completely positive programseaton.math.rpi.edu/faculty/mitchell/courses/matp6620/... · 2019. 4....

Completely Positive Programs

John E. Mitchell

Department of Mathematical Sciences

RPI, Troy, NY 12180 USA

May 2018

Mitchell Completely Positive Programs 1 / 11

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.


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.


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)


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.


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}.


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 ⇤


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

+}.


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

+}.


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).


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.


-

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.


completely positive programseaton.math.rpi.edu/faculty/mitchell/courses/matp6620/... · 2019. 4....

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