Convex Relaxations of Non-Convex
Mixed Integer Quadratically
Constrained Problems
Dr. Anureet Saxena
Associate, Research
Axioma Inc.
(Joint Work with Pierre Bonami and Jon Lee)
Dedicated to Prof. Egon Balas
MIQCP
aT
0x
m in
st
x T A i x + aT
i x + bi ·
xj 2
l· x ·
0; i = 1 : : : m
Z; j 2 NI
u
Integer Constrained
Variables
Symmetric Matrices
NOT necessarily
positive semidefinite
Anureet Saxena, Axioma Inc.
1
MIQCP
aT
0x
m in
st
A i :Y + aT
i x + bi ·
xj 2
l · x ·
Y =
0;
i = 1:::m
Z;
j 2 NI
u
xx T
yi j = x i x j
Anureet Saxena, Axioma Inc.
2
Research Question?
Determine lower bounds on the
optimal value of MIQCP by
constructing strong convex
relaxations of MIQCP.
Anureet Saxena, Axioma Inc.
3
Disjunctive Programming
Polyhedral Relaxation
Disjunction
P = f x j A x ¸ bg
Separation Problem
Given x2P show that x2PD or find an inequality which
is satisfied by all points in PD and is violated by x.
Anureet Saxena, Axioma Inc.
4
Disjunctive Programming
T heorem : x 2 PD if and only if t he opt im al
value of t he f ollow ing cut generat ing linear program ( CGL P ) is non-negat ive.
m i n ®x ¡ ¯
s:t
® = ut A + vt D t
8t = 1 : : : q
¯ · u t b + v t dt
8t = 1 : : : q
ut ; vt ¸ 0
8t = 1 : : : q
CGLP
Pq
t
t t
t= 1( u » + v » ) = 1
Anureet Saxena, Axioma Inc.
5
Disjunctive Programming
Polyhedral Relaxation
Disjunction
P = f x j A x ¸ bg
Outer Approximation
of MIQCP defined by
the incumbent solution
Anureet Saxena, Axioma Inc.
6
Disjunctive Programming
Polyhedral Relaxation
Disjunction
P = f x j A x ¸ bg
What are the sources
of non-convexity in
MIQCP?
Anureet Saxena, Axioma Inc.
7
Disjunctive Programming
Polyhedral Relaxation
Disjunction
P = f x j A x ¸ bg
Integrality Constraints
Y=xxT
• xj2 Z j2 NI
• Elementary 0-1 disjunction
?
(xj · 0) OR (xj ¸ 1)
• Split Disjunctions
• GUB Disjunctions
Anureet Saxena, Axioma Inc.
8
Y=xxT
Y=xxT
All eigenvalues of
Y-xxT are equal to
zero.
Eigenvectors of Y-xxT
associated with non-zero
eigenvalues can be used as
sources of cuts
Anureet Saxena, Axioma Inc.
9
Y=xxT
Ohh!!
I don’t like fractional
components. I can use
them to get good cuts
MILP
Anureet Saxena, Axioma Inc.
10
Y=xxT
Ohh!!
I don’t like non-zero
eigenvalues. I can use
them to get good cuts
MIQCP
Anureet Saxena, Axioma Inc.
11
Negative Eigenvalues of Y-xxT
^ ¡ x^ x^ T ) c =
If ( Y
¸ c w here ¸ < 0 t hen
² ( cT x) 2 · Y:ccT is a c o n v ex quadrat ic cut
^)
w hich cut s o® ( x^ ; Y
² equivalent t o im p osing t he SD P condit ion
Y ¡ xx T ¸ SD P 0 by SO CP cut s.
Anureet Saxena, Axioma Inc.
12
Positive Eigenvalues of Y-xxT
^ ¡ x^ x^ T ) c =
If ( Y
¸ c w here ¸ > 0 t hen
Y:ccT · ( cT x) 2
is a n o n - c o n v ex quadrat ic cut w hich cut s o®
^ ).
( x^ ; Y
Y:ccT ·
t =
Univariate non-convex
expression
Anureet Saxena, Axioma Inc.
t2
cT x
13
Positive Eigenvalues of Y-xxT
m i n ( x;Y ) 2 OA cT x
m ax ( x;Y ) 2 OA cT x
( cT x) 2
cT x
Y:ccT · ( cT x) 2
Anureet Saxena, Axioma Inc.
14
Positive Eigenvalues of Y-xxT
p( cT x) + q
Secant
Approximation
Y.ccT· p(cTx) + q
cT x
Anureet Saxena, Axioma Inc.
15
Positive Eigenvalues of Y-xxT
µL
µ
p1 ( cT x) + q1
µU
p2 ( cT x) + q2
cT x
Anureet Saxena, Axioma Inc.
16
Positive Eigenvalues of Y-xxT
"
µL ( c) · cT x · µ
Y:ccT · p1 ( cT x) + q1
#
"
W
Anureet Saxena, Axioma Inc.
µ · cT x · µU ( c)
Y:ccT · p2 ( cT x) + q2
17
#
Cutting Plane Algorithm
^)
( x^ ; Y
¸ < 0
( cT x) 2 · Y:ccT
E xt ract E igenvalues
and E igenvect ors of
^ ¡ x^ x^ T .
Y
¸ > 0
Y:ccT · ( cT x) 2
Convex
Quadratic Cut
Derive Disjunction
CGLP
Derive Disjunctive
Cut
Anureet Saxena, Axioma Inc.
18
Sequential Convexification
T h eor em L et c1 ; : : : ; cn denot e a set of m ut uallyort hogonal unit vect ors in R n , and let
8
>
<
S0 =
¯
¯ A :Y + aT x + b · 0
¯
i
i
i
¯
( x; Y ) ¯
l· x· u
>
¯
:
T
¯
Y ¡ xx ¸ SD P 0
9
i = 1:::m >
=
>
;
T · (cT x)2
Y.cc
S = cl conv S
\ ( x; Y ) j Y:c c · ( c x)
f or j = 1 : : : n
Can we improve the disjunctive
S
= cl conv S
\ ( x; Y ) j x 2 f 0; 1g
f or j = 1 : : : p
cuts by choosing c more
T he f ollow ing st at em ent s hold t rue:
carefully?
A :Y + a x + b · 0
i = 1:::m
³
n
j
T
j j
j¡ 1
³
n
n+ j
n+ j ¡ 1
8
>
<
Sn =
cl conv
Sn+ p =
cl conv
>
:
8
>
>
>
>
<
>
>
>
>
:
T
j
2
o´
o´
j
¯
T
¯
¯
i
i
i
¯
( x; Y ) ¯
l · x· u
¯
¯
Y ¡ xx T = 0
¯
¯
¯ A i :Y + aT
i x + bi · 0
¯
¯
l · x· u
( x; Y ) ¯¯
Y ¡ xx T = 0
¯
¯
¯
x j 2 f 0; 1g
9
>
=
>
;
9
i = 1:::m >
>
>
>
=
>
>
>
>
j = 1:::p ;
19
Improving Disjunctions?
W e are searching f or vect ors c w hich sat isf y,
^ :ccT > ( cT x^ ) 2
1. Y
2. m ax ( x;Y ) 2 OA cT x ¡
sm all as possible
This condition is always satisfied
if c belongs to vector space
spanned by eigenvectors of YxxT associated with positive
eigenvalues.
m i n ( x;Y ) 2 OA cT x is as
This can be calculated by
solving a linear program
whose right hand side is a
linear function of c
Anureet Saxena, Axioma Inc.
20
Improving Disjunctions?
W e are searching f or vect ors c w hich sat isf y,
1.
This condition is always satisfied
if c belongs
toavector
space
This
problem
can
be
formulated
as
mixed
T
T
2
^ :cc > ( c x^ )
Y
spanned by eigenvectors of Yinteger linear program!!
xxT associated with positive
eigenvalues.
Univariate Expression
Generating Mixed IntegerTProgram
T
2. m ax ( x;Y ) 2 OA c x ¡ m i n ( x;Y ) 2 OA c (UGMIP)
x is as
sm all as possible
This can be calculated by
solving a linear program
whose right hand side is a
linear function of c
Anureet Saxena, Axioma Inc.
21
Cutting Plane Algorithm
^)
( x^ ; Y
¸ < 0
( cT x) 2 · Y:ccT
E xt ract E igenvalues
and E igenvect ors of
^ ¡ x^ x^ T .
Y
¸ > 0
Y:ccT · ( cT x) 2
UGMIP
Convex
Quadratic Cut
Derive Disjunction
CGLP
Derive Disjunctive
Cut
Anureet Saxena, Axioma Inc.
22
MIQCP Reformulations
MIQCP (x,Y)
RLT + SDP
Disjunctive Cuts
Strengthening
MIQCP (x,Y)
Projection
Lifting
Strengthening ?
MIQCP (x)
Heavy
Relaxation
MIQCP (x)
Projected Ineq
Anureet Saxena, Axioma Inc.
B&B
Light
Relaxation
23
MIQCP Reformulations
MIQCP (x,Y)
RLT + SDP
Disjunctive Cuts
Strengthening
MIQCP (x,Y)
Lifting
Heavy
Relaxation
Convex Relaxations of Non-Convex Mixed
Integer Quadratically Constrained
Projection
Problems: Projected
Formulations
B&B
A. Saxena, P. Bonami and J. Lee
Strengthening ?
MIQCP (x)
MIQCP (x)
Projected Ineq
Anureet Saxena, Axioma Inc.
Light
Relaxation
24
Projecting the RLT Formulation
A k :Y + aT
k x + bk ·
³
Yi j ¡
0 k = 1:::m
´
l x + uj x i ¡ l i uj
³ i j
´
Yi j ¡ u i x j + l j x i ¡ u i l j
³
´
Yi j ¡ u i x j + u j x i ¡ u i u j
³
´
Yi j ¡ l i x j + l j x i ¡ l i l j
·
0 8i ; j
·
0 8i ; j
¸
0 8i ; j
¸
0 8i ; j
RLT Inequalities
yi¡j ( x) =
m ax f u i x j + u j x i ¡ u i u j ; l i x j + l j x i ¡ l i l j g
8i ; j
yi+j ( x) =
m in f l i x j + u j x i ¡ l i u j ; u i x j + l j x i ¡ u i l j g
8i ; j
Anureet Saxena, Axioma Inc.
25
Projecting the RLT Formulation
A k :Y + aT
k x + bk ·
yi¡j ( x) ·
Yi j ·
yi+j ( x)
0 k = 1:::m
8i ; j
n
Qx =
P( x;Y )
o
x j 9Y s.t . ( x; Y ) 2 P( x;Y )
Separation Problem
Given x show that x2Qx or find an inequality which is
satisfied by all points in Qx and is violated by x.
Anureet Saxena, Axioma Inc.
26
Projecting the RLT Formulation
mi n
´
aT
^ + bk
kx
·
¡ A k :Y + ´
yi¡j ( x^ ) ·
Yi j
·
yi+j ( x^ )
k = 1:::m
8i ; j
ProjLP
T h eor em x^ 2 Q x if and only if t he opt im al
value of P rojL P is non-posit ive.
Anureet Saxena, Axioma Inc
27
Projecting the RLT Formulation
Dual
Solution
(u, B, C)
mi n
X ¡
i ;j
´
aT
^ + bk
kx
·
¡ A k :Y + ´
yi¡j ( x^ ) ·
Yi j
·
B i j yi¡j
(x) ¡
Ci j yi+j
k = 1:::m
yi+j ( x^ )
¢
X
(x) +
uk
8i ; j
¡
aTk x
¢
+ bk · 0
k2 M
Projected
Inequality
Anureet Saxena, Axioma Inc
28
Projecting the RLT Formulation
mi n
• A linear
´
aT
^ + bk
kx
·
¡ A k :Y + ´
yi¡j ( x^ ) ·
Yi j
·
yi+j ( x^ )
k = 1:::m
8i ; j
programming separation algorithm
• Handles large number O(n2) of RLT inequalities as bound constraints
• No of constraints = No of quadratic constraints in the original problem
Anureet Saxena, Axioma Inc
29
Surrogate Constraints
x T A 1 x + aT
1 x + b1 · 0
u1
x T A m x + aT
m x + bm · 0
um
x T A x + aT x + b · 0
A=B–C
B, C ¸ 0
yi¡j ( x) · x i x j · yi+j ( x)
X ¡
i ;j
B i j yi¡j
(x) ¡
Ci j yi+j
¢
X
(x) +
uk
k2 M
¡
aTk x
Surrogate Constraint
Can we extract the
convex part of the
surrogate constraint
¢
+ bk · 0
Surrogate Constraints
x T A 1 x + aT
1 x + b1 · 0
u1
x T A m x + aT
m x + bm · 0
um
x T A x + aT x + b · 0
Surrogate Constraint
What happens if we
add all such convex
quadratic cuts?
A=B+C–D
B ¸SDP 0
C, D ¸ 0
T
x Bx +
X ¡
i ;j
Ci j yi¡j
(x) ¡
D i j yi+j
¢
X
(x) +
uk
k2 M
¡
aTk x
¢
+ bk · 0
Projecting the SDP Formulation
min ´
s.t .
¡ A k :Y + ´ ¸ aTk x^ + bk ; 8k 2 M
yi¡j ( x^ ) · Yi j · yi+j ( x^ ); 8i ; j 2 N
Y + ´ I ¡ x^ x^ T ¸ SD P 0
Dual
Solution
(u, B, C, D)
T
x Bx +
X ¡
Ci j yi¡j
(x) ¡
D i j yi+j
¢
X
(x) +
i ;j
uk
¡
ProjSDP
aTk x
¢
+ bk · 0
k2 M
T h eor em x^ 2 Q +x if and only if t he opt im al
value of P rojSD P is non-p osit ive. Separation Problem
is a SDP 
Anureet Saxena, Axioma Inc
32
Projecting the SDP Formulation
T h eor em x^ 2 Q +x if and only if t he opt im al
value of t he f ollow ing piecew ise linear convex
opt im zat ion problem is non-p osit ive.
m ax f F ( u; B ) j u 2 § M ; B ¸ SD P 0g ;
w here § M = f u j
F ( u; B ) =
Unconstrained Convex
Optimization Problem over
the Cartessian product of a
simplex and cone of PSD
matrices
P
k2 M u k = 1; u ¸ 0g and
³P
´+ ³
´
P
¡
k
yi j ( x^ ) ¡ x^ i x^ j
i ;j
k2 M u k A i j ¡ B i j
³P
´¡ ³
´
P
+
k
+ i ;j
yi j ( x^ ) ¡ x^ i x^ j
k2 M u k A i j ¡ B i j
³
´
P
P
T
T
+ k2 M u k ( x^ A k x^ ) +
^ + bk
k2 M u k ak x
Anureet Saxena, Axioma Inc
33
Projecting the SDP Formulation
A = B + C ¡ D
B ¸ SD P 0; C; D ¸
x T A x + aT x + b · 0
0
Projected Sub Gradient Heuristic
1. Initialize B = Projection of A to the cone of PSD matrices
2. Compute a sub gradient of F(u,B) at B
3. Perform line search along the sub gradient direction
4. Update B and goto 2
Anureet Saxena, Axioma Inc.
34
Limitations of Projection Theorems
x T A 1 x + aT
1 x + b1 · 0
u1
x T A m x + aT
m x + bm · 0
um
x T A x + aT x + b · 0
Surrogate Constraint
Once the surrogate constraint has been produced very little
global information is used in the convexification process
Anureet Saxena, Axioma Inc.
35
Limitations of Projection Theorems
m in x 3
s.t .
x1x2 ¡ x1 ¡ x2 ¡ x3 · 0
¡ 6x 1 + 8x 2 · 3
3x 1 ¡ x 2 · 3
0 · x 1 ; x 2 · 1:5
(
P1 = cl conv
• st_e23 instances from GlobalLib
• OPT = -1.08
• RLT = -3
• SDP + RLT = -1.5
• x = ( 0.811, 0.689, -1.500)
¯
)
¯ x x The
¡ xnon-convex
x3 · 0
¯
1 ¡ x 2 ¡ quadratic
x ¯ 1 2 constraint
¯
0 · x 1 ; x 2and
· the
1:5bound
constraints cannot cut off x
( 1:5; 0; ¡ 1:5) 2 P1
0:5407 ( 1:5; 0; ¡ 1:5) +
0:5407
+
( 0; 1:5; ¡ 1:5) 2 P1
0:4593 ( 0; 1:5; ¡ 1:5) =
0:4593
=
Anureet Saxena, Axioma Inc.
x^
1
36
Limitations of Projection Theorems
• st_e23 instances from GlobalLib
• OPT = -1.08
• RLT = -3
Global Information
• SDP + RLT = -1.5
• xfor
= (engaging
0.811, 0.689, -1.500)
We need a technique
additional constraints in the
problem during the convexification
process
Anureet Saxena, Axioma Inc.
37
Limitations of Projection Theorems
x1x2 ¡ x1 ¡ x2 ¡ x3 · 0
Spect r al" D ecomposi# t i on of
0 0:5
0:5 0
1
2
( x 1 + x 2 ) 2 ¡ x 1 ¡ x 2 ¡ x 3 · 12 ( x 1 ¡ x 2 ) 2
Univariate non-convex
expression
Anureet Saxena, Axioma Inc.
38
Limitations of Projection Theorems
m in ( x 1 ¡ x 2 )
s:t
¡ 6x 1 + 8x 2 · 3
3x 1 ¡ x 2 · 3
0 · x 1 ; x 2 · 1:5
( x1 ¡ x2) 2
m ax ( x 1 ¡ x 2 )
s:t
¡ 6x 1 + 8x 2 · 3
3x 1 ¡ x 2 · 3
0 · x 1 ; x 2 · 1:5
( x1 ¡ x2)
: : : · ( x1 ¡ x2) 2
Anureet Saxena, Axioma Inc.
39
Limitations of Projection Theorems
( x 1 ¡ x 2 ) 2 · 0:625( x 1 ¡ x 2 ) + 0:375
Secant
Approximation
( x1 ¡ x2)
Anureet Saxena, Axioma Inc.
40
Limitations of Projection Theorems
x1x2 ¡ x1 ¡ x2 ¡ x3 · 0
Spect ral D ecom posit ion
1
2
( x 1 + x 2 ) 2 ¡ x 1 ¡ x 2 ¡ x 3 · 12 ( x 1 ¡ x 2 ) 2
Secant
1
2
Cuts off the
incumbent
A pproxim
at ion
solution
( x 1 + x 2 ) 2 ¡ x 1 ¡ x 2 ¡ x 3 · 12 ( 0:625( x 1 ¡ x 2 ) + 0:375 )
Anureet Saxena, Axioma Inc.
41
Eigen Reformulation
A =
x T A x + aT x + b · 0
P
³
T
¸ k > 0 ¸ k vk x
´2
P
T
¸
v
v
j j j j
P
T
+ a x + b+
¸ k < 0 ¸ k sk
yk = vkT x
8 k : ¸k < 0
sk = yk2
8 k : ¸k < 0
Anureet Saxena, Axioma Inc.
·
0
42
Eigen Reformulation
A =
x T A x + aT x + b · 0
P
³
T
¸ k > 0 ¸ k vk x
Directions of maximal
non-convexity
´2
P
T
¸
v
v
j j j j
P
T
+ a x + b+
¸ k < 0 ¸ k sk
yk = vkT x
8 k : ¸k < 0
sk = yk2
8 k : ¸k < 0
Anureet Saxena, Axioma Inc.
·
0
43
Eigen Reformulation
m in aT
0x
s.t .
x T A k x + aT
k x + bk · 0 ;
x j 2 Z ; 8j 2 N 1
P
³
8k 2 M
´2
P
T
T
vkj x + ak x + bk +
¸ kj < 0 ¸ kj skj · 0 ;
¸ kj > 0 ¸ kj
T x ;
ykj = vkj
8 j : ¸ kj < 0; k 2 M
2 ;
skj = ykj
8 j : ¸ kj < 0; k 2 M
L kj · ykj · Ukj ;
8k 2 M
Geometric
correlations along
directions of maximal
8 j : ¸ kj < 0; k 2 Mnon-convexity
:
Anureet Saxena, Axioma Inc.
44
Eigen Reformulation
y2
• st_glmp_kky instances from GlobalLib
• OPT = -2.5
• RLT = RLT+SDP = -3.0
y1
x4x5 + x6x7
y1 = p1 x 4 ¡
2
y2 = p1 x 6 ¡
2
Projection along y1 and y2
¡ x3 ¡ z · 0
p1 x 5
2
p1 x 7
2
Can we exploit these
correlations in deriving
strong cutting planes?
Anureet Saxena, Axioma Inc.
45
Polarity Cuts
s1 ¡ ( L + U) y1 ¡ L U · 0
1. Projection
2. Determine Extreme Points
3. Lifting
L
U
y1
4. Convexification
s1 · y12
Anureet Saxena, Axioma Inc.
46
Polarity Cuts
1. Projection
2. Determine Extreme Points
3. Lifting
y1
4. Convexification
max y1
mi n y1
Projection
Anureet Saxena, Axioma Inc.
47
Polarity Cuts
1. Projection
2. Determine Extreme Points
3. Lifting
L
U
y1
4. Convexification
Anureet Saxena, Axioma Inc.
48
Polarity Cuts
( U; U 2 )
1. Projection
2. Determine Extreme Points
( L ; L 2)
3. Lifting
L
U
y1
4. Convexification
Anureet Saxena, Axioma Inc.
49
Polarity Cuts
Facet
( U; U 2 )
1. Projection
2. Determine Extreme Points
( L ; L 2)
3. Lifting
L
U
y1
m in ®k y^k + ¯ k ^sk ¡ °
s.t .
®k L k + ¯ k ( L k ) 2 ¡ ° ¸ 0
®k Uk + ¯ k ( Uk ) 2 ¡ ° ¸ 0
®k ¡ ®+k + ®¡k = 0
®+k + ®¡k ¡ ¯ k = 1
®+k ¸ 0; ®¡k ¸ 0; ¯ k · 0
4. Convexification
Polar Program
Anureet Saxena, Axioma Inc.
50
Polar Program
Polarity
Convex
Relaxation
Extreme points
of the projected set
• Additional problem constraints induce
geometric correlations along directions of
maximal non-convexity
min
s.t .
P
P
• Polarity uses these correlations to derive
strong cutting planes for MIQCP
³
(®k y^k + ¯k s^k ) ¡ °
2
¯k (ykt )
´
®k ykt +
¡ ° ¸ 0; 8t
¯k · 0; 8k 2 S
+
¡
®
¡
®
+
®
k
¡k + k ¡= 0; ¢8k 2 S
P
k 2 S ®k + ®k ¡ ¯k = 1
+
®k ¸ 0; ®k¡ ¸ 0
k2 S
• Projection mechanism identifies such
correlations
k2 S
Anureet Saxena, Axioma Inc
51
Eigen Reformulation
y2
• st_glmp_kky instances from GlobalLib
• OPT = -2.5
• RLT = RLT+SDP = -3.0
y1
x4x5 + x6x7
y1 = p1 x 4 ¡
2
y2 = p1 x 6 ¡
2
Projection along y1 and y2
¡ x3 ¡ z · 0
p1 x 5
2
p1 x 7
2
Can we exploit these
correlations in deriving
strong cutting planes?
Anureet Saxena, Axioma Inc.
52
Eigen Reformulation
• st_glmp_kky instances from GlobalLib
• OPT = -2.5
• RLT = RLT+SDP = -3.0
y1
Polarity cuts close 99.62% of
the duality gap!!
Projection along y1 and y2
Can we exploit these
correlations in deriving
strong cutting planes?
Anureet Saxena, Axioma Inc.
53
Relaxations of MIQCP
RLT + SDP
Projection
RLT
Disjunctive
Cuts
Projection
RLT + SDP
Eigen
Reformulation
Anureet Saxena, Axioma Inc.
Low Width
Disjunctions
Polarity
54
Computational Results
Solvers
• Convex Relaxations – IpOpt
• Eigenvalue Computations – LAPACK
• Linear Programs & Mixed Integer Programs– CPLEX 11
• COIN-OR / Bonmin based implementation
Test Bed
• 160 GlobalLIB Instances
• 4 Chemical Process Design instances from Lee & Grossman
• 90 Box QP Instances
Anureet Saxena, Axioma Inc.
55
Computational Results
Experiment Setup
• 1 Hour Time limit on each instance
opt ( F i nal Rel axat i on) ¡ RL T
D ual i t y Gap =
£ 100
Opt ( M I QCP ) ¡ RL T
Anureet Saxena, Axioma Inc.
56
GlobalLIB Instances
160 = 129 + 24 + 7
• All MIQCP Instances with
upto 50 variables
• x1 x2 x3 x4 x5
Numerical Problems
Zero Duality Gap
• (x1+x2)/x3 ¸ 2x1
• x0.75
Non-Zero Duality Gap
Anureet Saxena, Axioma Inc.
57
Computational Results
(Extended Formulations)
V1
V2
SDP
V1
Disjunctive
Cuts
Y:ccT · ( cT x) 2
V3
V2
Anureet Saxena, Axioma Inc.
UGMIP
58
GlobalLIB Instances
Summary of % Duality Gap Closed (129 Instances with non-zero Duality
Gap)
>99.99 %
98-99.99 %
75-98 %
25-75 %
0-25 %
Average Gap Closed
V1
16
1
10
11
91
V2
23
44
23
22
17
V3
23
52
21
20
13
24.80%
76.49%
80.86%
Anureet Saxena, Axioma Inc.
59
GlobalLIB Instances
Summary of % Duality Gap Closed (129 Instances with non-zero Duality
Gap)
>99.99 %
98-99.99 %
75-98 %
25-75 %
0-25 %
Average Gap Closed
V1
16
1
10
11
91
V2
23
44
23
22
17
V3
23
52
21
20
13
24.80%
76.49%
80.86%
Anureet Saxena, Axioma Inc.
60
Version (2,3) vs Version 1
Instance
st_qpc-m3a
st_ph13
st_ph11
ex3_1_4
st_jcbpaf2
st_ph12
ex2_1_9
prob05
st_glmp_kky
st_e24
st_ph15
st_bsj4
st_ph14
st_e08
st_ht
st_pan2
ex2_1_1
st_fp1
st_pan1
ex5_2_4
st_e02
st_kr
% Duality Gap Closed
V1
V2
V3
0.00%
98.10%
99.16%
0.00%
99.38%
98.80%
0.00%
99.46%
98.19%
0.00%
86.31%
99.57%
0.00%
99.47%
99.61%
0.00%
99.49%
99.62%
0.00%
98.79%
99.73%
0.00%
99.78%
99.49%
0.00%
99.80%
99.71%
0.00%
99.81%
99.81%
0.00%
99.83%
99.81%
0.00%
99.86%
99.80%
0.00%
99.85%
99.86%
0.00%
99.81%
99.89%
0.00%
99.81%
99.89%
0.00%
68.54%
99.91%
0.00%
72.62%
99.92%
0.00%
72.62%
99.92%
0.00%
99.72%
99.92%
0.00%
79.31%
99.92%
0.00%
99.88%
99.95%
0.00%
99.93%
99.95%
Instance
st_e33
st_z
st_qpc-m0
st_phex
st_e26
st_m1
ex2_1_6
st_fp6
st_e07
st_glmp_kk92
st_ph3
st_ph20
st_qpk1
st_bsj2
st_ph2
st_ph1
ex2_1_5
st_fp5
ex3_1_3
st_bpv2
st_qpc-m1
st_qpc-m3b
Anureet Saxena, Axioma Inc.
% Duality Gap Closed
V1
V2
V3
0.00%
99.94%
99.95%
0.00%
99.96%
99.95%
0.00%
99.96%
99.96%
0.00%
99.96%
99.96%
0.00%
99.96%
99.96%
0.00%
99.96%
99.96%
0.00%
99.95%
99.97%
0.00%
99.92%
99.97%
0.00%
99.97%
99.97%
0.00%
99.98%
99.98%
0.00%
99.98%
99.98%
0.00%
99.98%
99.98%
0.00%
99.98%
99.98%
0.00%
99.98%
99.96%
0.00%
99.98%
99.98%
0.00%
99.98%
99.98%
0.00%
99.98%
99.99%
0.00%
99.98%
99.99%
0.00%
99.99%
99.99%
0.00%
99.99%
99.99%
0.00%
99.99%
99.98%
0.00% 100.00% 100.00%
61
Version (2,3) vs Version 1
% Duality Gap Closed
% Duality Gap Closed
Instance
V1
V2
V3
Instance
V1
V2
V3
st_e33
0.00%
99.94%
99.95%
st_qpc-m3a
0.00%
98.10%
99.16%
st_z
0.00%
99.96%
99.95%
st_ph13
0.00%
99.38%
98.80%
st_qpc-m0
0.00%
99.96%
99.96%
st_ph11
0.00%
99.46%
98.19%
st_phex
0.00%
99.96%
99.96%
ex3_1_4
0.00%
86.31%
99.57%
st_e26
0.00%
99.96%
99.96%
st_jcbpaf2
0.00%
99.47%
99.61%
st_m1
0.00%
99.96%
99.96%
st_ph12
0.00%
99.49%
99.62%
Either
version
2
or
version
3
closes
>99%
of
the
duality
gap
on
44
ex2_1_6
0.00%
99.95%
99.97%
ex2_1_9
0.00%
98.79%
99.73%
which99.78%
version
1 is unable
gap. 99.92% 99.97%
st_fp6 to close any0.00%
prob05instances on0.00%
99.49%
st_e07
0.00%
99.97%
99.97%
st_glmp_kky
0.00%
99.80%
99.71%
st_glmp_kk92
0.00%
99.98%
99.98%
st_e24
0.00%
99.81%
99.81%
The relaxation
by adding
disjunctive cuts
be 99.98%
st_ph3
0.00% can
99.98%
st_ph15
0.00%obtained
99.83%
99.81%
st_ph20
st_bsj4substantially
0.00%
99.86% than
99.80%the SDP
stronger
relaxation!!0.00% 99.98% 99.98%
st_qpk1
0.00%
99.98%
99.98%
st_ph14
0.00%
99.85%
99.86%
st_bsj2
0.00%
99.98%
99.96%
st_e08
0.00%
99.81%
99.89%
st_ph2
0.00%
99.98%
99.98%
st_ht
0.00%
99.81%
99.89%
st_ph1
0.00%
99.98%
99.98%
st_pan2
0.00%
68.54%
99.91%
ex2_1_5
0.00%
99.98%
99.99%
ex2_1_1
0.00%
72.62%
99.92%
st_fp5
0.00%
99.98%
99.99%
st_fp1
0.00%
72.62%
99.92%
ex3_1_3
0.00%
99.99%
99.99%
st_pan1
0.00%
99.72%
99.92%
st_bpv2
0.00%
99.99%
99.99%
ex5_2_4
0.00%
79.31%
99.92%
st_qpc-m1
0.00%
99.99%
99.98%
st_e02
0.00%
99.88%
99.95%
st_qpc-m3b
0.00% 100.00% 100.00%
st_kr
0.00%
99.93%
99.95%
Observation
Anureet Saxena, Axioma Inc.
62
Linear Complementarity Disjunctions
• Some problems have linear complementarity constraints
xi x j = 0
• These constraints can be used to derive the linear complementarity
disjunctions
(xi=0) OR (xj=0)
which can be used with the medley of other disjunctions to derive
disjunctive cuts
Anureet Saxena, Axioma Inc.
65
Linear Complementarity Disjunctions
Instance
ex9_1_4
ex9_2_1
ex9_2_2
ex9_2_3
ex9_2_4
ex9_2_6
ex9_2_7
Without Using LCD
V2
V3
0.00%
1.55%
60.04%
92.02%
88.29%
98.06%
0.00%
47.17%
99.87%
99.89%
87.93%
62.00%
51.47%
86.25%
Using LCD
V2
100.00%
99.95%
100.00%
99.99%
99.99%
80.22%
99.97%
V3
99.97%
99.95%
100.00%
99.99%
100.00%
92.09%
99.95%
Observation
Linear Complementarity conditions can be exploited effectively
within a disjunctive programming framework to derive strong
cuts
Anureet Saxena, Axioma Inc.
66
Y=xxT
Y=xxT
All eigenvalues of
Y-xxT are equal to
zero.
What is the effect of
these disjunctive cuts
on the spectrum of
Y-xxT ?
Eigenvectors of Y-xxT
associated with non-zero
eigenvalues can be used as
sources of cuts
Anureet Saxena, Axioma Inc.
72
Spectrum of Y-xxT
% Duality Gap closed by
V1
< 10 %
> 40%
< 10%
V2
> 90 %
< 60%
< 10%
Anureet Saxena, Axioma Inc.
Instance Chosen
st_jcbpaf2
ex9_2_7
ex7_3_1
73
Version 1, 0% Gap Closed
1.20
1.00
0.80
Sum of Positive
Eigen Values of Y-xxT
0.60
Sum of Negative
Eigen Values of Y-xxT
0.40
0.20
0.00
-0.20
-0.40
-0.60
0
5
10
15
20
Anureet Saxena, Axioma Inc.
25
30
35
74
Version 2, 99.47% Gap Closed
1.20
1.00
0.80
0.60
0.40
0.20
0.00
-0.20
-0.40
-0.60
0
50
100
150
Anureet Saxena, Axioma Inc.
200
250
300
75
Version 3, 99.61% Gap Closed
1.20
1.00
0.80
0.60
0.40
0.20
0.00
-0.20
-0.40
-0.60
0
20
40
60
80
Anureet Saxena, Axioma Inc.
100
120
76
Computational Results
(Projected Formulations)
W1
W2
ProjLP
Disjunctive
Cuts
W1
Polarity
Cuts
All experiments were done using the eigen reformulation
Anureet Saxena, Axioma Inc.
77
Computational Results: GlobalLib (Projected)
>99.99 % gap closed
98-99.99 % gap closed
75-98 % gap closed
25-75 % gap closed
0-25 % gap closed
0-(-0.22) % gap closed
Average Gap Closed
Average Time taken (sec)
Disjunctive Cuts
ProjLP +
ProjLP
PolarLP
19
23
22
31
35
33
34
23
14
14
4
4
70.65%
76.06%
4.616
19.462
ProjLP
19
5
17
26
57
4
40.92%
0.893
Anureet Saxena, Axioma Inc
ProjLP +
PolarLP
23
21
18
32
30
4
60.48%
0.814
Extended
SDP + RLT +
SDP + RLT
Dsj
16
23
1
44
10
23
11
22
91
17
0
0
24.80%
76.49%
198.043
978.140
78
Computational Results: GlobalLib (Projected)
>99.99 % gap closed
98-99.99 % gap closed
75-98 % gap closed
25-75 % gap closed
0-25 % gap closed
0-(-0.22) % gap closed
Average Gap Closed
Average Time taken (sec)
Disjunctive Cuts
ProjLP +
ProjLP
PolarLP
19
23
22
31
35
33
34
23
14
14
4
4
70.65%
76.06%
4.616
19.462
ProjLP
19
5
17
26
57
4
40.92%
0.893
Anureet Saxena, Axioma Inc
ProjLP +
PolarLP
23
21
18
32
30
4
60.48%
0.814
Extended
SDP + RLT +
SDP + RLT
Dsj
16
23
1
44
10
23
11
22
91
17
0
0
24.80%
76.49%
198.043
978.140
79
Computational Results: GlobalLib (Projected)
>99.99 % gap closed
98-99.99 % gap closed
75-98 % gap closed
25-75 % gap closed
0-25 % gap closed
0-(-0.22) % gap closed
Average Gap Closed
Average Time taken (sec)
Disjunctive Cuts
ProjLP +
ProjLP
PolarLP
19
23
22
31
35
33
34
23
14
14
4
4
70.65%
76.06%
4.616
19.462
ProjLP
19
5
17
26
57
4
40.92%
0.893
Anureet Saxena, Axioma Inc
ProjLP +
PolarLP
23
21
18
32
30
4
60.48%
0.814
Extended
SDP + RLT +
SDP + RLT
Dsj
16
23
1
44
10
23
11
22
91
17
0
0
24.80%
76.49%
198.043
978.140
80
Computational Results: GlobalLib (Projected)
Disjunctive Cuts
Extended
ProjLP +
ProjLP +
SDP + RLT +
ProjLP
ProjLP
SDP + RLT
PolarLP
PolarLP
Dsj
>99.99 % gap closed
19
23
19
23
16
23
98-99.99 % gap closed
22
31
5
21
1
44
75-98 % gap closed
35
33
17
18
10
23
25-75 % gap closed
34
23
26
32
11
22
We can
relaxations
of x 91
0-25 % gap closed
14 generate14
57 in the space
30
17
0-(-0.22) % gap closed variables
4 which are4almost as4 strong as4 those in 0
0
Average Gap Closed
76.06%even40.92%
60.48%
24.80%
76.49%
the70.65%
extended space
though our
computing
Average Time taken (sec)
4.616 are 100
19.462
0.893
978.140
times
times smaller
on0.814
average.198.043
Observation
Anureet Saxena, Axioma Inc
81
Computational Results: GlobalLib (Projected)
>99.99 % gap closed
98-99.99 % gap closed
75-98 % gap closed
25-75 % gap closed
0-25 % gap closed
0-(-0.22) % gap closed
Average Gap Closed
Average Time taken (sec)
Disjunctive Cuts
ProjLP +
ProjLP
PolarLP
19
23
22
31
35
33
34
23
14
14
4
4
70.65%
76.06%
4.616
19.462
ProjLP
19
5
17
26
57
4
40.92%
0.893
ProjLP +
PolarLP
23
21
18
32
30
4
60.48%
0.814
Extended
SDP + RLT +
SDP + RLT
Dsj
16
23
1
44
10
23
11
22
91
17
0
0
24.80%
76.49%
198.043
978.140
30%
16%
Anureet Saxena, Axioma Inc
82
Computational Results: GlobalLib (Projected)
>99.99 % gap closed
98-99.99 % gap closed
75-98 % gap closed
25-75 % gap closed
0-25 % gap closed
0-(-0.22) % gap closed
Average Gap Closed
Average Time taken (sec)
Disjunctive Cuts
Extended
ProjLP +
ProjLP +
SDP + RLT +
ProjLP
ProjLP
SDP + RLT
PolarLP
PolarLP
Dsj
19
23
19
23
16
23
22
31
5
21
1
44
35
33
17
18
10
23
34
23
26
32
11
22
Polarity
cuts14can capture
14
57 a portion
30 of
91
17
4
4
4 cuts
0
0
strengthening
derived
from4 disjunctive
70.65%
76.06%
40.92%
60.48%
24.80%
76.49%
4.616 Global
19.462
0.893at work!!
0.814
198.043
978.140
Information
Observation
30%
16%
Anureet Saxena, Axioma Inc
83
Computational Results: Box QP (Projected)
W3
ProjLP
W3-SDP
ProjLP
Projected
Gradient
Heuristic
All experiments were done using the eigen reformulation
Anureet Saxena, Axioma Inc
84
Computational Results: Box QP (Projected)
Instances
spar20*
spar30*
spar40*
spar50*
spar60*
spar70*
spar80*
spar90*
spar100*
Average
% Duality Gap Closed
W3
W3-SDP
94.60 - 99.97
91.54 - 99.91
89.87 - 99.99
51.41 - 98.79
87.85 - 99.60
21.78 - 89.63
87.88 - 97.53
11.38 - 50.15
85.78 - 90.99
0.00 - 0.00
89.78 - 99.36
0.00 - 53.67
88.13 - 97.49
2.94 - 56.23
89.44 - 96.60
5.73 - 50.13
92.15 - 96.46
8.17 - 51.79
95.19%
50.01%
W3
2.49 - 408.36
12.33 - 565.88
35.77 - 134.8
50.22 - 180.96
121.83 - 226.11
191.12 - 693.28
257.62 - 892.96
408.73 - 991.04
538.03 - 1509.96
280.50
Anureet Saxena, Axioma Inc
Time Taken (sec)
W3-SDP
0.84 - 2.46
1.74 - 14.38
4.16 - 65.28
8.76 - 99.13
111.07 - 127.47
22.02 - 202.98
34.77 - 67.66
46.98 - 95.66
75.49 - 112.69
37.89
W3 (Adjusted)
0.51 - 1.60
3.32 - 14.49
13.75 - 49.76
28.95 - 76.19
86.28 - 141.77
92.63 - 143.35
121.62 - 230.53
184.63 - 294.92
279.41 - 385.64
101.57
85
Computational Results: Box QP (Projected)
Instances
spar20*
spar30*
spar40*
spar50*
spar60*
spar70*
spar80*
spar90*
spar100*
Average
% Duality Gap Closed
W3
W3-SDP
94.60 - 99.97
91.54 - 99.91
89.87 - 99.99
51.41 - 98.79
87.85 - 99.60
21.78 - 89.63
87.88 - 97.53
11.38 - 50.15
85.78 - 90.99
0.00 - 0.00
89.78 - 99.36
0.00 - 53.67
88.13 - 97.49
2.94 - 56.23
89.44 - 96.60
5.73 - 50.13
92.15 - 96.46
8.17 - 51.79
95.19%
50.01%
W3
2.49 - 408.36
12.33 - 565.88
35.77 - 134.8
50.22 - 180.96
121.83 - 226.11
191.12 - 693.28
257.62 - 892.96
408.73 - 991.04
538.03 - 1509.96
280.50
Anureet Saxena, Axioma Inc
Time Taken (sec)
W3-SDP
0.84 - 2.46
1.74 - 14.38
4.16 - 65.28
8.76 - 99.13
111.07 - 127.47
22.02 - 202.98
34.77 - 67.66
46.98 - 95.66
75.49 - 112.69
37.89
W3 (Adjusted)
0.51 - 1.60
3.32 - 14.49
13.75 - 49.76
28.95 - 76.19
86.28 - 141.77
92.63 - 143.35
121.62 - 230.53
184.63 - 294.92
279.41 - 385.64
101.57
86
Computational Results: Box QP (Projected)
Instances
spar20*
spar30*
spar40*
spar50*
spar60*
spar70*
spar80*
spar90*
spar100*
Average
% Duality Gap Closed
Time Taken (sec)
W3
W3-SDP
W3
W3-SDP
94.60 - 99.97
91.54 - 99.91
2.49 - 408.36
0.84 - 2.46
89.87 - 99.99
51.41 - 98.79
12.33 - 565.88
1.74 - 14.38
87.85 - 99.60
21.78 - 89.63
35.77 - 134.8
4.16 - 65.28
87.88 - 97.53
11.38 - 50.15
50.22 - 180.96
8.76 - 99.13
85.78 - 90.99
0.00 - 0.00
121.83 - 226.11 111.07 - 127.47
Convex 0.00
quadratic
generated
89.78 - 99.36
- 53.67 cuts
191.12
- 693.28 using
22.02the
- 202.98
88.13 - 97.49
2.94gradient
- 56.23 heuristic
257.62 - 892.96
34.77 -a67.66
projected
can capture
89.44substantial
- 96.60
5.73
- 50.13of strengthening
408.73 - 991.04 derived
46.98 -from
95.66
portion
92.15 - 96.46
8.17
538.03
- 1509.96 75.49 - 112.69
the- 51.79
SDP+RLT
relaxations
95.19%
50.01%
280.50
37.89
Observation
Anureet Saxena, Axioma Inc
W3 (Adjusted)
0.51 - 1.60
3.32 - 14.49
13.75 - 49.76
28.95 - 76.19
86.28 - 141.77
92.63 - 143.35
121.62 - 230.53
184.63 - 294.92
279.41 - 385.64
101.57
87
Computational Results: Box QP (Projected)
Instances
spar20*
spar30*
spar40*
spar50*
spar60*
spar70*
spar80*
spar90*
spar100*
Average
% Duality Gap Closed
Time Taken (sec)
W3
W3-SDP
W3
W3-SDP
W3 (Adjusted)
94.60 - 99.97
91.54 - 99.91
2.49 - 408.36
0.84 - 2.46
0.51 - 1.60
89.87 - 99.99
51.41 - 98.79
12.33 - 565.88
1.74 - 14.38
3.32 - 14.49
87.85 - 99.60
21.78 - 89.63
35.77 - 134.8
4.16 - 65.28
13.75 - 49.76
87.88 - 97.53
11.38 - 50.15
50.22 - 180.96
8.76 - 99.13
28.95 - 76.19
85.78 - 90.99
0.00 - 0.00
121.83 - 226.11 111.07 - 127.47
86.28 - 141.77
Convex 0.00
quadratic
generated
89.78 - 99.36
- 53.67 cuts
191.12
- 693.28 using
22.02the
- 202.98
92.63 - 143.35
88.13 - 97.49
2.94gradient
- 56.23 heuristic
257.62 - 892.96
34.77 -a67.66
121.62 - 230.53
projected
can capture
89.44substantial
- 96.60
5.73
- 50.13of strengthening
408.73 - 991.04 derived
46.98 -from
95.66
184.63 - 294.92
portion
92.15 - 96.46
8.17
538.03
- 1509.96 75.49 - 112.69
279.41 - 385.64
the- 51.79
SDP+RLT
relaxations
the eigen 101.57
95.19%
50.01%
280.50 Just using
37.89
Observation
reformulation with
ProjLP closes 50%
of the duality gap !!
Anureet Saxena, Axioma Inc
88
Computational Results: Box QP (Projected)
Instances
spar20*
spar30*
spar40*
spar50*
spar60*
spar70*
spar80*
spar90*
spar100*
Average
% Duality Gap Closed
W3
W3-SDP
94.60 - 99.97
91.54 - 99.91
89.87 - 99.99
51.41 - 98.79
87.85 - 99.60
21.78 - 89.63
87.88 - 97.53
11.38 - 50.15
85.78 - 90.99
0.00 - 0.00
89.78 - 99.36
0.00 - 53.67
88.13 - 97.49
2.94 - 56.23
89.44 - 96.60
5.73 - 50.13
92.15 - 96.46
8.17 - 51.79
95.19%
50.01%
W3
2.49 - 408.36
12.33 - 565.88
35.77 - 134.8
50.22 - 180.96
121.83 - 226.11
191.12 - 693.28
257.62 - 892.96
408.73 - 991.04
538.03 - 1509.96
280.50
Anureet Saxena, Axioma Inc
Time Taken (sec)
W3-SDP
0.84 - 2.46
1.74 - 14.38
4.16 - 65.28
8.76 - 99.13
111.07 - 127.47
22.02 - 202.98
34.77 - 67.66
46.98 - 95.66
75.49 - 112.69
37.89
W3 (Adjusted)
0.51 - 1.60
3.32 - 14.49
13.75 - 49.76
28.95 - 76.19
86.28 - 141.77
92.63 - 143.35
121.62 - 230.53
184.63 - 294.92
279.41 - 385.64
101.57
89
Computational Results: Box QP (Projected)
Instances
spar20*
spar30*
spar40*
spar50*
spar60*
spar70*
spar80*
spar90*
spar100*
Average
% Duality Gap Closed
W3
W3-SDP
94.60 - 99.97
91.54 - 99.91
89.87 - 99.99
51.41 - 98.79
87.85 - 99.60
21.78 - 89.63
87.88 - 97.53
11.38 - 50.15
85.78 - 90.99
0.00 - 0.00
89.78 - 99.36
0.00 - 53.67
88.13 - 97.49
2.94 - 56.23
89.44 - 96.60
5.73 - 50.13
92.15 - 96.46
8.17 - 51.79
95.19%
50.01%
W3
2.49 - 408.36
12.33 - 565.88
35.77 - 134.8
50.22 - 180.96
121.83 - 226.11
191.12 - 693.28
257.62 - 892.96
408.73 - 991.04
538.03 - 1509.96
280.50
Time Taken (sec)
W3-SDP
0.84 - 2.46
1.74 - 14.38
4.16 - 65.28
8.76 - 99.13
111.07 - 127.47
22.02 - 202.98
34.77 - 67.66
46.98 - 95.66
75.49 - 112.69
37.89
W3 (Adjusted)
0.51 - 1.60
3.32 - 14.49
13.75 - 49.76
28.95 - 76.19
86.28 - 141.77
92.63 - 143.35
121.62 - 230.53
184.63 - 294.92
279.41 - 385.64
101.57
spar100-075-1 Instance
• 95.84% gap closed in 1509 sec
• 94.84% gap closed in 366 sec
Assessing the Tailing off
Behaviour
Anureet Saxena, Axioma Inc
90
Computational Results: Box QP (Projected)
Instances
spar20*
spar30*
spar40*
spar50*
spar60*
spar70*
spar80*
spar90*
spar100*
Average
% Time spent on
Cut Generation
W3
W3-SDP
26.28 - 95.77
0.12 - 0.24
17.78 - 91.48
0.00 - 0.21
27.5 - 78.37
0.01 - 0.13
51.02 - 79.73
0.01 - 0.11
46.29 - 56.61
0.10 - 0.12
71.13 - 87.7
0.01 - 0.11
76.37 - 84.44
0.01 - 0.02
73.44 - 88.25
0.01 - 0.02
77.49 - 92.3
0.01 - 0.23
66.05%
0.04%
Time Spent on Cut Generation
• Increases with version W3
reaching 75% for larger instances
• Remains less than 0.25% for all
instances with W3-SDP
• ProjLP can be solved very
efficiently
Anureet Saxena, Axioma Inc
91
Computational Results: Box QP (Projected)
% Duality Gap Closed
Instances
SDPLR
SDPA
Proj
spar20*
99.67 - 100 99.67 - 99.99 94.6 - 99.97
spar30*
97.81 - 100 97.81 - 99.99 89.87 - 99.99
spar40*
96.6 - 100
96.6 - 99.99 87.85 - 99.6
spar50*
95.55 - 100 95.55 - 99.99 87.88 - 97.53
spar60*
98.69 - 100 98.69 - 99.99 85.78 - 90.99
spar70*
98.46 - 100 98.46 - 99.99 89.78 - 99.36
spar80*
97.85 - 100 97.84 - 99.99 88.13 - 97.49
spar90* 97.83 - 99.99 97.83 - 99.99 89.44 - 96.6
spar100* 98.17 - 99.38 98.17 - 99.38 92.15 - 96.46
Average
99.40%
99.40%
95.19%
Time Taken (sec)
SDPLR
SDPA
Proj
0.97 - 56.37
1.98 - 3.39
2.48 - 408.35
3.57 - 243.3
16.66 - 29.33
12.33 - 565.88
10.3 - 515.73
105.68 - 157.83
35.77 - 134.8
41.72 - 926.15
438.77 - 589.17
50.21 - 180.95
88.05 - 532.45
1150.06 - 1408.32
121.83 - 226.1
133.07 - 3600.75
2769.98 - 3721.34
191.11 - 693.27
965.18 - 5413.02
6618.79 - 8285.12
257.61 - 892.95
2403.62 - 7049.49 12838.46 - 17048.98 408.73 - 991.04
5355.2 - 10295.88 23509.13 - 28604.12 538.02 - 1509.96
1741.20
5247.04
280.50
Anureet Saxena, Axioma Inc
Time to solve
last relaxation
Proj
0.05 - 0.32
0.06 - 0.89
0.16 - 1.18
0.13 - 0.86
0.53 - 1.55
0.48 - 1.1
0.56 - 2.02
0.77 - 1.51
0.82 - 2
0.67
92
Computational Results: Box QP (Projected)
% Duality Gap Closed
Instances
SDPLR
SDPA
Proj
spar20*
99.67 - 100 99.67 - 99.99 94.6 - 99.97
spar30*
97.81 - 100 97.81 - 99.99 89.87 - 99.99
spar40*
96.6 - 100
96.6 - 99.99 87.85 - 99.6
spar50*
95.55 - 100 95.55 - 99.99 87.88 - 97.53
spar60*
98.69 - 100 98.69 - 99.99 85.78 - 90.99
spar70*
98.46 - 100 98.46 - 99.99 89.78 - 99.36
spar80*
97.85 - 100 97.84 - 99.99 88.13 - 97.49
spar90* 97.83 - 99.99 97.83 - 99.99 89.44 - 96.6
spar100* 98.17 - 99.38 98.17 - 99.38 92.15 - 96.46
Average
99.40%
99.40%
95.19%
Time Taken (sec)
SDPLR
SDPA
Proj
0.97 - 56.37
1.98 - 3.39
2.48 - 408.35
3.57 - 243.3
16.66 - 29.33
12.33 - 565.88
10.3 - 515.73
105.68 - 157.83
35.77 - 134.8
41.72 - 926.15
438.77 - 589.17
50.21 - 180.95
88.05 - 532.45
1150.06 - 1408.32
121.83 - 226.1
133.07 - 3600.75
2769.98 - 3721.34
191.11 - 693.27
965.18 - 5413.02
6618.79 - 8285.12
257.61 - 892.95
2403.62 - 7049.49 12838.46 - 17048.98 408.73 - 991.04
5355.2 - 10295.88 23509.13 - 28604.12 538.02 - 1509.96
1741.20
5247.04
280.50
Anureet Saxena, Axioma Inc
Time to solve
last relaxation
Proj
0.05 - 0.32
0.06 - 0.89
0.16 - 1.18
0.13 - 0.86
0.53 - 1.55
0.48 - 1.1
0.56 - 2.02
0.77 - 1.51
0.82 - 2
0.67
93
Computational Results: Box QP (Projected)
% Duality Gap Closed
Instances
SDPLR
SDPA
Proj
spar20*
99.67 - 100 99.67 - 99.99 94.6 - 99.97
spar30*
97.81 - 100 97.81 - 99.99 89.87 - 99.99
spar40*
96.6 - 100
96.6 - 99.99 87.85 - 99.6
spar50*
95.55 - 100 95.55 - 99.99 87.88 - 97.53
spar60*
98.69 - 100 98.69 - 99.99 85.78 - 90.99
spar70*
98.46 - 100 98.46 - 99.99 89.78 - 99.36
spar80*
97.85 - 100 97.84 - 99.99 88.13 - 97.49
spar90* 97.83 - 99.99 97.83 - 99.99 89.44 - 96.6
spar100* 98.17 - 99.38 98.17 - 99.38 92.15 - 96.46
Average
99.40%
99.40%
95.19%
Time Taken (sec)
SDPLR
SDPA
Proj
0.97 - 56.37
1.98 - 3.39
2.48 - 408.35
3.57 - 243.3
16.66 - 29.33
12.33 - 565.88
10.3 - 515.73
105.68 - 157.83
35.77 - 134.8
41.72 - 926.15
438.77 - 589.17
50.21 - 180.95
88.05 - 532.45
1150.06 - 1408.32
121.83 - 226.1
133.07 - 3600.75
2769.98 - 3721.34
191.11 - 693.27
965.18 - 5413.02
6618.79 - 8285.12
257.61 - 892.95
2403.62 - 7049.49 12838.46 - 17048.98 408.73 - 991.04
5355.2 - 10295.88 23509.13 - 28604.12 538.02 - 1509.96
1741.20
5247.04
280.50
Anureet Saxena, Axioma Inc
Time to solve
last relaxation
Proj
0.05 - 0.32
0.06 - 0.89
0.16 - 1.18
0.13 - 0.86
0.53 - 1.55
0.48 - 1.1
0.56 - 2.02
0.77 - 1.51
0.82 - 2
0.67
94
Computational Results: Box QP (Projected)
% Duality Gap Closed
Instances
SDPLR
SDPA
Proj
spar20*
99.67 - 100 99.67 - 99.99 94.6 - 99.97
spar30*
97.81 - 100 97.81 - 99.99 89.87 - 99.99
spar40*
96.6 - 100
96.6 - 99.99 87.85 - 99.6
spar50*
95.55 - 100 95.55 - 99.99 87.88 - 97.53
spar60*
98.69 - 100 98.69 - 99.99 85.78 - 90.99
spar70*
98.46 - 100 98.46 - 99.99 89.78 - 99.36
spar80*
97.85 - 100 97.84 - 99.99 88.13 - 97.49
spar90* 97.83 - 99.99 97.83 - 99.99 89.44 - 96.6
spar100* 98.17 - 99.38 98.17 - 99.38 92.15 - 96.46
Average
99.40%
99.40%
95.19%
Time Taken (sec)
SDPLR
SDPA
Proj
0.97 - 56.37
1.98 - 3.39
2.48 - 408.35
3.57 - 243.3
16.66 - 29.33
12.33 - 565.88
10.3 - 515.73
105.68 - 157.83
35.77 - 134.8
41.72 - 926.15
438.77 - 589.17
50.21 - 180.95
88.05 - 532.45
1150.06 - 1408.32
121.83 - 226.1
133.07 - 3600.75
2769.98 - 3721.34
191.11 - 693.27
965.18 - 5413.02
6618.79 - 8285.12
257.61 - 892.95
2403.62 - 7049.49 12838.46 - 17048.98 408.73 - 991.04
5355.2 - 10295.88 23509.13 - 28604.12 538.02 - 1509.96
1741.20
5247.04
280.50
Anureet Saxena, Axioma Inc
Time to solve
last relaxation
Proj
0.05 - 0.32
0.06 - 0.89
0.16 - 1.18
0.13 - 0.86
0.53 - 1.55
0.48 - 1.1
0.56 - 2.02
0.77 - 1.51
0.82 - 2
0.67
95
Computational Results: Box QP (Projected)
% Duality Gap Closed
Instances
SDPLR
SDPA
Proj
spar20*
99.67 - 100 99.67 - 99.99 94.6 - 99.97
spar30*
97.81 - 100 97.81 - 99.99 89.87 - 99.99
spar40*
96.6 - 100
96.6 - 99.99 87.85 - 99.6
spar50*
95.55 - 100 95.55 - 99.99 87.88 - 97.53
spar60*
98.69 - 100 98.69 - 99.99 85.78 - 90.99
spar70*
98.46 - 100 98.46 - 99.99 89.78 - 99.36
spar80*
97.85 - 100 97.84 - 99.99 88.13 - 97.49
spar90* 97.83 - 99.99 97.83 - 99.99 89.44 - 96.6
spar100* 98.17 - 99.38 98.17 - 99.38 92.15 - 96.46
Average
99.40%
99.40%
95.19%
Instances
spar100-025-1
spar100-025-2
spar100-025-3
spar100-050-1
spar100-050-2
spar100-050-3
spar100-075-1
spar100-075-2
spar100-075-3
No. Variables
SDP
Proj
5151
203
5151
201
5151
201
5151
201
5151
201
5151
201
5151
201
5151
201
5151
199
Time Taken (sec)
SDPLR
SDPA
Proj
0.97 - 56.37
1.98 - 3.39
2.48 - 408.35
3.57 - 243.3
16.66 - 29.33
12.33 - 565.88
10.3 - 515.73
105.68 - 157.83
35.77 - 134.8
41.72 - 926.15
438.77 - 589.17
50.21 - 180.95
88.05 - 532.45
1150.06 - 1408.32
121.83 - 226.1
133.07 - 3600.75
2769.98 - 3721.34
191.11 - 693.27
965.18 - 5413.02
6618.79 - 8285.12
257.61 - 892.95
2403.62 - 7049.49 12838.46 - 17048.98 408.73 - 991.04
5355.2 - 10295.88 23509.13 - 28604.12 538.02 - 1509.96
1741.20
5247.04
280.50
No. Constraints
Linear
Convex (Non-Linear)
SDP
Proj
SDP
Proj
20201
156
1
119
20201
151
1
95
20201
150
1
114
20201
150
1
98
20201
150
1
113
20201
150
1
97
20201
150
1
131
20201
150
1
109
20201
147
1
90
Computing Time (sec)
SDP
Proj
5719.42
1.14
10185.65
1.52
5407.09
1.24
10139.57
1.07
5355.20
1.26
7281.26
0.82
9660.79
2.00
6576.10
1.23
10295.88
0.87
Time to solve
last relaxation
Proj
0.05 - 0.32
0.06 - 0.89
0.16 - 1.18
0.13 - 0.86
0.53 - 1.55
0.48 - 1.1
0.56 - 2.02
0.77 - 1.51
0.82 - 2
0.67
% Duality Gap Closed
SDP
Proj
98.93%
92.36%
99.09%
92.16%
99.33%
93.26%
98.17%
93.62%
98.57%
94.13%
99.39%
95.81%
99.19%
95.84%
99.18%
96.47%
99.19%
96.06%
Computational Results: Box QP (Projected)
% Duality Gap Closed
Instances
SDPLR
SDPA
Proj
spar20*
99.67 - 100 99.67 - 99.99 94.6 - 99.97
spar30*
97.81 - 100 97.81 - 99.99 89.87 - 99.99
spar40*
96.6 - 100
96.6 - 99.99 87.85 - 99.6
spar50*
95.55 - 100 95.55 - 99.99 87.88 - 97.53
spar60*
98.69 - 100 98.69 - 99.99 85.78 - 90.99
spar70*
98.46 - 100 98.46 - 99.99 89.78 - 99.36
spar80*
97.85 - 100 97.84 - 99.99 88.13 - 97.49
spar90* 97.83 - 99.99 97.83 - 99.99 89.44 - 96.6
spar100* 98.17 - 99.38 98.17 - 99.38 92.15 - 96.46
Average
99.40%
99.40%
95.19%
Instances
spar100-025-1
spar100-025-2
spar100-025-3
spar100-050-1
spar100-050-2
spar100-050-3
spar100-075-1
spar100-075-2
spar100-075-3
No. Variables
SDP
Proj
5151
203
5151
201
5151
201
5151
201
5151
201
5151
201
5151
201
5151
201
5151
199
Time Taken (sec)
SDPLR
SDPA
Proj
0.97 - 56.37
1.98 - 3.39
2.48 - 408.35
3.57 - 243.3
16.66 - 29.33
12.33 - 565.88
10.3 - 515.73
105.68 - 157.83
35.77 - 134.8
41.72 - 926.15
438.77 - 589.17
50.21 - 180.95
88.05 - 532.45
1150.06 - 1408.32
121.83 - 226.1
133.07 - 3600.75
2769.98 - 3721.34
191.11 - 693.27
965.18 - 5413.02
6618.79 - 8285.12
257.61 - 892.95
2403.62 - 7049.49 12838.46 - 17048.98 408.73 - 991.04
5355.2 - 10295.88 23509.13 - 28604.12 538.02 - 1509.96
1741.20
5247.04
280.50
Time to solve
last relaxation
Proj
0.05 - 0.32
0.06 - 0.89
0.16 - 1.18
0.13 - 0.86
0.53 - 1.55
0.48 - 1.1
0.56 - 2.02
0.77 - 1.51
0.82 - 2
0.67
No. Constraints
Linear
Convex (Non-Linear) Computing Time (sec)
% Duality Gap Closed
SDP
Proj
SDP
Proj
SDP
Proj
SDP
Proj
20201
156
1
119
5719.42
1.14
98.93%
92.36%
20201
151
1
95 Very
10185.65
1.52
99.09%
92.16%
little computational
20201
150
1
114 overheads
5407.09 at the1.24
99.33%
93.26%
nodes
20201
150
1
98
10139.57
1.07
98.17%
93.62%
of
the
enumeration
tree
20201
150
1
113
5355.20
1.26
98.57%
94.13%
20201
150
1
97
7281.26
0.82
99.39%
95.81%
20201
150
1
131
9660.79
2.00
99.19%
95.84%
20201
150
1
109
6576.10
1.23
99.18%
96.47%
20201
147
1
90
10295.88
0.87
99.19%
96.06%
Computational Results: Box QP (Projected)
% Duality Gap Closed
Time Taken (sec)
Instances
SDPLR
SDPA
Proj
SDPLR
SDPA
Proj
spar20*
99.67 - 100 99.67 - 99.99 94.6 - 99.97
0.97 - 56.37
1.98 - 3.39
2.48 - 408.35
spar30*
97.81 - 100 97.81 - 99.99 89.87 - 99.99
3.57 - 243.3
16.66 - 29.33
12.33 - 565.88
spar40*
96.6 - 100
96.6 - 99.99 87.85 - 99.6
10.3 - 515.73
105.68 - 157.83
35.77 - 134.8
spar50*
95.55 - 100 95.55 - 99.99 87.88 - 97.53
41.72 - 926.15
438.77 - 589.17
50.21 - 180.95
spar60*
98.69 - 100 98.69 - 99.99 85.78 - 90.99
88.05 - 532.45
1150.06 - 1408.32
121.83 - 226.1
spar70*
98.46 - 100 98.46 - 99.99 89.78 - 99.36 133.07 - 3600.75
2769.98 - 3721.34
191.11 - 693.27
spar80*
97.85 - 100 97.84 - 99.99 88.13 - 97.49 965.18 - 5413.02
6618.79 - 8285.12
257.61 - 892.95
spar90* 97.83 - 99.99 97.83 - 99.99 89.44 - 96.6 2403.62 - 7049.49 12838.46 - 17048.98 408.73 - 991.04
Strengthened relaxations produced by our code are
spar100* 98.17 - 99.38 98.17 - 99.38 92.15 - 96.46 5355.2 - 10295.88 23509.13 - 28604.12 538.02 - 1509.96
Average
99.40%
99.40%
95.19%
5247.04
almost as
strong as1741.20
the SDP+ RLT
relaxations280.50
and
Observation
Time to solve
last relaxation
Proj
0.05 - 0.32
0.06 - 0.89
0.16 - 1.18
0.13 - 0.86
0.53 - 1.55
0.48 - 1.1
0.56 - 2.02
0.77 - 1.51
0.82 - 2
0.67
can be solved in less than 2 sec; state of art SDP
solvers can take upto a couple of hours to solve these
relaxations in the extended space
Anureet Saxena, Axioma Inc
98
Research Question?
1978-1988
1988-1998
1998-2008
2008-2018
• Data Structures
• Theoretical Computer Science
• Linear Programming
• Mixed Integer Linear Programming
•?
Anureet Saxena, Axioma Inc
99
Research Question?
1978-1988
1988-1998
1998-2008
2008-2018
• Data Structures
• Theoretical Computer Science
• Linear Programming
• Mixed Integer Linear Programming
• Mixed Integer Non-Linear Programming
Anureet Saxena, Axioma Inc
100
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