vsdp: a software package for verified semidefinite ...neum/glopt/gicolag/talks/...optimization,...

Semidefinite ProgrammingVerified Error Bounds

VSDPNumerical Results

VSDP: A software package for VerifiedSemidefinite Programming

Christian Jansson

Hamburg University of Technology

Christian Jansson VSDP: A software package for Verified Semidefinite Programming

Semidefinite Programming:

Many Thanks to the organizers

Arnold NeumaierImmanuel BomzeChristodoulos FloudasFrederic BenhamouLaurence WolseyIoannis Emiris (Athens, GR)

Semidefinite Programming: Sn

SDP: a nonsmooth, nonlinear, convex extension of LP, acts in thelinear space Sn of symmetric matrices.Sn is equipped with:

I partial ordering: U � V if V − U is psdI This partial ordering is defined by convex cone Sn

+; similar to≤ in IRn that is defined by the convex cone IRn

I inner product: 〈U,V 〉 = traceUTV =∑i ,j

UijVij

I similar to inner product between vectors (view matrix U asvector: U =⇒ (U11, . . . ,Un1,U12,U22, . . . ,U1n, . . . ,Unn)

UijVij

Semidefinite Programming: block-diagonal form of SDP

Replace in standard LP variables aij , cj , xj by symmetric matricesAij ,Cj ,Xj ∈ S sj , and · by 〈, 〉

Primalmin 〈C1,X1〉+ · · ·+ 〈Cn,Xn〉s.t. 〈A11,X1〉+ · · ·+ 〈A1nXn〉 = b1

· · ·〈Am1,X1〉+ · · ·+ 〈Amn,Xn〉 = bm

Xj � 0

Dualmax bT ys.t. y1A11 + · · ·+ ymAm1 + Z1 = C1

· · ·y1A1n + · · ·+ ymAmn + Zn = Cn

Zj � 0

I The dual constraints are equivalent to

y1A1j + · · ·+ ymAmj � Cj for j = 1, . . . ,m

‘’Linear matrix inequalities (LMI)”

I Weak duality holds. Strong duality complicated.

· · ·〈Am1,X1〉+ · · ·+ 〈Amn,Xn〉 = bm

Xj � 0

· · ·y1A1n + · · ·+ ymAmn + Zn = Cn

Zj � 0

· · ·〈Am1,X1〉+ · · ·+ 〈Amn,Xn〉 = bm

Xj � 0

· · ·y1A1n + · · ·+ ymAmn + Zn = Cn

Zj � 0

· · ·〈Am1,X1〉+ · · ·+ 〈Amn,Xn〉 = bm

Xj � 0

· · ·y1A1n + · · ·+ ymAmn + Zn = Cn

Zj � 0

Semidefinite Programming: PSD-Cone (nonsmooth)

(x yy z

)∈ S2

x ≥ 0, z ≥ 0, xz ≥ y2

Applications: control theory, circuit design, combinatorialoptimization, robust optimization, signal processing, algebraicgeometry, quantum chemistry, atomic physics ...

Electronic Structure Calculations: Schrodinger’s equation

Electrons do not move around the nucleusin circular orbits. At any moment electronsmay

(i) exist at any arbitrary point

(ii) but more frequently in certain regions,described by probability amplitudes,wave functions Ψ(r , t)

Electronic distribution inNH3 molecule (Ammonia)

with unitary time evaluation

∂tΨ = HΨ, Schrodinger’s Equation,

H Hamiltonian (operator corresponding to energy)

Electronic Structure Calculations: interpretation,ground-stage-energy

(Born:) Interpretation of the N-particle probability amplitude Ψ:

|Ψ(r1, . . . , rN , t)|2·N∏

|drj | =

The probability for finding theN particles in the 3N dimensional

volumeN∏

j=1|drj | surrounding

the point (r1, . . . , rN).

Challenging, large-scale problem in atomic physics:ground-stage-energy of molecules; that is the smallest eigenvalueof Hamiltonian H

Electronic Structure Calculations: interpretation,ground-stage-energy

(Born:) Interpretation of the N-particle probability amplitude Ψ:

|Ψ(r1, . . . , rN , t)|2·N∏

|drj | =

The probability for finding theN particles in the 3N dimensional

volumeN∏

j=1|drj | surrounding

the point (r1, . . . , rN).

Challenging, large-scale problem in atomic physics:ground-stage-energy of molecules; that is the smallest eigenvalueof Hamiltonian H

Electronic Structure Calculations: HamiltonianN-electron system 1, . . . ,NEach single electron has r basis states i = 1, . . . , r (obtained bydiscretization of space coordinates, spin).It turns out (second quantization representation, Fock space) thatthe Hamiltonian (N-electron system involving one-body andtwo-body interactions) is:

H =∑i ,i ′

h1(i , i′)a+

i ai ′ +1

∑i ,j ;i ′,j ′

h2(i , j , i′, j ′)a+

i a+j aj ′ai ′

h1, h2 matrices derived from one-body and two-body interac-tions

a+i creation operator (creates particle in state |i〉)

ai annihilation operator (annihilates a particle in state |i〉)H linear combination of creation and annihilation operators

weighted by matrix elements depending on interactionsChristian Jansson VSDP: A software package for Verified Semidefinite Programming

H =∑i ,i ′

h1(i , i′)a+

i ai ′ +1

∑i ,j ;i ′,j ′

h2(i , j , i′, j ′)a+

i a+j aj ′ai ′

H =∑i ,i ′

h1(i , i′)a+

i ai ′ +1

∑i ,j ;i ′,j ′

h2(i , j , i′, j ′)a+

i a+j aj ′ai ′

H =∑i ,i ′

h1(i , i′)a+

i ai ′ +1

∑i ,j ;i ′,j ′

h2(i , j , i′, j ′)a+

i a+j aj ′ai ′

H =∑i ,i ′

h1(i , i′)a+

i ai ′ +1

∑i ,j ;i ′,j ′

h2(i , j , i′, j ′)a+

i a+j aj ′ai ′

Electronic Structure Calculations: ground state energy E

Few calculations yield the SDP:

Minimize E = trace(h1Γ1) +1

2trace(h2Γ2)

s.t. Γ1(i , i′) = (Ψ, a+

i ′ aiΨ) ∈ S r

Γ2(i , j ; i′, j ′) = (Ψ, a+

i ′ a+j ′ ajaiΨ) ∈ S r(r−1)/2

Ψ ground stage probability amplitude

Ψ unknown ⇒ RDM-matrices Γ1, Γ2 unknown, and must satisfyadditional linear equations and convex inequality conditions,derived from properties of creation and annihilation operators.(N-representability conditions)

2trace(h2Γ2)

s.t. Γ1(i , i′) = (Ψ, a+

i ′ aiΨ) ∈ S r

Γ2(i , j ; i′, j ′) = (Ψ, a+

i ′ a+j ′ ajaiΨ) ∈ S r(r−1)/2

2trace(h2Γ2)

s.t. Γ1(i , i′) = (Ψ, a+

i ′ aiΨ) ∈ S r

Γ2(i , j ; i′, j ′) = (Ψ, a+

i ′ a+j ′ ajaiΨ) ∈ S r(r−1)/2

Electronic Structure Calculations: N-representabilityconditions

I I � Γ1 � 0, λmax(Γ1) ≤ 1 (boundedness qualification)

I Γ2 � 0

I∑i ,j

Γ2(i , j ; i , j) = N(N − 1), λmax(Γ2) ≤ N(N − 1)

I G (i , j ; i ′, j ′) = 〈Ψ|a+j aia

+i ′ aj ′ |Ψ〉

= Γ2(i , j′; j , i ′) + δ(i , i ′)Γ1(j

′, j)G � 0 (G -condition), further P,Q,T -conditions from spinsymmetries, antisymmetry of fermions, ...

The problem can be converted into a SDP in block-diagonal form(existence of optimal solutions follows)

I Γ2 � 0

I∑i ,j

Γ2(i , j ; i , j) = N(N − 1), λmax(Γ2) ≤ N(N − 1)

I G (i , j ; i ′, j ′) = 〈Ψ|a+j aia

+i ′ aj ′ |Ψ〉

= Γ2(i , j′; j , i ′) + δ(i , i ′)Γ1(j

I Γ2 � 0

I∑i ,j

Γ2(i , j ; i , j) = N(N − 1), λmax(Γ2) ≤ N(N − 1)

I G (i , j ; i ′, j ′) = 〈Ψ|a+j aia

+i ′ aj ′ |Ψ〉

= Γ2(i , j′; j , i ′) + δ(i , i ′)Γ1(j

I Γ2 � 0

I∑i ,j

Γ2(i , j ; i , j) = N(N − 1), λmax(Γ2) ≤ N(N − 1)

I G (i , j ; i ′, j ′) = 〈Ψ|a+j aia

+i ′ aj ′ |Ψ〉

= Γ2(i , j′; j , i ′) + δ(i , i ′)Γ1(j

I Γ2 � 0

I∑i ,j

Γ2(i , j ; i , j) = N(N − 1), λmax(Γ2) ≤ N(N − 1)

I G (i , j ; i ′, j ′) = 〈Ψ|a+j aia

+i ′ aj ′ |Ψ〉

= Γ2(i , j′; j , i ′) + δ(i , i ′)Γ1(j

Electronic Structure Calculations: references

The concept of N-representability conditions was first stated byColeman (1987)...Fukuda, Braams, Nakata, Overton, Percus, Yamashita, Zhao(Math. Progr. to appear): Improved SDP form , SDPARA (aparallel version of SDPA installed on IBM RS 6000 SP, 16processors), ground state energy for 47 moleculesImproved SDP form is ill-conditioned (to be more efficient, someequations were written as inequalities)

Electronic Structure Calculations: results (1)

Molecule problems can be downloaded (SDPA format):http://www.is.titech.ac.jp/ mituhiro/

VSDP could not verify the extremely large-scale molecule problems(m > 7000 and more than 2.5 million variables, reason: SDPA andSDPT3 were stopped after 3 days without computing anapproximation.)

First example:

NH−: N = 9 electrons, r = 12, m = 948size of block matrices: 6, 6, 6, 6, 15, 15, 36, 15, 15, 36, 72, 36,36, 20, 90, 90, 20, 306, 306, 90, 90ground-stage energy E: 58.054 eV

First example:

Using SDPA as approximate solver:approximate relative accuracy of E: 5.7944e − 003guaranteed relative accuracy: 5.8028e − 003times: t = 418, t = 89, t = 432

Using SDPT3 as approximate solver:approximate relative accuracy: 1.6310e − 006guaranteed relative accuracy: 3.0067e − 005times: t = 496, t = 1139, t = 6

Using SDPA as approximate solver:approximate relative accuracy of E: 5.7944e − 003guaranteed relative accuracy: 5.8028e − 003times: t = 418, t = 89, t = 432

Using SDPT3 as approximate solver:approximate relative accuracy: 1.6310e − 006guaranteed relative accuracy: 3.0067e − 005times: t = 496, t = 1139, t = 6

Electronic Structure Calculations: short summary

I Accuracy and computational effort of VSDP depends stronglyon the quality of the approximations.

I Fukuda et al (to appear) write: The SDP problem must besolved to high accuracy - typically 7 digits for the optimalvalue-

I Ill-conditioned problems due to modelling

I boundedness qualifications for the matrices Xj

I Nakata et al (J.Chem.Phys. 2001) report numericalinaccuracies

Electronic Structure Calculations: Numerical inaccuracies

Nakata, Nakatsuji, Masahiro, Fukuda, Fujisawa (J.of Chem.Physics, 2001) report for a slightly altered SDP with fewerrepresentability conditions:

As the number of constraints increases drastically in theP+Q+G calculations, the numerical accuracy becomesmuch worse in the DM(P+Q+G) results. The worst fiveare HF, OH-, LiH, CH-, and NH-. We notice that theyhave the DM(P+Q+G) energies higher than the full-CIones, though these values must be lower than the full-CIvalues.

Notice, this cannot happen with verified bounds.

Electronic Structure Calculations: results

Second example:

Ammonia NH3: N = 10 electrons, r = 16, m = 2964size of block matrices: 8, 8, 8, 8, 28, 28, 64, 28, 28, 64, 128, 64,64, 56, 224, 224, 56, 736, 736, 224, 224ground-stage energy: 67.92487 eV

Using SDPA as approximate solver:approximate relative accuracy: 4.9906e − 009guaranteed relative accuracy: 1.2502e − 007times: t = 6.59h, t = 0.54h, t = 6.67h

D.Chaykin implements a C++ code for verified results usingSDPARA for solving the problems with m > 5000 and n > 1500

Using SDPA as approximate solver:approximate relative accuracy: 4.9906e − 009guaranteed relative accuracy: 1.2502e − 007times: t = 6.59h, t = 0.54h, t = 6.67h

D.Chaykin implements a C++ code for verified results usingSDPARA for solving the problems with m > 5000 and n > 1500

Verified Error Bounds: floating-point arithmetic

I Floating-point arithmetic, may cause erroneousapproximations, especially for ill-conditioned or ill-posedproblems.

I Neumaier and Shcherbina (2004) presented aninnocent-looking integer problem where the high qualitycommercial codesCPLEX, BONSAIG, GLPK, XPRESS, XPRESS-MP,MINLP failed due to roundoff. Only Fort MP solved theproblem correctly.Reason: ill-conditioned relaxations must be solved

Verified Error Bounds: Ill-posed Problems?

How frequently are ill-posed problems?Fritz John (Translation Editor’s Preface in Tikhonov, Arsenin:Solutions of Ill-posed Problems, 1977):

One might say that the majority of applied problems are,and always have been, ill-posed, particularly when theyrequire numerical answers.

Ordonez and Freund (2003):71% (real life!) problems of NETLIB LP collection

are ill-posedFreund, Ordonez and Toh (2006):32 problems out of 80 problems of the SDPLIB

are ill-posed

Verified Error Bounds: SDP lower bound

How can we obtain rigorous results?Theoremy ∈ IRm approximate Lagrange parameter,

Dj := Cj −m∑

i=1yiAij , d j ≤ λmin(Dj), lj ≥ number of negative

eigenvalues of Dj . (d j measures violations)

λmax(Xj) ≤ x j ∈ IR+ ∪ {+∞} for optimal (Xj) (primalboundedness qualification). ⇒

f ∗p ≥ bT y +n∑

j=1ljd

−j x j =: f ∗p , where d−

j = min(0, d j)

Dj := Cj −m∑

j=1ljd

j = min(0, d j)

Dj := Cj −m∑

j=1ljd

j = min(0, d j)

Dj := Cj −m∑

j=1ljd

j = min(0, d j)

Dj := Cj −m∑

j=1ljd

j = min(0, d j)

Verified Error Bounds: SDP upper bound

Theorem(Xj) primal approximation, at most kj negative eigenvalues.

ri ≥ |bi −n∑

j=1〈Aij , Xj〉|, i = 1, . . . ,m,

λj ≤ λmin(Xj), j = 1, . . . , n,

%j ≥ sup{λmax(Cj −m∑

i=1yiAij) : −y ≤ y ≤ y},

where y > 0 upper bound of dual optimal solution. Then

f ∗d ≤n∑

j=1〈Cj , Xj〉 −

n∑j=1

kjλ−j %j +

m∑j=1

riy i =: f∗d

I Linear System must be solved only approximately, not rigorous

Semidefinite Programming: Algorithm

I Rigorous postprocessing, does not require error bounds fromthe solver, only the approximations are used

I Quality of rigorous bounds depends mainly on the accuracy ofthe solver

I Safety: comparing rigorous lower and upper bounds one canknow whether the numerical results are reliable

I Overestimation for computing d j is small

VSDP: Verified Semi-Definite Programming

MATLAB software package using INTLAB (Rump). Main features:

I verified lower and upper bounds of the optimal value,

I proves existence of feasible solutions, also for LMI’s,

I provides rigorous certificates of infeasibility,

I facilitates to solve approximately the problem by usingdifferent well-known semidefinite programming solvers(SDPT3, SDPA),

I several applications,

I can handle several formats,

I allows the use of interval data .

VSDP: Input with cell-arrays

>> DELTA = 1e-4;

>> C{1} = [ 0 1/2 0; 1/2 DELTA 0; 0 0 DELTA ];>> A{1,1} = [ 0 -1/2 0; -1/2 0 0; 0 0 0];>> A{2,1} = [ 1 0 0 ; 0 0 0; 0 0 0];>> A{3,1} = [ 0 0 1 ; 0 0 0; 1 0 0];>> A{4,1} = [ 0 0 0 ; 0 0 1; 0 1 0];

>> b = [1; 2*DELTA; 0; 0];>> blk{1,1} = ’s’; blk{1,2} = 3;

DELTA > 0, f ∗p = −1/2, well-posedDELTA = 0, f ∗p = +∞, f ∗d = −1, ill-posed, duality gap

VSDP: Approximate solvers SDPT3 and SDPA

>> [objt,Xt,yt,Zt,info] = mysdps(blk,A,C,b);

Output: approximations of

I primal and dual optimal value both stored in objt,

I primal and dual solutions Xt, yt, Zt,

I information about termination and performance stored ininfo.

>> objt, termination = info(1),objt =

-5.000322560803474e-001 -5.000000062262615e-001termination =

VSDP: Verified lower bound

>> [fL, Y, dl] = vsdplow(blk,A,C,b,Xt,yt,Zt)

fL, Y, dl are lower bound, certificate of dual feasibility,eigenvalues of defect, respectively.

fL =-5.000000062262615e-001

Y =-9.016817444180712e-005-2.499549190259066e+003

dl =4.319659051028185e-013

VSDP: Verified lower bound

>> [fL, Y, dl] = vsdplow(blk,A,C,b,Xt,yt,Zt)

fL, Y, dl are lower bound, certificate of dual feasibility,eigenvalues of defect, respectively.

fL =-5.000000062262615e-001

Y =-9.016817444180712e-005-2.499549190259066e+003

dl =4.319659051028185e-013

VSDP: Verified upper bound

>> [fU, X, lb] = vsdpup(blk,A,C,b,Xt,yt,Zt);

fU, X, lb are upper bound, certificate of primal feasibility,eigenvalues of certificate, respectively.

>> fU, Xout = X{1}; Xout(:,1), lb

fU =-4.999677693282600e-001

intval ans =< 2.000000000000000e-004, 2.710505431213762e-020>< -1.000000000000000e+000, 0.000000000000000e+000>< 0.000000000000000e+000, 0.000000000000000e+000>

lb =1.289076539560735e-008

VSDP: Verified upper bound

>> [fU, X, lb] = vsdpup(blk,A,C,b,Xt,yt,Zt);

fU, X, lb are upper bound, certificate of primal feasibility,eigenvalues of certificate, respectively.

>> fU, Xout = X{1}; Xout(:,1), lb

fU =-4.999677693282600e-001

intval ans =< 2.000000000000000e-004, 2.710505431213762e-020>< -1.000000000000000e+000, 0.000000000000000e+000>< 0.000000000000000e+000, 0.000000000000000e+000>

lb =1.289076539560735e-008

VSDP: Summary

Summarizing, we have verified strong duality for the above problemby using the SDPT3 approximations. Moreover, the inequality

−5.000000062e−001 = f ∗p ≤ f ∗d = f ∗p ≤ f∗d = −4.999677693e−001,

is fulfilled, and certificates of strictly primal and strictly dualfeasible solutions are computed.

Numerical Results of VSDP for SDPLIB

I 92 problems out of control and system theory, truss topologydesign, graph partitioning, max cut, ...

I removed 2 very large problems, where SDPT3 and SDPA outof memory

I largest rigorously solved problems: theta G51(6910, 1001),max G32(2000, 2000)

Using SDPT3:

I median of guaranteed relative accuracy: 7.0 · 10−7

I median (t/t) = 0.085

Using SDPT3:

I VSDP has computed f∗d = +∞ for 32 ill-posed problems,

reflecting that distance to primal infeasibility is zero.

I For all other problems finite bounds are computed, and strongduality is proved, with exception of hinf2

I SDPT3 gave 7 warnings , 3 warnings for well-posed problems,thus not reflecting the difficulty of the problems.

Numerical Results of VSDP for SDPLIB with boundednessSolution of ill-posed problems requires additional assumptions:

I existence of solution within reasonable bounds

Kahan:

“Nobody can know even roughly how wrong acomputation is without knowing at least roughly whatwould have been right.”

x j = µ · λmax(Xj) for j = 1, . . . , n,and y i = µ · |yi | for i = 1, . . . ,m,

µ = 10 conservative factor,

I median of guaranteed relative accuracy: 2.52 10−6

Kahan:

Combinatorial Optimization: Lifting

Tight semidefinite relaxations are known: Max-Cut, GraphPartitioning, Coloring, ...There x ∈ {−1, 1}n. Lifting: X = xxT ⇒

X � 0, diag(X ) = e, rank(X ) = 1

Theorem (Laurent, Poljak 1995)

If ... ⇒ −1 ≤ Xij ≤ 1, and if Xij ∈ {−1, 1} then X = xxT ,x ∈ {−1, 1}n.

Hence, ... implies λmax(X ) ≤ n =: x (PBQ) .

X � 0, diag(X ) = e, rank(X ) = 1

Graph Partitioning: algebraic formulation

I Partition graph G (n nodes, aij

weights) into K sets of equalcardinality which minimizes the sum ofweights of the joining edges (HereK = 2)

I VLSI placement and routing, FEM, ...

I NP-hard (Branch-Bound-and-Cut)

x ∈ {−1, 1}n partitioning, 1− xixj = 0 iff i , j in same set

f ∗p = min∑i<j

aij1− xixj

2st. x ∈ {−1, 1}n,

n∑j=1

xj = 0

(parity condition)Quadratic integer problem

Graph Partitioning: algebraic formulation

I Partition graph G (n nodes, aij

weights) into K sets of equalcardinality which minimizes the sum ofweights of the joining edges (HereK = 2)

I VLSI placement and routing, FEM, ...

I NP-hard (Branch-Bound-and-Cut)

x ∈ {−1, 1}n partitioning, 1− xixj = 0 iff i , j in same set

f ∗p = min∑i<j

aij1− xixj

2st. x ∈ {−1, 1}n,

n∑j=1

xj = 0

(parity condition)Quadratic integer problem

Graph Partitioning: ill-posed algebraic formulation

Gruber, Rendl 2002: Laplace matrix L := Diag(Ae)− A,

X := xxT , 〈L,X 〉 := traceLTX

f ∗p = min1

4〈L,X 〉 s.t. diag(X ) = e, eTXe = 0,X � 0, rank(X ) = 1

I X � 0, eTXe = 0 ⇒ X singular, Slater’s ConstraintQualification not fulfilled, hence ill-posed: arbitrarily smallperturbations yield infeasibility

I Drop nonlinear constraint rank(X ) = 1: ⇒convex ill-posedsemidefinite relaxation

f ∗p = min1

Graph Partitioning: rigorous lower boundTight rigorous lower bounds by using Theorem (PBQ):

Corollary

y approximation, D :=1

4L−Diag(y1:n)− yn+1ee

f ∗p ≥n∑

i=1yi + l · n · d− =: f ∗p

d ≤ λmin(D), d− = min{0, d}, l ≥ number of negativeeigenvalues of D.

I y dual optimal, d = λmin(D) ⇒ no overestimation

I f ∗p can be computed efficiently in (0(n3)), compared to

(0(n9/2)) for the approximate solution

I yields verified solutions in Branch-and-Bound

Graph Partitioning: rigorous lower boundTight rigorous lower bounds by using Theorem (PBQ):

Corollary

y approximation, D :=1

4L−Diag(y1:n)− yn+1ee

f ∗p ≥n∑

i=1yi + l · n · d− =: f ∗p

d ≤ λmin(D), d− = min{0, d}, l ≥ number of negativeeigenvalues of D.

I y dual optimal, d = λmin(D) ⇒ no overestimation

I f ∗p can be computed efficiently in (0(n3)), compared to

(0(n9/2)) for the approximate solution

I yields verified solutions in Branch-and-Bound

Graph Partitioning: numerical results

Problems from Gruber and Rendl: Matlab m-files can be found athttp://uni-klu.ac.at/groups/math/optimization/

SDP-solver SDPT3 (Tutuncu, Toh, Todd)INTLAB (Rump)

n f ∗ f ∗p µ(f ∗, f ∗p ) t t

200 -1.04285e+004 -1.04285e+004 6.86788e-008 8.81 0.19400 -3.01393e+004 -3.01393e+004 3.82904e-007 41.27 0.89600 -5.57876e+004 -5.57876e+004 1.05772e-006 131.47 2.69

µ(f ∗, f ∗p) relative errorn = 600 : 601 constraints, 180300 variables

n f ∗ f ∗p µ(f ∗, f ∗p ) t t

200 -1.04285e+004 -1.04285e+004 6.86788e-008 8.81 0.19400 -3.01393e+004 -3.01393e+004 3.82904e-007 41.27 0.89600 -5.57876e+004 -5.57876e+004 1.05772e-006 131.47 2.69

n f ∗ f ∗p µ(f ∗, f ∗p ) t t

200 -1.04285e+004 -1.04285e+004 6.86788e-008 8.81 0.19400 -3.01393e+004 -3.01393e+004 3.82904e-007 41.27 0.89600 -5.57876e+004 -5.57876e+004 1.05772e-006 131.47 2.69

Conclusion

I Fukuda et al write in their conclusion:And finally, optimizers have the challenge of solvinglarger SDPs with m > 20000 and n > 3000 withhigh accuracy.

I and there is hope to verify this high accuracy

Conclusion

I Fukuda et al write in their conclusion:And finally, optimizers have the challenge of solvinglarger SDPs with m > 20000 and n > 3000 withhigh accuracy.

I and there is hope to verify this high accuracy

VSDP: References

VSDP: A software package for Verified SemiDefiniteProgrammig: http://www.ti3.tu-harburg.de/jansson/vsdp

Cheap rigorous error bounds are available, which can be used for

ill-conditioned and even ill-posed problems:

I NEUMAIER and SHCHERBINA 2004, MILP

I JANSSON 2004, LP and Convex Programming

I KEIL and JANSSON 2006, NETLIB LP library

I JANSSON, CHAYKIN and KEIL 2007, SDP

I JANSSON, Ill-posed SDP

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