A fresh CP look at MINLP { new formulations and ...€¦ · A fresh CP look at MINLP {new...

A fresh CP look at MINLP –

new formulations and relaxations

Immanuel Bomze, Universitat Wien

COST MINLP Workshop@IMUS Seville, 31 March 2015

Overview

1. Ternary fractional QPs

Overview


2. Everything is quadratic !

Overview



3. ... and therefore linearly copositive

Overview




4. New bounds and ...

Overview




4. New bounds and ...

5. ... new approximation hierarchies

Ternary decisions

Do you like Copositive Optimization ?

−

{ ∈ }

{−

Ternary decisions


Yes !

−

{ ∈ }

{−

Ternary decisions


Yes ! +1,

−

{ ∈ }

{−

Ternary decisions


Yes ! +1,

No ! −1,

{ ∈ }

{−

Ternary decisions


Yes ! +1,

No ! −1,

don’t know/tell 0.

{ ∈ }

{− }

Ternary decisions


Yes ! +1,

No ! −1,


Ternary optimization:

min {f(x) : x ∈ Tn}

with T = {−1,0,1}.

Ternary decisions


Yes ! +1,

No ! −1,


Ternary optimization:

min {f(x) : x ∈ Tn}

with T = {−1,0,1}.

What for ? Eg., graph tri-partitioning problems as in PageRank

for Folksonomy in social media/Semantic Web [Hotho et al.’06].

Predicting ternary decisions – TFQP

Given dissimilarities Dij > 0 (D adjacency of aversion graph):coherence within three groups/separation between them:

small x>Dx =∑i,jDijxixj



small x>Dx =∑i,jDijxixj ? ... too simple, ignores outside option.




Better: relative to density, so

min

(x>Dx)/∑i

x2i : x ∈ Tn \ {o}

which is APX-hard [Bhaskara et al.’12].





min

(x>Dx)/∑i

x2i : x ∈ Tn \ {o}

which is APX-hard [Bhaskara et al.’12]. More general

z∗T� := min

{f(x)

g(x): x ∈ Tn \ {o}

}for quadratic f , g.





min

(x>Dx)/∑i

x2i : x ∈ Tn \ {o}

which is APX-hard [Bhaskara et al.’12]. More general

z∗T� := min

{f(x)

g(x): x ∈ Tn \ {o}

}for quadratic f , g. Easier variant if g(x) 6= 0 for all x ∈ Tn:

z∗T := min

{f(x)

g(x): x ∈ Tn

}.

Ternary fractional quadratic problems (TFQPs) ...

... are special cases of (Mixed-)Binary constrained fractional QPs

Theorem [Amaral/B.’15]: Both z∗T� and z∗T are in fact MBFQP

min

{f(v)

g(v): v ∈ Rd+ , Cv = c , vi ∈ {0,1} for all i ∈ B

},

with d = 3n and Burer’s key condition satisfied.

∈ ⇐⇒ − { }




min

{f(v)


},


Proof. x ∈ T ⇐⇒ x = y − z with {y, z} ⊆ {0,1} and y + z ≤ 1.




min

{f(v)


},



... latter reminds on Densest Subgraph problem (poly-time),

but also includes APX-hard Max-Cut, k-Densest Subgraph ...




min

{f(v)


},



... latter reminds on Densest Subgraph problem (poly-time),

but also includes APX-hard Max-Cut, k-Densest Subgraph ...

Application: e.g., gene annotation graphs [Saha et al.’10] !

Burer’s key condition ? Copositive formulation !

... key condition: linear constraints force binary var.s into [0,1].

∈C{ • • • − ∈ B}

•



Used in [Burer’09];

∈C{ • • • − ∈ B}

•



Used in [Burer’09]; also here, for

Theorem [Amaral/B.’15]: Any Mixed-integer FQP with linear

constraints is, under key condition, equivalent to the linear COP

z∗cop = minY∈Cd+1

{A • Y : B • Y = 1 , Cc • Y = 0, Y0i − Yii = 0 , all i ∈ B} ,

•








with A coming from f , B from g, Cc from constraints Cv = c,

duality A • Y =∑i,j AijYij and

Y ∈ Cd+1 = conv{yy> : y ∈ Rd+1

+

}

> >








with A coming from f , B from g, Cc from constraints Cv = c,

duality A • Y =∑i,j AijYij and

Y ∈ Cd+1 = conv{yy> : y ∈ Rd+1

+

}the completely positive matrix cone.

Reason: lifting A • (yy>) = y>Ay, homogenization [Shor ’87].

General form of conic linear optimization

Let K be a convex cone of symmetric d× d matrices X.

min {C • X : Ai • X = bi , i∈ [1:m] , X ∈ K}



min {C • X : Ai • X = bi , i∈ [1:m] , X ∈ K} , barrier ??




Familiar cases:

K = N ={X = X> : X≥O

}= N ∗ . . . LP, barrier:−

∑i,jlogXij ,

K




Familiar cases:

K = N ={X = X> : X≥O

}= N ∗ . . . LP, barrier:−

∑i,jlogXij ,

and

K = P ={X = X> : X�O

}= P∗ . . .SDP, barrier:−

∑ilogλi(X) .




Familiar cases:

K = N ={X = X> : X≥O

}= N ∗ . . . LP, barrier:−

∑i,jlogXij ,

and

K = P ={X = X> : X�O


∑ilogλi(X) .

In above cases, the dual cone of K,

K∗ ={S = S> : S • X ≥ 0 for all X ∈ K

}coincides with K (self-duality)




Familiar cases:

K = N ={X = X> : X≥O

}= N ∗ . . . LP, barrier:−

∑i,jlogXij ,

and

K = P ={X = X> : X�O


∑ilogλi(X) .

In above cases, the dual cone of K,

K∗ ={S = S> : S • X ≥ 0 for all X ∈ K

}coincides with K (self-duality), but in general K∗ differs from K.

Copositive optimization (COP), duality

A very special matrix cone:

K = C = conv{xx> : x ∈ Rn, x ≥ o

},

the cone of completely positive matrices,

K∗ C∗ ≥

}




},

the cone of completely positive matrices, with its dual cone

K∗ = C∗ ={S = S> is copositive; means: x>Sx ≥ 0 if x ≥ o

}6= C .

C ⊂ P ∩N ⊂ P N ⊂ C∗ ≥

∗ { • • ∈ ∈ C}

∗∈

∗ ∗




},



}6= C .

Well known relations:

C ⊂ P ∩N P N ⊂ C∗

∗ { • • ∈ ∈ C}

∗∈

∗ ∗




},



}6= C .


C ⊂ P ∩N ⊂ P +N

∗ { • • ∈ ∈ C}

∗

∗ ∗




},



}6= C .


C ⊂ P ∩N ⊂ P +N ⊂ C∗

∗ { • • ∈ ∈ C}

∗

∗ ∗




},



}6= C .


C ⊂ P ∩N ⊂ P +N ⊂ C∗ . . . strict for n ≥ 5 .

∗ { • • ∈ ∈ C}

∗

∗ ∗




},



}6= C .



Primal-dual pair in (COP):

p∗ = inf {C • X : Ai • X = bi , i∈ [1:m] , X ∈ C}and

d∗ = supy∈Rm{b>y : C−

∑i yiAi ∈ C∗

}.




},



}6= C .



Primal-dual pair in (COP):

p∗ = inf {C • X : Ai • X = bi , i∈ [1:m] , X ∈ C}and

d∗ = supy∈Rm{b>y : C−

∑i yiAi ∈ C∗

}.

Usual weak (d∗ ≤ p∗) and strong (d∗ = p∗) duality results hold.

Semidefinite cone P

Immanuel

Schreibmaschinentext

Why duality in (COP) ? And how ?

General rule: any dually feasible solution y ∈ Rm gives lower bound

b>y ≤ d∗ ≤ p∗

if S(y) = C−∑i yiAi ∈ C∗.

≤ ∗ ≤ ∗

−

∈ C∗



b>y ≤ d∗ ≤ p∗

if S(y) = C−∑i yiAi ∈ C∗. For MBFQP

yB ≤ d∗ ≤ z∗cop = z∗

if

S(y) = A− yBB + yCCc +∑i∈B

yi

0 1 01 −1 00 0 0

∈ C∗ .



b>y ≤ d∗ ≤ p∗



if


yi

0 1 01 −1 00 0 0

∈ C∗ .Bad news: checking S ∈ C∗ is NP-hard [Dickinson/Gijben ’14];



b>y ≤ d∗ ≤ p∗



if


yi

0 1 01 −1 00 0 0


good news: checking S ∈ P +N ⊂ C∗ is poly-time.



b>y ≤ d∗ ≤ p∗



if


yi

0 1 01 −1 00 0 0


good news: checking S ∈ P +N ⊂ C∗ is poly-time. Even better:

resulting bound beats Lagrangian dual bounds (S ∈ P) ...

From linear to quadratic constraints – some thoughts

Everything is linear


Everything is linear – really ?

Nono – everything is nonlinear –



Nono – everything is nonlinear – c’mon

Ahh ... everything is quadratic !




Ahh ... everything is quadratic ! – wait ...

Binarity: x(1− x) = 0, quadratic, ok; also x ≥ 0 ⇔ x = t2






Monomial: x3 = x1x2, build∏ixmii recursively, ok.







So polynomial optimization is quadratic







So polynomial optimization is quadratic, and hence everything.

Then lift quadratics to linear objective, so everything is linearcopositive ;-)







So polynomial optimization is quadratic, and hence everything.

Then lift quadratics to linear objective, so everything is linearcopositive ;-)

Time to get serious: [Pena/Vera/Zuluaga ’15] show indeed that

any polynomial optimization problem is linear copositive.

Quadratically/linearly constrained QPs

Consider m+ 1 (possibly nonconvex) quadratic functions

qi(x) = x>Qix− 2b>i x + ci , i∈ [0:m] ,

and the problem to minimize q0 over two different sets:

F := {x ∈ Rn : qi(x) ≤ 0 , all i∈ [1:m]} ,

∗∈ ∈



qi(x) = x>Qix− 2b>i x + ci , i∈ [0:m] ,


F := {x ∈ Rn : qi(x) ≤ 0 , all i∈ [1:m]}



qi(x) = x>Qix− 2b>i x + ci , i∈ [0:m] ,


F := {x ∈ Rn : qi(x) ≤ 0 , all i∈ [1:m]} , or

F+ :={x ∈ Rn+ : qi(x) ≤ 0 , all i∈ [1:m]

}.

∗∈

∈‖



qi(x) = x>Qix− 2b>i x + ci , i∈ [0:m] ,


F := {x ∈ Rn : qi(x) ≤ 0 , all i∈ [1:m]} , or

F+ :={x ∈ Rn+ : qi(x) ≤ 0 , all i∈ [1:m]

}.

So consider two QCQPs

z∗ := infx∈F

q0(x) and z∗+ := infx∈F+

q0(x) .



qi(x) = x>Qix− 2b>i x + ci , i∈ [0:m] ,


F := {x ∈ Rn : qi(x) ≤ 0 , all i∈ [1:m]} , or

F+ :={x ∈ Rn+ : qi(x) ≤ 0 , all i∈ [1:m]

}.


z∗ := infx∈F


q0(x) .

Other linear constraints: slack var.s, Ax = a and x ∈ Rn+ can be

treated similarly by additional qm+1(x) = ‖Ax− a‖2 ≤ 0 .



qi(x) = x>Qix− 2b>i x + ci , i∈ [0:m] ,


F := {x ∈ Rn : qi(x) ≤ 0 , all i∈ [1:m]} , or

F+ :={x ∈ Rn+ : qi(x) ≤ 0 , all i∈ [1:m]

}.


z∗ := infx∈F


q0(x) .

Other linear constraints: slack var.s, Ax = a and x ∈ Rn+ can be

treated similarly by additional qm+1(x) = ‖Ax− a‖2 ≤ 0 .

Important: explicit sign constraints xj ≥ 0 in F+.

(Full) Lagrangian relaxation ...

... for z∗+:

L(x; u, v) = x>Hux− 2d>u x− 2v>x + c>u ,


... for z∗+:


∇xL(x; u, v) = 2[Hux− du − v]


... for z∗+:


∇xL(x; u, v) = 2[Hux− du − v] and

D2xL(x; u, v) = 2Hu for all (x; u, v) ∈ Rn × Rm+ × Rn+ ,

with Hessian Hu = Q0 +m∑i=1

uiQi and du = b0 +m∑i=1

uibi.


... for z∗+:






uibi.

... for z∗ similar, just drop v:

L0(x; u) = x>Hux− 2d>u x + c>u = L(x; u, o) ,


... for z∗+:






uibi.

... for z∗ similar, just drop v:

L0(x; u) = x>Hux− 2d>u x + c>u = L(x; u, o) ,

∇xL0(x; u) = 2[Hux− du] and

D2xL0(x; u) = 2Hu for all (x; u) ∈ Rn × Rm+ .

(Semi-)Lagrangian dual functions and problems

... for z∗, no explicit sign constraints:

Θ0(u) := inf {L0(x; u) : x ∈ Rn} and z∗dual := sup{

Θ0(u) : u ∈ Rm+}




Θ0(u) : u ∈ Rm+}

... for z∗+ several options:




Θ0(u) : u ∈ Rm+}

... for z∗+ several options: full Lagrangian

Θ(u, v) := inf {L(x; u, v) : x ∈ Rn}with dual problem

z∗dual,+ := sup{

Θ(u, v) : (u, v) ∈ Rm+ × Rn+}




Θ0(u) : u ∈ Rm+}



z∗dual,+ := sup{

Θ(u, v) : (u, v) ∈ Rm+ × Rn+}

or Semi-Lagrangian dual

Θsemi(u) := inf{L0(x; u) : x ∈ Rn+

}




Θ0(u) : u ∈ Rm+}



z∗dual,+ := sup{

Θ(u, v) : (u, v) ∈ Rm+ × Rn+}

or Semi-Lagrangian dual

Θsemi(u) := inf{L0(x; u) : x ∈ Rn+

}with

z∗semi := sup{

Θsemi(u) : u ∈ Rn+}.

Semi-Lagrangian bounds are always tighter

General principle: for any (u, v) ∈ Rm+ × Rn+

Θ(u, v) = inf {L(x; u, v) : x ∈ Rn}



Θ(u, v) = inf {L(x; u, v) : x ∈ Rn}

≤ inf{L(x; u, v) : x ∈ Rn+

}



Θ(u, v) = inf {L(x; u, v) : x ∈ Rn}

≤ inf{L(x; u, v) : x ∈ Rn+

}= inf

{L0(x, u)− 2v>x : x ∈ Rn+

}



Θ(u, v) = inf {L(x; u, v) : x ∈ Rn}

≤ inf{L(x; u, v) : x ∈ Rn+

}= inf

{L0(x, u)− 2v>x : x ∈ Rn+

}≤ inf

{L0(x, u) : x ∈ Rn+

}



Θ(u, v) = inf {L(x; u, v) : x ∈ Rn}

≤ inf{L(x; u, v) : x ∈ Rn+

}= inf

{L0(x, u)− 2v>x : x ∈ Rn+

}≤ inf

{L0(x, u) : x ∈ Rn+

}= Θsemi(u) .



Θ(u, v) = inf {L(x; u, v) : x ∈ Rn}

≤ inf{L(x; u, v) : x ∈ Rn+

}= inf

{L0(x, u)− 2v>x : x ∈ Rn+

}≤ inf

{L0(x, u) : x ∈ Rn+

}= Θsemi(u) .

Therefore in fact

z∗dual,+ ≤ z∗semi ≤ z

∗+ .



Θ(u, v) = inf {L(x; u, v) : x ∈ Rn}

≤ inf{L(x; u, v) : x ∈ Rn+

}= inf

{L0(x, u)− 2v>x : x ∈ Rn+

}≤ inf

{L0(x, u) : x ∈ Rn+

}= Θsemi(u) .

Therefore in fact

z∗dual,+ ≤ z∗semi ≤ z

∗+ .

Both bounds have conic representations !

SDP formulation of Lagrangian bounds

Select (d+1)×(d+1) matrices Mi such that qi(x) = y>Miy = Mi•Ywhere Y = [1, x>]>[1, x>].

• > > >

∗ { • • ≤ ∈ • }

{ • • ≤ ∈ • }



Y is psd. and satisfies J0 • Y = 1 where J0 = [1, o>][1, o>]>.



Y is psd. and satisfies J0 • Y = 1 where J0 = [1, o>][1, o>]>. Thus

z∗ = infY psd.

{M0 • Y : Mi • Y ≤ 0, i∈ [1:m] , J0 • Y = 1, rk Y = 1} .




z∗ = infY psd.

{M0 • Y : Mi • Y ≤ 0, i∈ [1:m] , J0 • Y = 1, rk Y = 1} .

SDP relaxation: drop rank constraint, keep psd. condition,

z∗SDP,primal = infY psd.

{M0 • Y : Mi • Y ≤ 0, i∈ [1:m], J0 • Y = 1} .




z∗ = infY psd.

{M0 • Y : Mi • Y ≤ 0, i∈ [1:m] , J0 • Y = 1, rk Y = 1} .

SDP relaxation: drop rank constraint, keep psd. condition,

z∗SDP,primal = infY psd.

{M0 • Y : Mi • Y ≤ 0, i∈ [1:m], J0 • Y = 1} .

Dual SDP problem: slack matrix S(y) = M0 − y0J0 +m∑i=1

uiMi,

z∗SDP,dual := sup{y0 ∈ R : S(y) psd., y = [y0, u

>]> ∈ R× Rm+}.

COP formulation of Semi-Lagrangian bounds

Theorem [Shor <’83, ’87]:

z∗dual = z∗SDP,dual ≤ z∗SDP,primal ≤ z

∗ .

>




∗ .

Further, no duality/relaxation gap,

z∗dual = z∗ if and only if




∗ .


z∗dual = z∗ if and only if S(y) is psd.

for y> = [q0(x), u>] at a KKT pair (x, u) ∈ F × Rm+ for z∗.




∗ .




... sufficient condition for global optimality of KKT point x.




∗ .




... sufficient condition for global optimality of KKT point x.

Theorem [B.’14]: For the Semi-Lagrangian bound we have

z∗semi = z∗COP,dual

with the copositive optimization representation

z∗COP,dual := sup{y0 ∈ R : S(y) copositive, y = [y0, u

>]> ∈ R× Rm+}.

Copositive optimization - conic primal/dual pair

Lifting again: if x ∈ Rn+ and y> = [1, x>], then

Y = yy> is completely positive (cp.)



Y = yy> is completely positive (cp.), so reformulation of z∗+:

z∗+ = infY cp.

{M0 • Y : Mi • Y ≤ 0, i∈ [1:m], J0 • Y = 1, rk Y = 1} ,




z∗+ = infY cp.

{M0 • Y : Mi • Y ≤ 0, i∈ [1:m], J0 • Y = 1, rk Y = 1} ,

and its natural relaxation: drop rank constraint but keep Y cp.

z∗COP,primal = infY cp.

{M0 • Y : Mi • Y ≤ 0, i∈ [1:m], J0 • Y = 1} .




z∗+ = infY cp.

{M0 • Y : Mi • Y ≤ 0, i∈ [1:m], J0 • Y = 1, rk Y = 1} ,

and its natural relaxation: drop rank constraint but keep Y cp.

z∗COP,primal = infY cp.

{M0 • Y : Mi • Y ≤ 0, i∈ [1:m], J0 • Y = 1} .

The conic dual of this problem is exactly z∗COP,dual.

Strong duality for copositive primal/dual pair

Theorem [B.’14]: Consider the copositive relaxation of z∗+.

If Qi is strictly copositive for one i∈ [0:m] and if F ◦+ 6= ∅, then

z∗COP,dual = z∗COP,primal and both values are attained.

> > ∈ ×¯





Further, the Semi-Lagrangean/copositive relaxation is tight,

z∗semi = z∗+ if and only if






z∗semi = z∗+ if and only if S(y) is copositive

for y> = [q0(x), u>] at a generalized KKT pair (x, u) ∈ F+ × Rm+:








i.e., Hux = du + v with vjxj = 0 and uiqi(x) = 0 for all i, j,

but without requiring vj ≥ 0.








i.e., Hux = du + v with vjxj = 0 and uiqi(x) = 0 for all i, j,

but without requiring vj ≥ 0.

Again have sufficient condition for global optimality of x.

Ok. But how to employ it ?

So we know, under weak conditions,

z∗dual,+ ≤ z∗semi = z∗COP,dual = z∗COP,primal ≤ z

∗+ .




∗+ .

[Dickinson/Gijben ’14]: the copositive cone C?d and its dual cone

Cd are both intractable, checking membership is NP-hard.




∗+ .



But can employ approximation hierarchies

Drd ⊂ C?d for all r = 0,1,2, . . .

of tractable cones Drd.




∗+ .



But can employ approximation hierarchies

Drd ⊂ Dr+1d ⊂ C?d for all r = 0,1,2, . . .

of tractable cones Drd.

Approximation hierarchies by tractable cones

Dual cones [Drd]? shrink towards cp. cone Cd, e.g., if

[Drd]? = {Y ∈ Lrd : Y is psd.}

with Lrd polyhedral cones of d×d matrices with no negative entries

such that∞⋂r=0

Lrd = Cd .





such that∞⋂r=0

Lrd = Cd .

Theorem [B.’14]: The tractable bounds

z∗D,r = sup{y0 ∈ R : S(y) ∈ Drn+1, y = [y0, u

>]> ∈ R× Rm+}

increase with r





such that∞⋂r=0

Lrd = Cd .

Theorem [B.’14]: The tractable bounds

z∗D,r = sup{y0 ∈ R : S(y) ∈ Drn+1, y = [y0, u

>]> ∈ R× Rm+}

increase with r and are tighter than the Lagrangian bounds.

Survey of approximation constructions

Name symbol mode method remarksB./de Klerk E outer LP rational grid for ∆n

Yıldırım Y outer LP Y ⊂ E, gridBundfuss/Dur B outer LP simplicial partitionB./Dur/Teo B(M) outer LP M⊃ K∗Bundfuss/Dur D inner LP simplicial partitionSponsel et al. D(M) inner LP M⊂ K∗

Parrilo et al. I inner LP coeff p(d)S ≥ o

Parrilo et al. S inner SDP p(d)S is a s.o.s.

Pena et al. Q inner SDP I ⊂ Q ⊂ SLasserre L(µ, T ) outer SDP µ-moments over TDickinson/Povh L(µ, T ;A) outer SDP L(µ, T ;A) ⊂ L(µ, T )

Survey of approximation constructions

Name symbol mode method remarks

Bundfuss/Dur D inner LP simplicial partitionSponsel et al. D(M) inner LP M⊂ K∗

Parrilo et al. I inner LP coeff p(d)S ≥ o

Parrilo et al. S inner SDP p(d)S is a s.o.s.

Pena et al. Q inner SDP I ⊂ Q ⊂ S

Improvement even at approximation level zero !

Starting hierarchy with r = 0: take

L0d =

{X = X>d× d : min

i,jXij ≥ 0

}.

Then D0d = P+N is cone of nonnegatively decomposable (NND)

matrices [Jarre/Lieder ’13] and its dual [ (NND)? = DNN ]

[D0d ]? =

{X = X>d× d psd. : min

i,jXij ≥ 0

}is familiar doubly nonnegative (DNN) cone P ∩N .

Theorem [B.’14]: Even at zero approximation level, with abovechoice,

z∗dual,+ ≤ z∗D,0 ≤ z

∗semi = z∗COP ≤ z

∗+ .

Hopefully [experiments needed] first inequality is significant.

Selected references in chronological order

[Shor ’87] Quadratic optimization problems, Izv. Akad. Nauk SSSR Tekhn.Kibernet. 22, 128–139.

[Hotho et al.’06] Information retrieval in folksonomies: Search and ranking,in The Semantic Web: Research and Applications, 411-426. Springer,Heidelberg.

[Burer ’09] On the copositive representation of binary and continuous non-convex quadratic programs, Math. Programming 120, 479–495.

[Saha et al.’10] Dense Subgraphs with Restrictions and Applications to GeneAnnotation Graphs, in Research in Computational Molecular Biology,456–472, Springer, Berlin.

[Bhaskara et al.’12] On quadratic programming with a ratio objective, inProc.ICALP’12, 109-120. Springer, Berlin.

[Jarre/Lieder ’13] Computing a nonnegative decomposition of a matrix,talk presented on 1 August 2013 at ICCOPT, Lisbon.

Selected references, continued

[Dickinson/Gijben ’14] On the computational complexity of membershipproblems for the completely positive cone and its dual,Comput. Optim. Appl. 57, 403–415.

[B. ’14] Copositive relaxation beats Lagrangian dual bounds in quadraticallyand linearly constrained QPs,Isaac Newton Institute Preprint series NI13064-POP, submitted.

[Amaral/B.’15] Copositivity-based approximations for mixed-integer fractio-nal quadratic optimization, Pacific J Opt., to appear.

[Pena/Vera/Zuluaga ’15] Completely positive reformulations for polynomialoptimization, Math.Programming, DOI 10.1007/s10107-014-0822-9, toappear.

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