation.
Two examples of matrices that are completely positive:
X =
1 2
2 5
�=
1 0
2 1
� 1 2
0 1
�
X =
1 xxT xxT
�=
1
x
� ⇥1 xT ⇤
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E. ' s t a i n : -it=L: i f f : : Itc::3= f;]C ' ' I t [ i fC '
o ] + [E)c o 23
= L : : al::D
Properties of completely positive matrices
Some properties of CP:
Theorem
1 If X is completely positive then it is doubly nonnegative:1 it is symmetric and positive semidefinite, and2 every entry in X is nonnegative.
2 If X 2 IRq⇥q is doubly nonnegative and q 4 then X is completelypositive.
3 The dual cone to CPq is the cone of copositive matrices
COPq := {X 2 IRq⇥q : dT Xd � 0 8 d 2 IRq
+}.
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Properties of completely positive matrices
Some properties of CP:
Theorem
1 If X is completely positive then it is doubly nonnegative:1 it is symmetric and positive semidefinite, and2 every entry in X is nonnegative.
2 If X 2 IRq⇥q is doubly nonnegative and q 4 then X is completelypositive.
3 The dual cone to CPq is the cone of copositive matrices
COPq := {X 2 IRq⇥q : dT Xd � 0 8 d 2 IRq
+}.
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Properties of completely positive matrices
Some properties of CP:
Theorem
1 If X is completely positive then it is doubly nonnegative:1 it is symmetric and positive semidefinite, and2 every entry in X is nonnegative.
2 If X 2 IRq⇥q is doubly nonnegative and q 4 then X is completelypositive.
3 The dual cone to CPq is the cone of copositive matrices
COPq := {X 2 IRq⇥q : dT Xd � 0 8 d 2 IRq
+}.
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Properties of completely positive matrices
Some properties of CP:
Theorem
1 If X is completely positive then it is doubly nonnegative:1 it is symmetric and positive semidefinite, and2 every entry in X is nonnegative.
2 If X 2 IRq⇥q is doubly nonnegative and q 4 then X is completelypositive.
3 The dual cone to CPq is the cone of copositive matrices
COPq := {X 2 IRq⇥q : dT Xd � 0 8 d 2 IRq
+}.
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Properties of completely positive matrices
Some properties of CP:
Theorem
1 If X is completely positive then it is doubly nonnegative:1 it is symmetric and positive semidefinite, and2 every entry in X is nonnegative.
2 If X 2 IRq⇥q is doubly nonnegative and q 4 then X is completelypositive.
3 The dual cone to CPq is the cone of copositive matrices
COPq := {X 2 IRq⇥q : dT Xd � 0 8 d 2 IRq
+}.
Mitchell Completely Positive Programs 8 / 11
Properties of completely positive matrices
Some properties of CP:
Theorem
1 If X is completely positive then it is doubly nonnegative:1 it is symmetric and positive semidefinite, and2 every entry in X is nonnegative.
2 If X 2 IRq⇥q is doubly nonnegative and q 4 then X is completelypositive.
3 The dual cone to CPq is the cone of copositive matrices
COPq := {X 2 IRq⇥q : dT Xd � 0 8 d 2 IRq
+}.
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C P ' E SSI,c - c o p e
The relaxation is exact
Under some simple assumptions, the completely positive formulation
(3) is equivalent to the quadratic integer program (1), so it is not just a
relaxation.
Theorem (Burer, 2009)
Assume the constraints x � 0, Ax = b imply 0 xj 1 8j 2 B. Thenthe quadratic integer program (1) is equivalent to its completelypositive relaxation (3), in that
1 their optimal values agree, and2 if (x⇤,X ⇤) is optimal for (3) then x⇤ is in the convex hull of optimal
solutions to (1).
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The relaxation is exact
Under some simple assumptions, the completely positive formulation
(3) is equivalent to the quadratic integer program (1), so it is not just a
relaxation.
Theorem (Burer, 2009)
Assume the constraints x � 0, Ax = b imply 0 xj 1 8j 2 B. Thenthe quadratic integer program (1) is equivalent to its completelypositive relaxation (3), in that
1 their optimal values agree, and2 if (x⇤,X ⇤) is optimal for (3) then x⇤ is in the convex hull of optimal
solutions to (1).
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The relaxation is exact
Under some simple assumptions, the completely positive formulation
(3) is equivalent to the quadratic integer program (1), so it is not just a
relaxation.
Theorem (Burer, 2009)
Assume the constraints x � 0, Ax = b imply 0 xj 1 8j 2 B. Thenthe quadratic integer program (1) is equivalent to its completelypositive relaxation (3), in that
1 their optimal values agree, and2 if (x⇤,X ⇤) is optimal for (3) then x⇤ is in the convex hull of optimal
solutions to (1).
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The relaxation is exact
Thus, we have a conic optimization problem that is NP-hard.
No nice barrier function is known, so it is hard to work directly with the
cone of completely positive matrices.
Even the separation problem is NP-complete:
Separation problem for CPq: given a doubly nonnegativeX 2 Sq, determine whether X 2 CP
q.
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-
Solution approaches and extensions
Cutting plane and column generation methods have been developed.
The results of Burer have been extended to show that any
quadratically constrained quadratic program (convex or
nonconvex) can be expressed as an equivalent completely positive
program, with a slightly broadened definition of a completely positive
matrix.
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