Convex Relaxations of Non-Convex Mixed Integer Quadratically Constrained Problems
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Transcript of Convex Relaxations of Non-Convex Mixed Integer Quadratically Constrained Problems
Convex Relaxations of Non-Convex Mixed Integer Quadratically
Constrained Problems
Dr. Anureet SaxenaAssociate, Research
Axioma Inc.
(Joint Work with Pierre Bonami and Jon Lee)
Dedicated to Prof. Egon Balas
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MIQCP
min aT0xst
xTA ix + aTi x + bi · 0; i = 1 : : :mxj 2 Z; j 2 N I
l · x · u
Integer Constrained VariablesSymmetric Matrices
NOT necessarily positive semidefinite
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MIQCP
min aT0xst
A i:Y + aTi x + bi · 0; i = 1 : : :mxj 2 Z; j 2 N I
l · x · uY = xxT
yi j = xi xj
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Research Question?
Determine lower bounds on the optimal value of MIQCP by constructing strong convex relaxations of MIQCP.
Disjuncti
ve Programming
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Disjunctive Programming
P = f x j Ax ¸ bg
DisjunctionPolyhedral Relaxation
Separation Problem
Given x2P show that x2PD or find an inequality which is satisfied by all points in PD and is violated by x.
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Disjunctive Programming
Anureet Saxena, Axioma Inc.
CGLP
T heorem: x 2 PD if and only if the optimalvalue of the following cut generating linear pro-gram (CGLP ) is non-negative.
min ®x ¡ ¯s:t
®= utA + vtD t 8t = 1 : : :q
¯ · utb+ vtdt 8t = 1 : : :q
ut;vt ¸ 0 8t = 1 : : :q
P qt=1(u
t»+ vt»t) = 1
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Disjunctive Programming
P = f x j Ax ¸ bg
DisjunctionPolyhedral Relaxation
Outer Approximation of MIQCP defined by the incumbent solution
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Disjunctive Programming
P = f x j Ax ¸ bg
DisjunctionPolyhedral Relaxation
What are the sources of non-convexity in
MIQCP?
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Disjunctive Programming
P = f x j Ax ¸ bg
DisjunctionPolyhedral Relaxation
• xj2 Z j2 NI
• Elementary 0-1 disjunction
(xj · 0) OR (xj ¸ 1)
• Split Disjunctions
• GUB Disjunctions
Integrality Constraints
?
Y=xxT
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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
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Y=xxT
Ohh!!I don’t like fractional components. I can use them to get good cuts
MILP
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Y=xxT
MIQCP
Ohh!!I don’t like non-zero eigenvalues. I can use them to get good cuts
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Negative Eigenvalues of Y-xxT
If (Y ¡ xxT )c = ¸c where ¸ < 0 then
² (cTx)2 · Y:ccT is a convex quadratic cutwhich cuts o®(x; Y )
² equivalent to imposing the SDP conditionY ¡ xxT ¸ SDP 0 by SOCP cuts.
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Positive Eigenvalues of Y-xxT
If (Y ¡ xxT )c = ¸c where ¸ > 0 then
Y:ccT · (cTx)2
is a non-convex quadratic cut which cuts o®(x; Y ).
Univariate non-convex expression
Y:ccT · t2
t = cTx
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Positive Eigenvalues of Y-xxT
min(x;Y )2OA cTx max(x;Y )2OA cTx
cTx
Y:ccT · (cTx)2
(cTx)2
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Positive Eigenvalues of Y-xxT
cTx
p(cTx) + q
Y.ccT· p(cTx) + q
SecantApproximation
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Positive Eigenvalues of Y-xxT
cTx
p1(cTx) + q1 p2(cTx) + q2
µµL µU
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Positive Eigenvalues of Y-xxT
cTx
"µL (c) · cTx · µ
Y:ccT · p1(cTx) + q1
#W
"µ · cTx · µU(c)
Y:ccT · p2(cTx) + q2
#
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Cutting Plane Algorithm
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(x; Y )
(cTx)2 · Y:ccT Y:ccT · (cTx)2
Derive Disjunction
Derive DisjunctiveCut
CGLP
Convex Quadratic Cut
¸ > 0¸ < 0
Extract E igenvaluesand E igenvectors ofY ¡ xxT .
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Sequential ConvexificationT heorem Let c1; : : : ; cn denote a set of mutually-orthogonal unit vectors in Rn, and let
S0 =
8><
>:(x;Y )
¯¯¯¯¯¯¯
A i:Y + aTi x + bi · 0 i = 1 : : :ml · x · u
Y ¡ xxT ¸ SDP 0
9>=
>;
Sj = clconv³Sj ¡ 1 \
n(x;Y ) j Y:cj cTj · (cTj x)
2o´
f or j = 1 : : :n
Sn+j = clconv³Sn+j ¡ 1 \
n(x;Y ) j xj 2 f 0;1g
o´f or j = 1 : : :p
T he following statements hold true:
Sn = clconv
8><
>:(x;Y )
¯¯¯¯¯¯¯
A i:Y + aTi x + bi · 0 i = 1 : : :ml · x · u
Y ¡ xxT = 0
9>=
>;
Sn+p = clconv
8>>>><
>>>>:
(x;Y )
¯¯¯¯¯¯¯¯¯¯
A i:Y + aTi x + bi · 0 i = 1 : : :ml · x · u
Y ¡ xxT = 0xj 2 f 0;1g j = 1 : : :p
9>>>>=
>>>>;
Y.ccT · (cT x)2
Can we improve the disjunctive cuts by choosing c more
carefully?
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We are searching for vectors c which satisfy,
1. Y :ccT > (cT x)2
2. max(x;Y )2OAcTx ¡ min(x;Y )2OAc
Tx is assmall as possible
Improving Disjunctions?
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This condition is always satisfied if c belongs to vector space spanned by eigenvectors of Y-xxT associated with positive eigenvalues.
This can be calculated by solving a linear program whose right hand side is a linear function of c
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We are searching for vectors c which satisfy,
1. Y :ccT > (cT x)2
2. max(x;Y )2OAcTx ¡ min(x;Y )2OAc
Tx is assmall as possible
Improving Disjunctions?
Anureet Saxena, Axioma Inc.
This condition is always satisfied if c belongs to vector space spanned by eigenvectors of Y-xxT associated with positive eigenvalues.
This can be calculated by solving a linear program whose right hand side is a linear function of c
This problem can be formulated as a mixed integer linear program!!
Univariate Expression Generating Mixed Integer Program (UGMIP)
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Cutting Plane Algorithm
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Extract E igenvaluesand E igenvectors ofY ¡ xxT .
(x; Y )
(cTx)2 · Y:ccT Y:ccT · (cTx)2
Derive Disjunction
Derive DisjunctiveCut
CGLP
Convex Quadratic Cut
¸ > 0¸ < 0
UGMIP
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MIQCP Reformulations
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MIQCP (x)
MIQCP (x,Y)MIQCP (x,Y)RLT + SDP
Disjunctive Cuts
B & B
MIQCP (x)Projected Ineq
Lifting
Strengthening
Strengthening ?
Heavy Relaxation
Light Relaxation
Projection
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MIQCP Reformulations
Anureet Saxena, Axioma Inc.
MIQCP (x)
MIQCP (x,Y)MIQCP (x,Y)RLT + SDP
Disjunctive Cuts
B & B
MIQCP (x)Projected Ineq
Lifting
Strengthening
Strengthening ?
Heavy Relaxation
Light Relaxation
Projection
Convex Relaxations of Non-Convex Mixed Integer Quadratically Constrained Problems: Projected Formulations
A. Saxena, P. Bonami and J. Lee
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Projecting the RLT Formulation
Ak:Y + aTkx + bk · 0 k = 1 : : :m
Yi j ¡³lixj + uj xi ¡ liuj
´· 0 8i; j
Yi j ¡³uixj + lj xi ¡ ui lj
´· 0 8i; j
Yi j ¡³uixj + uj xi ¡ uiuj
´¸ 0 8i; j
Yi j ¡³lixj + lj xi ¡ li lj
´¸ 0 8i; j
RLT Inequalities
y¡ij (x) = max f uixj + uj xi ¡ uiuj ; lixj + lj xi ¡ li lj g 8i; j
y+ij (x) = min f lixj + uj xi ¡ liuj ;uixj + lj xi ¡ ui lj g 8i; j
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Projecting the RLT Formulation
Ak:Y + aTk x + bk · 0 k = 1 : : :m
y¡ij (x) · Yi j · y+ij (x) 8i; j
P(x;Y )
Qx =nx j 9Y s.t. (x;Y ) 2 P(x;Y )
o
Separation Problem
Given x show that x2Qx or find an inequality which is satisfied by all points in Qx and is violated by x.
Projecting the RLT Formulation
min ´
aTk x + bk · ¡ Ak:Y + ´ k = 1 : : :m
y¡ij (x) · Yi j · y+ij (x) 8i; j
ProjLP
T heorem x 2 Qx if and only if the optimalvalue of P rojLP is non-positive.
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Projecting the RLT Formulation
min ´
aTk x + bk · ¡ Ak:Y + ´ k = 1 : : :m
y¡ij (x) · Yi j · y+ij (x) 8i; j
Dual Solution(u, B, C)
X
i ;j
¡B i j y¡i j (x) ¡ Ci j y+i j (x)
¢+X
k2M
uk¡aTk x+bk
¢· 0
Projected Inequality
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Projecting the RLT Formulation
min ´
aTk x + bk · ¡ Ak:Y + ´ k = 1 : : :m
y¡ij (x) · Yi j · y+ij (x) 8i; j
• A linear programming separation algorithm
• Handles large number O(n2) of RLT inequalities as bound
constraints
• No of constraints = No of quadratic constraints in the original problemAnureet Saxena, Axioma Inc 30
Surrogate Constraints
xTA1x + aT1x + b1 · 0
xTAmx + aTmx + bm · 0
u1
um
xTAx + aTx + b· 0 Surrogate Constraint
X
i ;j
¡B i j y¡i j (x) ¡ Ci j y+i j (x)
¢+X
k2M
uk¡aTk x+bk
¢· 0
A = B – CB, C ¸ 0
Surrogate Constraint
y¡ij (x) · xixj · y+ij (x)
Can we extract the convex part of the surrogate constraint
Surrogate Constraints
xTA1x + aT1x + b1 · 0
xTAmx + aTmx + bm · 0
u1
um
xTAx + aTx + b· 0 Surrogate Constraint
A = B + C – DB ¸SDP 0C, D ¸ 0
xT Bx+X
i ;j
¡Ci j y¡i j (x) ¡ Di j y+i j (x)
¢+X
k2M
uk¡aTk x+bk
¢· 0
What happens if we add all such convex
quadratic cuts?
Projecting the SDP Formulationmin ´s.t.¡ Ak:Y +´ ¸ aTk x +bk; 8k 2 My¡i j (x) · Yi j · y+i j (x); 8i; j 2 NY +´I ¡ xxT ¸ SD P 0
xT Bx+X
i ;j
¡Ci j y¡i j (x) ¡ Di j y+i j (x)
¢+X
k2M
uk¡aTk x+bk
¢· 0
Dual Solution
(u, B, C, D)
T heorem x 2 Q+x if and only if the optimal
value of P rojSDP is non-positive.
ProjSDP
Separation Problem is a SDP
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Projecting the SDP Formulation
T heorem x 2 Q+x if and only if the optimal
value of the following piecewise linear convexoptimzation problem is non-positive.
maxf F (u;B ) j u 2 § M ; B ¸ SDP 0g;
where § M = fu jPk2M uk = 1; u ¸ 0g and
F (u;B ) =Pi;j
³ Pk2M ukA
kij ¡ B i j
´+ ³y¡ij (x) ¡ xi xj
´
+Pi;j
³ Pk2M ukA
kij ¡ B i j
´¡ ³y+ij (x) ¡ xi xj
´
+Pk2M uk(x
TAkx) +Pk2M uk
³aTk x + bk
´
Unconstrained Convex Optimization Problem over the Cartessian product of a simplex and cone of PSD
matrices
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Projecting the SDP Formulation
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xTAx + aTx + b· 0A = B + C ¡ DB ¸ SDP 0; C;D ¸ 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
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Limitations of Projection Theorems
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xTA1x + aT1x + b1 · 0
xTAmx + aTmx + bm · 0
u1
um
xTAx + aTx + b· 0 Surrogate Constraint
Once the surrogate constraint has been produced very little global information is used in the convexification process
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Limitations of Projection Theorems
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min x3s.t.x1x2 ¡ x1 ¡ x2 ¡ x3 · 0¡ 6x1+ 8x2 · 33x1 ¡ x2 · 30 · x1;x2 · 1:5
• st_e23 instances from GlobalLib
• OPT = -1.08
• RLT = -3
• SDP + RLT = -1.5
• x = ( 0.811, 0.689, -1.500)
P1 = clconv
(
x
¯¯¯¯¯x1x2 ¡ x1 ¡ x2 ¡ x3 · 0
0 · x1;x2 · 1:5
)
(1:5;0; ¡ 1:5) 2 P1 (0;1:5; ¡ 1:5) 2 P10:5407 (1:5;0; ¡ 1:5) + 0:4593 (0;1:5; ¡ 1:5) = x0:5407 + 0:4593 = 1
The non-convex quadratic constraint and the bound constraints cannot cut off x
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Limitations of Projection Theorems
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min x3s.t.x1x2 ¡ x1 ¡ x2 ¡ x3 · 0¡ 6x1+ 8x2 · 33x1 ¡ x2 · 30 · x1;x2 · 1:5
• st_e23 instances from GlobalLib
• OPT = -1.08
• RLT = -3
• SDP + RLT = -1.5
• x = ( 0.811, 0.689, -1.500)
P1 = clconv
(
x
¯¯¯¯¯x1x2 ¡ x1 ¡ x2 ¡ x3 · 0
0 · x1;x2 · 1:5
)
(1:5;0; ¡ 1:5) 2 P1 (0;1:5; ¡ 1:5) 2 P10:5407 (1:5;0; ¡ 1:5) + 0:4593 (0;1:5; ¡ 1:5) = x0:5407 + 0:4593 = 1
Global Information
We need a technique for engaging additional constraints in the
problem during the convexification process
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Limitations of Projection Theorems
Anureet Saxena, Axioma Inc.
x1x2 ¡ x1 ¡ x2 ¡ x3 · 0
12 (x1 + x2)
2 ¡ x1 ¡ x2 ¡ x3 ·12 (x1 ¡ x2)
2
Spectral Decomposition of"0 0:50:5 0
#
Univariate non-convex expression
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Limitations of Projection Theorems
min (x1 ¡ x2)s:t¡ 6x1+ 8x2 · 33x1 ¡ x2 · 30 · x1;x2 · 1:5
: : : · (x1 ¡ x2)2
(x1 ¡ x2)
(x1 ¡ x2)2
max (x1 ¡ x2)s:t¡ 6x1+ 8x2 · 33x1 ¡ x2 · 30 · x1;x2 · 1:5
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Limitations of Projection Theorems
SecantApproximation
(x1 ¡ x2)
(x1 ¡ x2)2 · 0:625(x1 ¡ x2) + 0:375
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Limitations of Projection Theorems
Anureet Saxena, Axioma Inc.
x1x2 ¡ x1 ¡ x2 ¡ x3 · 0
12 (x1 + x2)
2 ¡ x1 ¡ x2 ¡ x3 ·12 (x1 ¡ x2)
2
12 (x1+ x2)
2¡ x1¡ x2¡ x3 ·12 (0:625(x1 ¡ x2) + 0:375)
Spectral Decomposition
Secant ApproximationCuts off the incumbent
solution
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Eigen Reformulation
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xTAx + aTx + b· 0 A =Pj ¸ j vj v
Tj
P¸k>0 ¸k
³vTk x
´2+ aTx + b+
P¸k<0 ¸ksk · 0
yk = vTk x 8 k : ¸k < 0
sk = y2k 8 k : ¸k < 0
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Eigen Reformulation
Anureet Saxena, Axioma Inc.
xTAx + aTx + b· 0 A =Pj ¸ j vj v
Tj
P¸k>0 ¸k
³vTk x
´2+ aTx + b+
P¸k<0 ¸ksk · 0
yk = vTk x 8 k : ¸k < 0
sk = y2k 8 k : ¸k < 0
Directions of maximal non-convexity
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Eigen Reformulation
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min aT0xs.t.xTAkx + aTk x + bk · 0 ; 8k 2 Mxj 2 Z ; 8j 2 N1
P¸kj>0 ¸kj
³vTkj x
´2+ aTk x + bk +
P¸kj<0 ¸kj skj · 0 ; 8k 2 M
ykj = vTkj x ; 8 j : ¸kj < 0; k 2 Mskj = y2kj ; 8 j : ¸kj < 0; k 2 ML kj · ykj · Ukj ; 8 j : ¸kj < 0; k 2 M :
Geometric correlations along
directions of maximal non-convexity
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Eigen Reformulation
Anureet Saxena, Axioma Inc.
y1
y2
• st_glmp_kky instances from GlobalLib
• OPT = -2.5
• RLT = RLT+SDP = -3.0
x4x5+ x6x7 ¡ x3 ¡ z · 0y1 =
1p2x4 ¡
1p2x5
y2 =1p2x6 ¡
1p2x7
Projection along y1 and y2 Can we exploit these correlations in deriving strong cutting planes?
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Polarity Cuts
1. Projection
2. Determine Extreme Points
3. Lifting
4. Convexification
y1
s1 · y21
L U
s1 ¡ (L + U)y1 ¡ LU · 0
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Polarity Cuts
1. Projection
2. Determine Extreme Points
3. Lifting
4. Convexification
y1
Projection
min y1 max y1
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Polarity Cuts
1. Projection
2. Determine Extreme Points
3. Lifting
4. Convexification
y1L U
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Polarity Cuts
1. Projection
2. Determine Extreme Points
3. Lifting
4. Convexification
y1L U
(L ;L 2)
(U;U2)
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Polarity Cuts
1. Projection
2. Determine Extreme Points
3. Lifting
4. Convexification
y1L U
(L ;L 2)
Facet
min ®kyk + ¯ksk ¡ °s.t.®kL k + ¯k(L k)
2 ¡ ° ¸ 0®kUk + ¯k(Uk)
2 ¡ ° ¸ 0®k ¡ ®+k + ®¡k = 0®+k + ®¡k ¡ ¯k = 1®+k ¸ 0; ®¡k ¸ 0; ¯k · 0
(U;U2)
Polar Program
Polar Program
Convex Relaxation
Extreme pointsof the projected set
minP
k2S (®k yk +¯k sk) ¡ °s.t.P
k2S
³®kytk +¯k (ytk)
2´¡ ° ¸ 0;8t
¯k · 0; 8k 2 S®k ¡ ®+k +®¡k =0; 8k 2 SP
k2S
¡®+k +®¡k ¡ ¯k
¢= 1
®+k ¸ 0; ®¡k ¸ 0
Polarity
• Additional problem constraints induce geometric correlations along directions of maximal non-convexity
• Projection mechanism identifies such correlations
• Polarity uses these correlations to derive strong cutting planes for MIQCP
Anureet Saxena, Axioma Inc 52
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Eigen Reformulation
Anureet Saxena, Axioma Inc.
y1
y2
• st_glmp_kky instances from GlobalLib
• OPT = -2.5
• RLT = RLT+SDP = -3.0
x4x5+ x6x7 ¡ x3 ¡ z · 0y1 =
1p2x4 ¡
1p2x5
y2 =1p2x6 ¡
1p2x7
Projection along y1 and y2 Can we exploit these correlations in deriving strong cutting planes?
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Eigen Reformulation
Anureet Saxena, Axioma Inc.
y1
y2
• st_glmp_kky instances from GlobalLib
• OPT = -2.5
• RLT = RLT+SDP = -3.0
x4x5+ x6x7 ¡ x3 ¡ z · 0y1 =
1p2x4 ¡
1p2x5
y2 =1p2x6 ¡
1p2x7
Projection along y1 and y2 Can we exploit these correlations in deriving strong cutting planes?
Polarity cuts close 99.62% of the duality gap!!
Anureet Saxena, Axioma Inc. 55
Relaxations of MIQCP
RLT + SDPDisjunctive
CutsLow Width
Disjunctions
Projection RLT
Projection RLT + SDP
EigenReformulation
Polarity
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Computational Results
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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
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Computational Results
Anureet Saxena, Axioma Inc.
Experiment Setup
• 1 Hour Time limit on each instance
Duality Gap=opt(F inal Relaxation) ¡ RLT
Opt(M I QCP ) ¡ RLT£ 100
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GlobalLIB Instances
Anureet Saxena, Axioma Inc.
160 = 129 + 24 + 7
• All MIQCP Instances with upto 50 variables
• x1 x2 x3 x4 x5
• (x1+x2)/x3 ¸ 2x1
• x0.75
Numerical Problems
Zero Duality Gap
Non-Zero Duality Gap
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Computational Results (Extended Formulations)
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V1 SDP
V2 Disjunctive CutsV1
Y:ccT · (cTx)2
V3 UGMIPV2
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GlobalLIB Instances
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Summary of % Duality Gap Closed (129 Instances with non-zero Duality Gap)
V1 V2 V3>99.99 % 16 23 2398-99.99 % 1 44 5275-98 % 10 23 2125-75 % 11 22 200-25 % 91 17 13
Average Gap Closed 24.80% 76.49% 80.86%
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GlobalLIB Instances
Anureet Saxena, Axioma Inc.
Summary of % Duality Gap Closed (129 Instances with non-zero Duality Gap)
V1 V2 V3>99.99 % 16 23 2398-99.99 % 1 44 5275-98 % 10 23 2125-75 % 11 22 200-25 % 91 17 13
Average Gap Closed 24.80% 76.49% 80.86%
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Version (2,3) vs Version 1
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% Duality Gap ClosedInstance V1 V2 V3
st_qpc-m3a 0.00% 98.10% 99.16%st_ph13 0.00% 99.38% 98.80%st_ph11 0.00% 99.46% 98.19%ex3_1_4 0.00% 86.31% 99.57%st_jcbpaf2 0.00% 99.47% 99.61%st_ph12 0.00% 99.49% 99.62%ex2_1_9 0.00% 98.79% 99.73%prob05 0.00% 99.78% 99.49%st_glmp_kky 0.00% 99.80% 99.71%st_e24 0.00% 99.81% 99.81%st_ph15 0.00% 99.83% 99.81%st_bsj4 0.00% 99.86% 99.80%st_ph14 0.00% 99.85% 99.86%st_e08 0.00% 99.81% 99.89%st_ht 0.00% 99.81% 99.89%st_pan2 0.00% 68.54% 99.91%ex2_1_1 0.00% 72.62% 99.92%st_fp1 0.00% 72.62% 99.92%st_pan1 0.00% 99.72% 99.92%ex5_2_4 0.00% 79.31% 99.92%st_e02 0.00% 99.88% 99.95%st_kr 0.00% 99.93% 99.95%
% Duality Gap ClosedInstance V1 V2 V3
st_e33 0.00% 99.94% 99.95%st_z 0.00% 99.96% 99.95%st_qpc-m0 0.00% 99.96% 99.96%st_phex 0.00% 99.96% 99.96%st_e26 0.00% 99.96% 99.96%st_m1 0.00% 99.96% 99.96%ex2_1_6 0.00% 99.95% 99.97%st_fp6 0.00% 99.92% 99.97%st_e07 0.00% 99.97% 99.97%st_glmp_kk92 0.00% 99.98% 99.98%st_ph3 0.00% 99.98% 99.98%st_ph20 0.00% 99.98% 99.98%st_qpk1 0.00% 99.98% 99.98%st_bsj2 0.00% 99.98% 99.96%st_ph2 0.00% 99.98% 99.98%st_ph1 0.00% 99.98% 99.98%ex2_1_5 0.00% 99.98% 99.99%st_fp5 0.00% 99.98% 99.99%ex3_1_3 0.00% 99.99% 99.99%st_bpv2 0.00% 99.99% 99.99%st_qpc-m1 0.00% 99.99% 99.98%st_qpc-m3b 0.00% 100.00% 100.00%
63
Version (2,3) vs Version 1
Anureet Saxena, Axioma Inc.
% Duality Gap ClosedInstance V1 V2 V3
st_qpc-m3a 0.00% 98.10% 99.16%st_ph13 0.00% 99.38% 98.80%st_ph11 0.00% 99.46% 98.19%ex3_1_4 0.00% 86.31% 99.57%st_jcbpaf2 0.00% 99.47% 99.61%st_ph12 0.00% 99.49% 99.62%ex2_1_9 0.00% 98.79% 99.73%prob05 0.00% 99.78% 99.49%st_glmp_kky 0.00% 99.80% 99.71%st_e24 0.00% 99.81% 99.81%st_ph15 0.00% 99.83% 99.81%st_bsj4 0.00% 99.86% 99.80%st_ph14 0.00% 99.85% 99.86%st_e08 0.00% 99.81% 99.89%st_ht 0.00% 99.81% 99.89%st_pan2 0.00% 68.54% 99.91%ex2_1_1 0.00% 72.62% 99.92%st_fp1 0.00% 72.62% 99.92%st_pan1 0.00% 99.72% 99.92%ex5_2_4 0.00% 79.31% 99.92%st_e02 0.00% 99.88% 99.95%st_kr 0.00% 99.93% 99.95%
% Duality Gap ClosedInstance V1 V2 V3
st_e33 0.00% 99.94% 99.95%st_z 0.00% 99.96% 99.95%st_qpc-m0 0.00% 99.96% 99.96%st_phex 0.00% 99.96% 99.96%st_e26 0.00% 99.96% 99.96%st_m1 0.00% 99.96% 99.96%ex2_1_6 0.00% 99.95% 99.97%st_fp6 0.00% 99.92% 99.97%st_e07 0.00% 99.97% 99.97%st_glmp_kk92 0.00% 99.98% 99.98%st_ph3 0.00% 99.98% 99.98%st_ph20 0.00% 99.98% 99.98%st_qpk1 0.00% 99.98% 99.98%st_bsj2 0.00% 99.98% 99.96%st_ph2 0.00% 99.98% 99.98%st_ph1 0.00% 99.98% 99.98%ex2_1_5 0.00% 99.98% 99.99%st_fp5 0.00% 99.98% 99.99%ex3_1_3 0.00% 99.99% 99.99%st_bpv2 0.00% 99.99% 99.99%st_qpc-m1 0.00% 99.99% 99.98%st_qpc-m3b 0.00% 100.00% 100.00%
Observation
Either version 2 or version 3 closes >99% of the duality gap on 44 instances on which version 1 is unable to close any gap.
The relaxation obtained by adding disjunctive cuts can be substantially stronger than the SDP relaxation!!
66
Linear Complementarity Disjunctions
Anureet Saxena, Axioma Inc.
• Some problems have linear complementarity constraints
xi xj = 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
67
Linear Complementarity Disjunctions
Anureet Saxena, Axioma Inc.
Without Using LCD Using LCDInstance V2 V3 V2 V3
ex9_1_4 0.00% 1.55% 100.00% 99.97%ex9_2_1 60.04% 92.02% 99.95% 99.95%ex9_2_2 88.29% 98.06% 100.00% 100.00%ex9_2_3 0.00% 47.17% 99.99% 99.99%ex9_2_4 99.87% 99.89% 99.99% 100.00%ex9_2_6 87.93% 62.00% 80.22% 92.09%ex9_2_7 51.47% 86.25% 99.97% 99.95%
ObservationLinear Complementarity conditions can be exploited effectively within a disjunctive programming framework to derive strong cuts
73Anureet Saxena, Axioma Inc.
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
What is the effect of these disjunctive cuts on the spectrum of
Y-xxT ?
Anureet Saxena, Axioma Inc. 74
Spectrum of Y-xxT
% Duality Gap closed by
V1 V2 Instance Chosen < 10 % > 90 % st_jcbpaf2 > 40% < 60% ex9_2_7 < 10% < 10% ex7_3_1
Anureet Saxena, Axioma Inc. 75
Version 1, 0% Gap Closed
0 5 10 15 20 25 30 35-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
Sum of Positive Eigen Values of Y-xxT
Sum of NegativeEigen Values of Y-xxT
Anureet Saxena, Axioma Inc. 76
Version 2, 99.47% Gap Closed
0 50 100 150 200 250 300-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
Anureet Saxena, Axioma Inc. 77
Version 3, 99.61% Gap Closed
0 20 40 60 80 100 120-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
78
Computational Results (Projected Formulations)
Anureet Saxena, Axioma Inc.
W1 ProjLP
W2
Disjunctive Cuts
W1 PolarityCuts
All experiments were done using the eigen reformulation
Computational Results: GlobalLib (Projected)
Disjunctive Cuts Extended
ProjLP ProjLP + PolarLP ProjLP ProjLP +
PolarLP SDP + RLT SDP + RLT + Dsj
>99.99 % gap closed 19 23 19 23 16 2398-99.99 % gap closed 22 31 5 21 1 4475-98 % gap closed 35 33 17 18 10 2325-75 % gap closed 34 23 26 32 11 220-25 % gap closed 14 14 57 30 91 170-(-0.22) % gap closed 4 4 4 4 0 0Average Gap Closed 70.65% 76.06% 40.92% 60.48% 24.80% 76.49%Average Time taken (sec) 4.616 19.462 0.893 0.814 198.043 978.140
Anureet Saxena, Axioma Inc 79
Computational Results: GlobalLib (Projected)
Disjunctive Cuts Extended
ProjLP ProjLP + PolarLP ProjLP ProjLP +
PolarLP SDP + RLT SDP + RLT + Dsj
>99.99 % gap closed 19 23 19 23 16 2398-99.99 % gap closed 22 31 5 21 1 4475-98 % gap closed 35 33 17 18 10 2325-75 % gap closed 34 23 26 32 11 220-25 % gap closed 14 14 57 30 91 170-(-0.22) % gap closed 4 4 4 4 0 0Average Gap Closed 70.65% 76.06% 40.92% 60.48% 24.80% 76.49%Average Time taken (sec) 4.616 19.462 0.893 0.814 198.043 978.140
Anureet Saxena, Axioma Inc 80
Computational Results: GlobalLib (Projected)
Disjunctive Cuts Extended
ProjLP ProjLP + PolarLP ProjLP ProjLP +
PolarLP SDP + RLT SDP + RLT + Dsj
>99.99 % gap closed 19 23 19 23 16 2398-99.99 % gap closed 22 31 5 21 1 4475-98 % gap closed 35 33 17 18 10 2325-75 % gap closed 34 23 26 32 11 220-25 % gap closed 14 14 57 30 91 170-(-0.22) % gap closed 4 4 4 4 0 0Average Gap Closed 70.65% 76.06% 40.92% 60.48% 24.80% 76.49%Average Time taken (sec) 4.616 19.462 0.893 0.814 198.043 978.140
Anureet Saxena, Axioma Inc 81
Computational Results: GlobalLib (Projected)
Disjunctive Cuts Extended
ProjLP ProjLP + PolarLP ProjLP ProjLP +
PolarLP SDP + RLT SDP + RLT + Dsj
>99.99 % gap closed 19 23 19 23 16 2398-99.99 % gap closed 22 31 5 21 1 4475-98 % gap closed 35 33 17 18 10 2325-75 % gap closed 34 23 26 32 11 220-25 % gap closed 14 14 57 30 91 170-(-0.22) % gap closed 4 4 4 4 0 0Average Gap Closed 70.65% 76.06% 40.92% 60.48% 24.80% 76.49%Average Time taken (sec) 4.616 19.462 0.893 0.814 198.043 978.140
ObservationWe can generate relaxations in the space of x
variables which are almost as strong as those in the extended space even though our computing
times are 100 times smaller on average.
Anureet Saxena, Axioma Inc 82
Computational Results: GlobalLib (Projected)
Disjunctive Cuts Extended
ProjLP ProjLP + PolarLP ProjLP ProjLP +
PolarLP SDP + RLT SDP + RLT + Dsj
>99.99 % gap closed 19 23 19 23 16 2398-99.99 % gap closed 22 31 5 21 1 4475-98 % gap closed 35 33 17 18 10 2325-75 % gap closed 34 23 26 32 11 220-25 % gap closed 14 14 57 30 91 170-(-0.22) % gap closed 4 4 4 4 0 0Average Gap Closed 70.65% 76.06% 40.92% 60.48% 24.80% 76.49%Average Time taken (sec) 4.616 19.462 0.893 0.814 198.043 978.140
16%
30%
Anureet Saxena, Axioma Inc 83
Computational Results: GlobalLib (Projected)
Disjunctive Cuts Extended
ProjLP ProjLP + PolarLP ProjLP ProjLP +
PolarLP SDP + RLT SDP + RLT + Dsj
>99.99 % gap closed 19 23 19 23 16 2398-99.99 % gap closed 22 31 5 21 1 4475-98 % gap closed 35 33 17 18 10 2325-75 % gap closed 34 23 26 32 11 220-25 % gap closed 14 14 57 30 91 170-(-0.22) % gap closed 4 4 4 4 0 0Average Gap Closed 70.65% 76.06% 40.92% 60.48% 24.80% 76.49%Average Time taken (sec) 4.616 19.462 0.893 0.814 198.043 978.140
16%
30%
ObservationPolarity cuts can capture a portion of
strengthening derived from disjunctive cuts
Global Information at work!!
Anureet Saxena, Axioma Inc 84
Computational Results: Box QP (Projected)
W3 ProjLPProjected GradientHeuristic
W3-SDP ProjLP
All experiments were done using the eigen reformulation
Anureet Saxena, Axioma Inc 85
Computational Results: Box QP (Projected)
% Duality Gap Closed Time Taken (sec)Instances W3 W3-SDP W3 W3-SDP W3 (Adjusted)spar20* 94.60 - 99.97 91.54 - 99.91 2.49 - 408.36 0.84 - 2.46 0.51 - 1.60spar30* 89.87 - 99.99 51.41 - 98.79 12.33 - 565.88 1.74 - 14.38 3.32 - 14.49spar40* 87.85 - 99.60 21.78 - 89.63 35.77 - 134.8 4.16 - 65.28 13.75 - 49.76spar50* 87.88 - 97.53 11.38 - 50.15 50.22 - 180.96 8.76 - 99.13 28.95 - 76.19spar60* 85.78 - 90.99 0.00 - 0.00 121.83 - 226.11 111.07 - 127.47 86.28 - 141.77spar70* 89.78 - 99.36 0.00 - 53.67 191.12 - 693.28 22.02 - 202.98 92.63 - 143.35spar80* 88.13 - 97.49 2.94 - 56.23 257.62 - 892.96 34.77 - 67.66 121.62 - 230.53spar90* 89.44 - 96.60 5.73 - 50.13 408.73 - 991.04 46.98 - 95.66 184.63 - 294.92spar100* 92.15 - 96.46 8.17 - 51.79 538.03 - 1509.96 75.49 - 112.69 279.41 - 385.64Average 95.19% 50.01% 280.50 37.89 101.57
Anureet Saxena, Axioma Inc 86
Computational Results: Box QP (Projected)
% Duality Gap Closed Time Taken (sec)Instances W3 W3-SDP W3 W3-SDP W3 (Adjusted)spar20* 94.60 - 99.97 91.54 - 99.91 2.49 - 408.36 0.84 - 2.46 0.51 - 1.60spar30* 89.87 - 99.99 51.41 - 98.79 12.33 - 565.88 1.74 - 14.38 3.32 - 14.49spar40* 87.85 - 99.60 21.78 - 89.63 35.77 - 134.8 4.16 - 65.28 13.75 - 49.76spar50* 87.88 - 97.53 11.38 - 50.15 50.22 - 180.96 8.76 - 99.13 28.95 - 76.19spar60* 85.78 - 90.99 0.00 - 0.00 121.83 - 226.11 111.07 - 127.47 86.28 - 141.77spar70* 89.78 - 99.36 0.00 - 53.67 191.12 - 693.28 22.02 - 202.98 92.63 - 143.35spar80* 88.13 - 97.49 2.94 - 56.23 257.62 - 892.96 34.77 - 67.66 121.62 - 230.53spar90* 89.44 - 96.60 5.73 - 50.13 408.73 - 991.04 46.98 - 95.66 184.63 - 294.92spar100* 92.15 - 96.46 8.17 - 51.79 538.03 - 1509.96 75.49 - 112.69 279.41 - 385.64Average 95.19% 50.01% 280.50 37.89 101.57
Anureet Saxena, Axioma Inc 87
Computational Results: Box QP (Projected)
% Duality Gap Closed Time Taken (sec)Instances W3 W3-SDP W3 W3-SDP W3 (Adjusted)spar20* 94.60 - 99.97 91.54 - 99.91 2.49 - 408.36 0.84 - 2.46 0.51 - 1.60spar30* 89.87 - 99.99 51.41 - 98.79 12.33 - 565.88 1.74 - 14.38 3.32 - 14.49spar40* 87.85 - 99.60 21.78 - 89.63 35.77 - 134.8 4.16 - 65.28 13.75 - 49.76spar50* 87.88 - 97.53 11.38 - 50.15 50.22 - 180.96 8.76 - 99.13 28.95 - 76.19spar60* 85.78 - 90.99 0.00 - 0.00 121.83 - 226.11 111.07 - 127.47 86.28 - 141.77spar70* 89.78 - 99.36 0.00 - 53.67 191.12 - 693.28 22.02 - 202.98 92.63 - 143.35spar80* 88.13 - 97.49 2.94 - 56.23 257.62 - 892.96 34.77 - 67.66 121.62 - 230.53spar90* 89.44 - 96.60 5.73 - 50.13 408.73 - 991.04 46.98 - 95.66 184.63 - 294.92spar100* 92.15 - 96.46 8.17 - 51.79 538.03 - 1509.96 75.49 - 112.69 279.41 - 385.64Average 95.19% 50.01% 280.50 37.89 101.57
ObservationConvex quadratic cuts generated using the projected gradient heuristic can capture a
substantial portion of strengthening derived from the SDP+RLT relaxations
Anureet Saxena, Axioma Inc 88
Computational Results: Box QP (Projected)
% Duality Gap Closed Time Taken (sec)Instances W3 W3-SDP W3 W3-SDP W3 (Adjusted)spar20* 94.60 - 99.97 91.54 - 99.91 2.49 - 408.36 0.84 - 2.46 0.51 - 1.60spar30* 89.87 - 99.99 51.41 - 98.79 12.33 - 565.88 1.74 - 14.38 3.32 - 14.49spar40* 87.85 - 99.60 21.78 - 89.63 35.77 - 134.8 4.16 - 65.28 13.75 - 49.76spar50* 87.88 - 97.53 11.38 - 50.15 50.22 - 180.96 8.76 - 99.13 28.95 - 76.19spar60* 85.78 - 90.99 0.00 - 0.00 121.83 - 226.11 111.07 - 127.47 86.28 - 141.77spar70* 89.78 - 99.36 0.00 - 53.67 191.12 - 693.28 22.02 - 202.98 92.63 - 143.35spar80* 88.13 - 97.49 2.94 - 56.23 257.62 - 892.96 34.77 - 67.66 121.62 - 230.53spar90* 89.44 - 96.60 5.73 - 50.13 408.73 - 991.04 46.98 - 95.66 184.63 - 294.92spar100* 92.15 - 96.46 8.17 - 51.79 538.03 - 1509.96 75.49 - 112.69 279.41 - 385.64Average 95.19% 50.01% 280.50 37.89 101.57
ObservationConvex quadratic cuts generated using the projected gradient heuristic can capture a
substantial portion of strengthening derived from the SDP+RLT relaxationsJust using the eigen
reformulation with ProjLP closes 50% of the duality gap !!
Anureet Saxena, Axioma Inc 89
Computational Results: Box QP (Projected)
% Duality Gap Closed Time Taken (sec)Instances W3 W3-SDP W3 W3-SDP W3 (Adjusted)spar20* 94.60 - 99.97 91.54 - 99.91 2.49 - 408.36 0.84 - 2.46 0.51 - 1.60spar30* 89.87 - 99.99 51.41 - 98.79 12.33 - 565.88 1.74 - 14.38 3.32 - 14.49spar40* 87.85 - 99.60 21.78 - 89.63 35.77 - 134.8 4.16 - 65.28 13.75 - 49.76spar50* 87.88 - 97.53 11.38 - 50.15 50.22 - 180.96 8.76 - 99.13 28.95 - 76.19spar60* 85.78 - 90.99 0.00 - 0.00 121.83 - 226.11 111.07 - 127.47 86.28 - 141.77spar70* 89.78 - 99.36 0.00 - 53.67 191.12 - 693.28 22.02 - 202.98 92.63 - 143.35spar80* 88.13 - 97.49 2.94 - 56.23 257.62 - 892.96 34.77 - 67.66 121.62 - 230.53spar90* 89.44 - 96.60 5.73 - 50.13 408.73 - 991.04 46.98 - 95.66 184.63 - 294.92spar100* 92.15 - 96.46 8.17 - 51.79 538.03 - 1509.96 75.49 - 112.69 279.41 - 385.64Average 95.19% 50.01% 280.50 37.89 101.57
Anureet Saxena, Axioma Inc 90
Computational Results: Box QP (Projected)
% Duality Gap Closed Time Taken (sec)Instances W3 W3-SDP W3 W3-SDP W3 (Adjusted)spar20* 94.60 - 99.97 91.54 - 99.91 2.49 - 408.36 0.84 - 2.46 0.51 - 1.60spar30* 89.87 - 99.99 51.41 - 98.79 12.33 - 565.88 1.74 - 14.38 3.32 - 14.49spar40* 87.85 - 99.60 21.78 - 89.63 35.77 - 134.8 4.16 - 65.28 13.75 - 49.76spar50* 87.88 - 97.53 11.38 - 50.15 50.22 - 180.96 8.76 - 99.13 28.95 - 76.19spar60* 85.78 - 90.99 0.00 - 0.00 121.83 - 226.11 111.07 - 127.47 86.28 - 141.77spar70* 89.78 - 99.36 0.00 - 53.67 191.12 - 693.28 22.02 - 202.98 92.63 - 143.35spar80* 88.13 - 97.49 2.94 - 56.23 257.62 - 892.96 34.77 - 67.66 121.62 - 230.53spar90* 89.44 - 96.60 5.73 - 50.13 408.73 - 991.04 46.98 - 95.66 184.63 - 294.92spar100* 92.15 - 96.46 8.17 - 51.79 538.03 - 1509.96 75.49 - 112.69 279.41 - 385.64Average 95.19% 50.01% 280.50 37.89 101.57
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 91
Computational Results: Box QP (Projected)
% Time spent onCut Generation
Instances W3 W3-SDPspar20* 26.28 - 95.77 0.12 - 0.24spar30* 17.78 - 91.48 0.00 - 0.21spar40* 27.5 - 78.37 0.01 - 0.13spar50* 51.02 - 79.73 0.01 - 0.11spar60* 46.29 - 56.61 0.10 - 0.12spar70* 71.13 - 87.7 0.01 - 0.11spar80* 76.37 - 84.44 0.01 - 0.02spar90* 73.44 - 88.25 0.01 - 0.02spar100* 77.49 - 92.3 0.01 - 0.23Average 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 92
Computational Results: Box QP (Projected)Time to solve
% Duality Gap Closed Time Taken (sec) last relaxationInstances SDPLR SDPA Proj SDPLR SDPA Proj Projspar20* 99.67 - 100 99.67 - 99.99 94.6 - 99.97 0.97 - 56.37 1.98 - 3.39 2.48 - 408.35 0.05 - 0.32spar30* 97.81 - 100 97.81 - 99.99 89.87 - 99.99 3.57 - 243.3 16.66 - 29.33 12.33 - 565.88 0.06 - 0.89spar40* 96.6 - 100 96.6 - 99.99 87.85 - 99.6 10.3 - 515.73 105.68 - 157.83 35.77 - 134.8 0.16 - 1.18spar50* 95.55 - 100 95.55 - 99.99 87.88 - 97.53 41.72 - 926.15 438.77 - 589.17 50.21 - 180.95 0.13 - 0.86spar60* 98.69 - 100 98.69 - 99.99 85.78 - 90.99 88.05 - 532.45 1150.06 - 1408.32 121.83 - 226.1 0.53 - 1.55spar70* 98.46 - 100 98.46 - 99.99 89.78 - 99.36 133.07 - 3600.75 2769.98 - 3721.34 191.11 - 693.27 0.48 - 1.1spar80* 97.85 - 100 97.84 - 99.99 88.13 - 97.49 965.18 - 5413.02 6618.79 - 8285.12 257.61 - 892.95 0.56 - 2.02spar90* 97.83 - 99.99 97.83 - 99.99 89.44 - 96.6 2403.62 - 7049.49 12838.46 - 17048.98 408.73 - 991.04 0.77 - 1.51spar100* 98.17 - 99.38 98.17 - 99.38 92.15 - 96.46 5355.2 - 10295.88 23509.13 - 28604.12 538.02 - 1509.96 0.82 - 2Average 99.40% 99.40% 95.19% 1741.20 5247.04 280.50 0.67
Anureet Saxena, Axioma Inc 93
Computational Results: Box QP (Projected)Time to solve
% Duality Gap Closed Time Taken (sec) last relaxationInstances SDPLR SDPA Proj SDPLR SDPA Proj Projspar20* 99.67 - 100 99.67 - 99.99 94.6 - 99.97 0.97 - 56.37 1.98 - 3.39 2.48 - 408.35 0.05 - 0.32spar30* 97.81 - 100 97.81 - 99.99 89.87 - 99.99 3.57 - 243.3 16.66 - 29.33 12.33 - 565.88 0.06 - 0.89spar40* 96.6 - 100 96.6 - 99.99 87.85 - 99.6 10.3 - 515.73 105.68 - 157.83 35.77 - 134.8 0.16 - 1.18spar50* 95.55 - 100 95.55 - 99.99 87.88 - 97.53 41.72 - 926.15 438.77 - 589.17 50.21 - 180.95 0.13 - 0.86spar60* 98.69 - 100 98.69 - 99.99 85.78 - 90.99 88.05 - 532.45 1150.06 - 1408.32 121.83 - 226.1 0.53 - 1.55spar70* 98.46 - 100 98.46 - 99.99 89.78 - 99.36 133.07 - 3600.75 2769.98 - 3721.34 191.11 - 693.27 0.48 - 1.1spar80* 97.85 - 100 97.84 - 99.99 88.13 - 97.49 965.18 - 5413.02 6618.79 - 8285.12 257.61 - 892.95 0.56 - 2.02spar90* 97.83 - 99.99 97.83 - 99.99 89.44 - 96.6 2403.62 - 7049.49 12838.46 - 17048.98 408.73 - 991.04 0.77 - 1.51spar100* 98.17 - 99.38 98.17 - 99.38 92.15 - 96.46 5355.2 - 10295.88 23509.13 - 28604.12 538.02 - 1509.96 0.82 - 2Average 99.40% 99.40% 95.19% 1741.20 5247.04 280.50 0.67
Anureet Saxena, Axioma Inc 94
Computational Results: Box QP (Projected)Time to solve
% Duality Gap Closed Time Taken (sec) last relaxationInstances SDPLR SDPA Proj SDPLR SDPA Proj Projspar20* 99.67 - 100 99.67 - 99.99 94.6 - 99.97 0.97 - 56.37 1.98 - 3.39 2.48 - 408.35 0.05 - 0.32spar30* 97.81 - 100 97.81 - 99.99 89.87 - 99.99 3.57 - 243.3 16.66 - 29.33 12.33 - 565.88 0.06 - 0.89spar40* 96.6 - 100 96.6 - 99.99 87.85 - 99.6 10.3 - 515.73 105.68 - 157.83 35.77 - 134.8 0.16 - 1.18spar50* 95.55 - 100 95.55 - 99.99 87.88 - 97.53 41.72 - 926.15 438.77 - 589.17 50.21 - 180.95 0.13 - 0.86spar60* 98.69 - 100 98.69 - 99.99 85.78 - 90.99 88.05 - 532.45 1150.06 - 1408.32 121.83 - 226.1 0.53 - 1.55spar70* 98.46 - 100 98.46 - 99.99 89.78 - 99.36 133.07 - 3600.75 2769.98 - 3721.34 191.11 - 693.27 0.48 - 1.1spar80* 97.85 - 100 97.84 - 99.99 88.13 - 97.49 965.18 - 5413.02 6618.79 - 8285.12 257.61 - 892.95 0.56 - 2.02spar90* 97.83 - 99.99 97.83 - 99.99 89.44 - 96.6 2403.62 - 7049.49 12838.46 - 17048.98 408.73 - 991.04 0.77 - 1.51spar100* 98.17 - 99.38 98.17 - 99.38 92.15 - 96.46 5355.2 - 10295.88 23509.13 - 28604.12 538.02 - 1509.96 0.82 - 2Average 99.40% 99.40% 95.19% 1741.20 5247.04 280.50 0.67
Anureet Saxena, Axioma Inc 95
Computational Results: Box QP (Projected)Time to solve
% Duality Gap Closed Time Taken (sec) last relaxationInstances SDPLR SDPA Proj SDPLR SDPA Proj Projspar20* 99.67 - 100 99.67 - 99.99 94.6 - 99.97 0.97 - 56.37 1.98 - 3.39 2.48 - 408.35 0.05 - 0.32spar30* 97.81 - 100 97.81 - 99.99 89.87 - 99.99 3.57 - 243.3 16.66 - 29.33 12.33 - 565.88 0.06 - 0.89spar40* 96.6 - 100 96.6 - 99.99 87.85 - 99.6 10.3 - 515.73 105.68 - 157.83 35.77 - 134.8 0.16 - 1.18spar50* 95.55 - 100 95.55 - 99.99 87.88 - 97.53 41.72 - 926.15 438.77 - 589.17 50.21 - 180.95 0.13 - 0.86spar60* 98.69 - 100 98.69 - 99.99 85.78 - 90.99 88.05 - 532.45 1150.06 - 1408.32 121.83 - 226.1 0.53 - 1.55spar70* 98.46 - 100 98.46 - 99.99 89.78 - 99.36 133.07 - 3600.75 2769.98 - 3721.34 191.11 - 693.27 0.48 - 1.1spar80* 97.85 - 100 97.84 - 99.99 88.13 - 97.49 965.18 - 5413.02 6618.79 - 8285.12 257.61 - 892.95 0.56 - 2.02spar90* 97.83 - 99.99 97.83 - 99.99 89.44 - 96.6 2403.62 - 7049.49 12838.46 - 17048.98 408.73 - 991.04 0.77 - 1.51spar100* 98.17 - 99.38 98.17 - 99.38 92.15 - 96.46 5355.2 - 10295.88 23509.13 - 28604.12 538.02 - 1509.96 0.82 - 2Average 99.40% 99.40% 95.19% 1741.20 5247.04 280.50 0.67
Anureet Saxena, Axioma Inc 96
Computational Results: Box QP (Projected)Time to solve
% Duality Gap Closed Time Taken (sec) last relaxationInstances SDPLR SDPA Proj SDPLR SDPA Proj Projspar20* 99.67 - 100 99.67 - 99.99 94.6 - 99.97 0.97 - 56.37 1.98 - 3.39 2.48 - 408.35 0.05 - 0.32spar30* 97.81 - 100 97.81 - 99.99 89.87 - 99.99 3.57 - 243.3 16.66 - 29.33 12.33 - 565.88 0.06 - 0.89spar40* 96.6 - 100 96.6 - 99.99 87.85 - 99.6 10.3 - 515.73 105.68 - 157.83 35.77 - 134.8 0.16 - 1.18spar50* 95.55 - 100 95.55 - 99.99 87.88 - 97.53 41.72 - 926.15 438.77 - 589.17 50.21 - 180.95 0.13 - 0.86spar60* 98.69 - 100 98.69 - 99.99 85.78 - 90.99 88.05 - 532.45 1150.06 - 1408.32 121.83 - 226.1 0.53 - 1.55spar70* 98.46 - 100 98.46 - 99.99 89.78 - 99.36 133.07 - 3600.75 2769.98 - 3721.34 191.11 - 693.27 0.48 - 1.1spar80* 97.85 - 100 97.84 - 99.99 88.13 - 97.49 965.18 - 5413.02 6618.79 - 8285.12 257.61 - 892.95 0.56 - 2.02spar90* 97.83 - 99.99 97.83 - 99.99 89.44 - 96.6 2403.62 - 7049.49 12838.46 - 17048.98 408.73 - 991.04 0.77 - 1.51spar100* 98.17 - 99.38 98.17 - 99.38 92.15 - 96.46 5355.2 - 10295.88 23509.13 - 28604.12 538.02 - 1509.96 0.82 - 2Average 99.40% 99.40% 95.19% 1741.20 5247.04 280.50 0.67
No. ConstraintsNo. Variables Linear Convex (Non-Linear) Computing Time (sec) % Duality Gap Closed
Instances SDP Proj SDP Proj SDP Proj SDP Proj SDP Projspar100-025-1 5151 203 20201 156 1 119 5719.42 1.14 98.93% 92.36%spar100-025-2 5151 201 20201 151 1 95 10185.65 1.52 99.09% 92.16%spar100-025-3 5151 201 20201 150 1 114 5407.09 1.24 99.33% 93.26%spar100-050-1 5151 201 20201 150 1 98 10139.57 1.07 98.17% 93.62%spar100-050-2 5151 201 20201 150 1 113 5355.20 1.26 98.57% 94.13%spar100-050-3 5151 201 20201 150 1 97 7281.26 0.82 99.39% 95.81%spar100-075-1 5151 201 20201 150 1 131 9660.79 2.00 99.19% 95.84%spar100-075-2 5151 201 20201 150 1 109 6576.10 1.23 99.18% 96.47%spar100-075-3 5151 199 20201 147 1 90 10295.88 0.87 99.19% 96.06%
Computational Results: Box QP (Projected)Time to solve
% Duality Gap Closed Time Taken (sec) last relaxationInstances SDPLR SDPA Proj SDPLR SDPA Proj Projspar20* 99.67 - 100 99.67 - 99.99 94.6 - 99.97 0.97 - 56.37 1.98 - 3.39 2.48 - 408.35 0.05 - 0.32spar30* 97.81 - 100 97.81 - 99.99 89.87 - 99.99 3.57 - 243.3 16.66 - 29.33 12.33 - 565.88 0.06 - 0.89spar40* 96.6 - 100 96.6 - 99.99 87.85 - 99.6 10.3 - 515.73 105.68 - 157.83 35.77 - 134.8 0.16 - 1.18spar50* 95.55 - 100 95.55 - 99.99 87.88 - 97.53 41.72 - 926.15 438.77 - 589.17 50.21 - 180.95 0.13 - 0.86spar60* 98.69 - 100 98.69 - 99.99 85.78 - 90.99 88.05 - 532.45 1150.06 - 1408.32 121.83 - 226.1 0.53 - 1.55spar70* 98.46 - 100 98.46 - 99.99 89.78 - 99.36 133.07 - 3600.75 2769.98 - 3721.34 191.11 - 693.27 0.48 - 1.1spar80* 97.85 - 100 97.84 - 99.99 88.13 - 97.49 965.18 - 5413.02 6618.79 - 8285.12 257.61 - 892.95 0.56 - 2.02spar90* 97.83 - 99.99 97.83 - 99.99 89.44 - 96.6 2403.62 - 7049.49 12838.46 - 17048.98 408.73 - 991.04 0.77 - 1.51spar100* 98.17 - 99.38 98.17 - 99.38 92.15 - 96.46 5355.2 - 10295.88 23509.13 - 28604.12 538.02 - 1509.96 0.82 - 2Average 99.40% 99.40% 95.19% 1741.20 5247.04 280.50 0.67
No. ConstraintsNo. Variables Linear Convex (Non-Linear) Computing Time (sec) % Duality Gap Closed
Instances SDP Proj SDP Proj SDP Proj SDP Proj SDP Projspar100-025-1 5151 203 20201 156 1 119 5719.42 1.14 98.93% 92.36%spar100-025-2 5151 201 20201 151 1 95 10185.65 1.52 99.09% 92.16%spar100-025-3 5151 201 20201 150 1 114 5407.09 1.24 99.33% 93.26%spar100-050-1 5151 201 20201 150 1 98 10139.57 1.07 98.17% 93.62%spar100-050-2 5151 201 20201 150 1 113 5355.20 1.26 98.57% 94.13%spar100-050-3 5151 201 20201 150 1 97 7281.26 0.82 99.39% 95.81%spar100-075-1 5151 201 20201 150 1 131 9660.79 2.00 99.19% 95.84%spar100-075-2 5151 201 20201 150 1 109 6576.10 1.23 99.18% 96.47%spar100-075-3 5151 199 20201 147 1 90 10295.88 0.87 99.19% 96.06%
Very little computational overheads at the nodes of the enumeration tree
Computational Results: Box QP (Projected)Time to solve
% Duality Gap Closed Time Taken (sec) last relaxationInstances SDPLR SDPA Proj SDPLR SDPA Proj Projspar20* 99.67 - 100 99.67 - 99.99 94.6 - 99.97 0.97 - 56.37 1.98 - 3.39 2.48 - 408.35 0.05 - 0.32spar30* 97.81 - 100 97.81 - 99.99 89.87 - 99.99 3.57 - 243.3 16.66 - 29.33 12.33 - 565.88 0.06 - 0.89spar40* 96.6 - 100 96.6 - 99.99 87.85 - 99.6 10.3 - 515.73 105.68 - 157.83 35.77 - 134.8 0.16 - 1.18spar50* 95.55 - 100 95.55 - 99.99 87.88 - 97.53 41.72 - 926.15 438.77 - 589.17 50.21 - 180.95 0.13 - 0.86spar60* 98.69 - 100 98.69 - 99.99 85.78 - 90.99 88.05 - 532.45 1150.06 - 1408.32 121.83 - 226.1 0.53 - 1.55spar70* 98.46 - 100 98.46 - 99.99 89.78 - 99.36 133.07 - 3600.75 2769.98 - 3721.34 191.11 - 693.27 0.48 - 1.1spar80* 97.85 - 100 97.84 - 99.99 88.13 - 97.49 965.18 - 5413.02 6618.79 - 8285.12 257.61 - 892.95 0.56 - 2.02spar90* 97.83 - 99.99 97.83 - 99.99 89.44 - 96.6 2403.62 - 7049.49 12838.46 - 17048.98 408.73 - 991.04 0.77 - 1.51spar100* 98.17 - 99.38 98.17 - 99.38 92.15 - 96.46 5355.2 - 10295.88 23509.13 - 28604.12 538.02 - 1509.96 0.82 - 2Average 99.40% 99.40% 95.19% 1741.20 5247.04 280.50 0.67
ObservationStrengthened relaxations produced by our code are almost as strong as the SDP+ RLT relaxations and 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 99
Research Question?
1978-1988 •Data Structures•Theoretical Computer Science
1988-1998 •Linear Programming
1998-2008 •Mixed Integer Linear Programming
2008-2018 • ?Anureet Saxena, Axioma Inc 100
Research Question?
1978-1988 •Data Structures•Theoretical Computer Science
1988-1998 •Linear Programming
1998-2008 •Mixed Integer Linear Programming
2008-2018• Mixed Integer Non-Linear
Programming
Anureet Saxena, Axioma Inc 101
Go Global for Global Optimization