Interpolation synthesis for quadratic polynomial inequalities and combination with EUF

0.59 MB·

by  Ting Gan

Interpolation synthesis for quadratic polynomial
inequalities and combination with EUF
TingGan1,LiyunDai1,BicanXia1,NaijunZhan2,DeepakKapur3,andMingshuaiChen2
1 LMAM&SchoolofMathematicalSciences,PekingUniversity
{gant,dailiyun,xbc}@pku.edu.cn,
6
2 StateKeyLab.ofComputerScience,InstituteofSoftware,CAS
1
[email protected]
0
2 3 DepartmentofComputerScience,UniversityofNewMexico
[email protected]
v
o
N
Abstract. Analgorithmforgeneratinginterpolantsforformulaswhicharecon-

0 junctionsofquadraticpolynomialinequalities(bothstrictandnonstrict)ispro-
1
posed. The algorithm is based on a key observation that quadratic polynomial

inequalitiescanbelinearizediftheyareconcave.AgeneralizationofMotzkin’s
]
O transpositiontheoremisproved,whichisusedtogenerateaninterpolantbetween
twomutuallycontradictoryconjunctionsofpolynomialinequalities,usingsemi-
L
definiteprogrammingintimecomplexityO(n3+nm))withagiventhreshold,
.
s wherenisthenumberofvariablesandmisthenumberofinequalities.Using
c
the framework proposed by [21] for combining interpolants for a combination
[
ofquantifier-freetheorieswhichhavetheirowninterpolationalgorithms,acom-

3 binationalgorithmisgivenforthecombinedtheoryofconcavequadraticpoly-
v nomialinequalitiesandtheequalitytheoryoveruninterpretedfunctionssymbols
2 (EUF).Theproposedapproachisapplicabletoallexistingabstractdomainslike
0 octagon,polyhedra,ellipsoidandsoon,thereforeitcanbeusedtoimprovethe
8
scalabilityofexistingverificationtechniquesforprogramsandhybridsystems.In
4
addition,wealsodiscusshowtoextendourapproachtoformulasbeyondconcave
0
quadraticpolynomialsusingGro¨bnerbasis.
.
1
0
6 Keywords: Programverification,Interpolant,Concavequadraticpolynomials,Motzin’s
1 theorem,Semi-definiteprogramming.
:
v
1 Introduction
i
X
InterpolantshavebeenpopularizedbyMcMillan[15]forautomaticallygenerating
r
invariantsofprograms.Sincethen,developingefficientalgorithmsforgeneratinginter-
a
polantsforvarioustheorieshasbecomeanactiveareaofresearch;inparticular,methods
havebeendevelopedforgeneratinginterpolantsforPresburgerarithmetic(bothforin-
tegers as well as for rationals/reals), theory of equality over uninterpreted symbols as
wellastheircombination.Mostofthesemethodsassumetheavailabilityofarefutation
proofofα∧βtogeneratea“reverse”interpolantof(α,β);calculihavebeenproposed
to label an inference node in a refutational proof depending upon whether symbols
of formulas on which the inference is applied are purely from α or β. For proposi-
tional calculus, there already existed methods for generating interpolants from reso-
lution proofs [11,16] prior to McMillan’s work, which generate different interpolants
fromthosedonebyMcMillan’smethod.ThisledD’Silvaetal[6]tostudystrengthsof
variousinterpolants.
In Kapur, Majumdar and Zarba [10], an intimate connection between interpolants
andquantifiereliminationwasestablished.Usingthisconnection,existenceofquantifier-
free as well as interpolants with quantifiers were shown for a variety of theories over
container data structures. A CEGAR based approach was generalized for verification
of programs over container data structures using interpolants. Using this connection
betweeninterpolantgenerationandquantifierelimination,Kapur[9]hasshownthatin-
terpolantsformalatticeorderedusingimplication,withtheinterpolantgeneratedfrom
αbeingthebottomofsuchalatticeandtheinterpolantgeneratedfromβ beingthetop
ofthelattice.
Nonlinearpolynomialsinequalitieshavebeenfoundusefultoexpressinvariantsfor
softwareinvolvingsophisticatednumbertheoreticfunctionsaswellashybridsystems;
an interested reader may see [27,28] where different controllers involving nonlinear
polynomialinequalitiesarediscussedforsomeindustrialapplications.
Weproposeanalgorithmtogenerateinterpolantsforquadraticpolynomialinequal-
ities (including strict inequalities). Based on the insight that for analyzing the solu-
tionspaceofconcavequadraticpolynomial(strict)inequalities,itsufficestolinearize
them. We prove a generalization of Motzkin’s transposition theorem to be applicable
for quadratic polynomial inequalities (including strict as well as nonstrict). Based on
thisresult,weprovetheexistenceofinterpolantsfortwomutuallycontradictorycon-
junctionsα,β ofconcavequadraticpolynomialinequalitiesandgiveanalgorithmfor
computing an interpolant using semi-definite programming. The algorithm is recur-
sivewiththebasisstepofthealgorithmrelyingonanadditionalconditiononconcave
quadraticpolynomialsappearinginnonstrictinequalitiesthatanynonpositiveconstant
combinationofthesepolynomialsisneveranonzerosumofsquarepolynomial(called
NSOSC). In this case, an interpolant output by the algorithm is either a strict in-
equalityoranonstrictinequalitymuchlikeinthelinearcase.Incase,thisconditionis
notsatisfiedbythenonstrictinequalities,i.e.,thereisanonpositiveconstantcombina-
tions of polynomials appearing as nonstrict inequalities that is a negative of a sum of
squares,thennewmutuallycontradictoryconjunctionsofconcavequadraticpolynomi-
als infewer variablesare derivedfrom the inputaugmented withthe equality relation
deduced, and the algorithm is recursively invoked on the smaller problem. The out-
putofthisalgorithmisingeneralaninterpolantthatisadisjunctionofconjunctionof
polynomial nonstrict or strict inequalities. The NSOSC condition can be checked in
polynomialtimeusingsemi-definiteprogramming.
Wealsoshowhowseparatingtermst−,t+ canbeconstructedusingcommonsym-
bols in α,β such that α ⇒ t− ≤ x ≤ t+ and β ⇒ t+ ≤ y ≤ t−, whenever
(α∧β) ⇒ x = y. Similar to the construction for interpolants, this construction has
thesamerecursivestructurewithconcavequadraticpolynomialssatisfyingNSOSCas
thebasisstep.Thisresultenablestheuseoftheframeworkproposedin[17]basedon
hierarchical theories and a combination method for generating interpolants by Yorsh
andMusuvathi,fromcombiningequalityinterpolatingquantifier-freetheoriesforgen-
erating interpolants for the combined theory of quadratic polynomial inequalities and
theoryofuninterpretedsymbols.
Obviously,ourresultsaresignificantinprogramverificationasallwell-knownab-
stractdomains,e.g.octagon,polyhedra,ellipsoid andsoon,whicharewidelyusedin
the verification of programs and hybrid systems, are quadratic and concave. In addi-
tion,wealsodiscussthepossibilitytoextendourresultstogeneralpolynomialformu-
lasbyallowingpolynomialequalitieswhosepolynomialsmaybeneitherconcavenor
quadraticusingGro¨bnerbasis.
Wedevelopacombinationalgorithmforgeneratinginterpolantsforthecombination
ofconcavequadraticpolynomialinequalitiesanduninterpretedfunctionsymbols.
In [5], Dai et al. gave an algorithm for generating interpolants for conjunctions of
mutually contradictory nonlinear polynomial inequalities based on the existence of a
witness guaranteed by Stengle’s Positivstellensatz [22] that can be computed using
semi-definiteprogramming.Theiralgorithmisincompleteingeneralbutifeveryvari-
ablesrangesoveraboundedinterval(calledArchimedeancondition),thentheiralgo-
rithm is complete. A major limitation of their work is that formulas α,β cannot have
uncommonvariables4.However,theydonotgiveanycombinationalgorithmforgener-
atinginterpolantsinthepresenceofuninterpretedfunctionsymbolsappearinginα,β.
Thepaperisorganizedasfollows.Afterdiscussingsomepreliminariesinthenext
section, Section 3 defines concave quadratic polynomials, their matrix representation
and their linearization. Section 4 presents the main contribution of the paper. A gen-
eralizationofMotzkin’stranspositiontheoremforquadraticpolynomialinequalitiesis
presented. Using this result, we prove the existence of interpolants for two mutually
contradictoryconjunctionsα,β ofconcavequadraticpolynomialinequalitiesandgive
analgorithm(Algorithm2)forcomputinganinterpolantusingsemi-definiteprogram-
ming. Section 5 extends this algorithm to the combined theory of concave quadratic
inequalitiesandEUFusingtheframeworkusedin[21,17].Implementationandexper-
imentalresultsusingtheproposedalgorithmsarebrieflyreviewedinSection6,andwe
concludeanddiscusfutureworkinSection7.
2 Preliminaries
Let N, Q and R be the set of natural, rational and real numbers, respectively. Let
R[x]bethepolynomialringoverRwithvariablesx=(x ,··· ,x ).Anatomicpoly-
1 n
nomial formula ϕ is of the form p(x)(cid:5)0, where p(x) ∈ R[x], and (cid:5) can be any of
=,>,≥,(cid:54)=;withoutanylossofgenerality,wecanassume(cid:5)tobeanyof>,≥.Anar-
bitrarypolynomialformulaisconstructedfromatomiconeswithBooleanconnectives
andquantificationsoverrealnumbers.LetPT(R)beafirst-ordertheoryofpolynomi-
als with real coefficient, In this paper, we are focusing on quantifier-free fragment of
PT(R).
Laterwediscussquantifier-freetheoryofequalityoftermsoveruninterpretedfunc-
tion symbols and its combination with the quantifier-free fragment of PT(R). Let Σ
beasetof(new)functionsymbols.LetPT(R)Σ betheextensionofthequantifier-free
theorywithuninterpretedfunctionsymbolsinΣ.
Forconvenience,weuse⊥tostandforfalseand(cid:62)fortrueinwhatfollows.
4Seehoweveranexpandedversionoftheirpaperunderpreparationwheretheyproposeheuris-
ticsusingprogramanalysisforeliminatinguncommonvariables.
Definition1. AmodelM = (M,f )ofPT(R)Σ consistsofamodelM ofPT(R)
M
andafunctionf :Rn →Rforeachf ∈Σ witharityn.
M
Definition2. LetφandψbeformulasofaconsideredtheoryT,then
– φisvalidw.r.t.T,writtenas|= φ,iffφistrueinallmodelsofT;
T
– φentailsψ w.r.t.T,writtenasφ |= ψ,iffforanymodelofT,ifψ istrueinthe
T
model,soisφ;
– φ is satisfiable w.r.t. T, iff there exists a model of T such that in which φ is true;
otherwiseunsatisfiable.
Notethatφisunsatisfiableiffφ|= ⊥.
T
Craig showed that given two formulas φ and ψ in a first-order theory T such that
φ |= ψ, there always exists an interpolant I over the common symbols of φ and ψ
suchthatφ|=I,I |=ψ.Intheverificationliterature,thisterminologyhasbeenabused
following[15],whereanreverseinterpolantI overthecommonsymbolsofφandψis
definedforφ∧ψ |=⊥as:φ|=I andI∧ψ |=⊥.
Definition3. Let φ and ψ be two formulas in a theory T such that φ∧ψ |= ⊥. A
T
formulaI saidtobea(reverse)interpolantofφandψifthefollowingconditionshold:
i φ|= I;
T
ii I∧ψ |= ⊥;and
T
iii I onlycontainscommonsymbolsandfreevariablessharedbyφandψ.
Ifψ isclosed,thenφ |= ψ iffφ∧¬ψ |= ⊥.Thus,I isaninterpolantofφand
T T
ψ iff I is a reverse interpolant of φ and ¬ψ. In this paper, we just deal with reveres
interpolant,andfromnowon,weabuseinterpolantandreverseinterpolant.
2.1 Motzkin’stranspositiontheorem
Motzkin’stranspositiontheorem[18]isoneofthefundamentalresultsaboutlinear
inequalities; it also served as a basis of the interpolant generation algorithm for the
quantifier-freetheoryoflinearinequalitiesin[17].Thetheoremhasseveralvariantsas
well.Belowwegivetwoofthem.
Theorem1 (Motzkin’s transposition theorem [18]). Let A and B be matrices and
letαandβbecolumnvectors.ThenthereexistsavectorxwithAx≥αandBx>β,
iff
forallrowvectorsy,z≥0:
(i)if yA+zB =0thenyα+zβ ≤0;
(ii)if yA+zB =0andz(cid:54)=0thenyα+zβ <0.
Corollary1. LetA ∈ Rr×n andB ∈ Rs×n bematricesandα ∈ Rr andβ ∈ Rs be
columnvectors.DenotebyA ,i = 1,...,r theithrowofAandbyB ,j = 1,...,s
i j
thejthrowofB.ThentheredoesnotexistavectorxwithAx ≥ αandBx > β,iff
thereexistrealnumbersλ ,...,λ ≥0andη ,η ,...,η ≥0suchthat
1 r 0 1 s
r s
(cid:88) (cid:88)
λ (A x−α )+ η (B x−β )+η ≡0, (1)
i i i j j j 0
i=1 j=1
s
(cid:88)
η >0. (2)
j
j=0
Proof. The“if”partisobvious.Belowweprovethe“onlyif”part.
ByTheorem1,ifAx≥αandBx>βhavenocommonsolution,thenthereexist
tworowvectorsy∈Rr andz∈Rswithy≥0andz≥0suchthat
(yA+zB =0∧yα+zβ >0)∨(yA+zB =0∧z(cid:54)=0∧yα+zβ ≥0).
Letλ =y ,i=1,...,r,η =z ,j =1,...,sandη =yα+zβ.Thenitiseasyto
i i j j 0
checkthatEqs.(1)and(2)hold. (cid:116)(cid:117)
3 Concavequadraticpolynomialsandtheirlinearization
Definition4 (ConcaveQuadratic).Apolynomialf ∈R[x]iscalledconcavequadratic
(CQ),ifthefollowingtwoconditionshold:
(i) f hastotaldegreeatmost2,i.e.,ithastheformf =xTAx+2αTx+a,whereA
isarealsymmetricmatrix,αisacolumnvectoranda∈Risaconstant;
(ii) thematrixAisnegativesemi-definite,writtenasA(cid:22)0.5
Example1. Letg =−x2+2x −x2+2x −y2,thenitcanbeexpressedas
1 1 1 2 2
 T    T 
x −1 0 0 x 1 x
1 1 1
g1 =x2  0 −1 0 x2+21 x2.
y 0 0 −1 y 0 y
 
−1 0 0
Thedegreeofg1is2,andthecorrespondingA= 0 −1 0 (cid:22)0.Thus,g1isCQ.
0 0 −1
Itiseasytoseethatiff ∈ R[x]islinear,thenf isCQbecauseitstotaldegreeis1
andthecorrespondingAis0whichisofcoursenegativesemi-definite.
Aquadraticpolynomialcanalsoberepresentedasaninnerproductofmatrices(cf.
(cid:28) (cid:18)1 xT (cid:19)(cid:29)
[13]),i.e.,f(x)= P, .
xxxT
5A being negative semi-definite has many equivalent characterizations: for every vector x,
xTAx ≤ 0;everykthminorofA≤ 0ifk isoddand≥ 0otherwise;aHermitianmatrix
whoseeigenvaluesarenonpositive.
3.1 Linearization
Considerquadraticpolynomialsf andg (i=1,...,r,j =1,...,s),
i j
f =xTA x+2αTx+a ,
i i i i
g =xTB x+2βTx+b ,
j j j j
whereA ,B aresymmetricn×nmatrices,α ,β ∈ Rn,anda ,b ∈ R;letP :=
i j i j i j i
(cid:18)a αT(cid:19) (cid:18)b βT(cid:19)
i i , Q := j j be(n+1)×(n+1)matrices,then
α A j β B
i i j j
(cid:28) (cid:18)1 xT (cid:19)(cid:29) (cid:28) (cid:18)1 xT (cid:19)(cid:29)
f (x)= P , , g (x)= Q , .
i i xxxT j j xxxT
ForCQpolynomialsf sandg sinwhicheachA (cid:22)0,B (cid:22)0,define
i j i j
K ={x∈Rn |f (x)≥0,...,f (x)≥0,g (x)>0,...,g (x)>0}. (3)
1 r 1 s
(cid:28) (cid:18)1 xT (cid:19)(cid:29)
Givenaquadraticpolynomialf(x)= P, ,itslinearizationisdefined
xxxT
(cid:28) (cid:18)1 xT(cid:19)(cid:29) (cid:18)1 xT(cid:19)
asf(x)= P, ,where (cid:23)0.
x X x X
Let
X =(X ,X ,X ,...,X ,...,X ,...,X ,...,X )
(1,1) (2,1) (2,2) (k,1) (k,k) (n,1) (n,n)
be the vector variable with n(n+1) dimensions corresponding to the matrix X. Since
2
(cid:28) (cid:18)1 xT(cid:19)(cid:29)
X isasymmetricmatrix, P, isalinearexpressioninx,X.
x X
Now,let
(cid:18)1 xT(cid:19) (cid:28) (cid:18)1 xT(cid:19)(cid:29)
K ={x| (cid:23)0, ∧r P , ≥0,
1 x X i=1 i x X
(cid:28) (cid:18)1 xT(cid:19)(cid:29)
∧s Q , >0, forsomeX}, (4)
j=1 j x X
whichisthesetofallx∈Rnonlinearizationsoftheabovef sandg s.
i j
In[7,13],whenK andK aredefinedonlywithf withoutg ,i.e.,onlywithnon-
1 i j
strictinequalities,itisprovedthatK =K . BythefollowingTheorem2,weshowthat
1
K =K alsoholdseveninthepresenceofstrictinequalitieswhenf andg areCQ.So,
1 i j
whenf andg areCQ,theCQpolynomialinequalitiescanbetransformedequivalently
i j
toasetoflinearinequalityconstraintsandapositivesemi-definiteconstraint.
Theorem2. Let f ,...,f and g ,...,g be CQ polynomials, K and K as above,
1 r 1 s 1
thenK =K .
1
Proof. Foranyx ∈ K,letX = xxT.Thenitiseasytoseethatx,X satisfy(4).So
x∈K ,thatisK ⊆K .
1 1
Next,weproveK ⊆ K.Letx ∈ K ,thenthereexistsasymmetricn×nmatrix
1 1
(cid:18)1 xT(cid:19)
X satisfying(4).Because (cid:23)0,wehaveX−xxT (cid:23)0.Thenbythelasttwo
x X
conditionsin(4),wehave
(cid:28) (cid:18)1 xT (cid:19)(cid:29) (cid:28) (cid:18)1 xT(cid:19)(cid:29) (cid:28) (cid:18)0 0 (cid:19)(cid:29)
f (x)= P , = P , + P ,
i i xxxT i x X i 0xxT −X
=(cid:28)P ,(cid:18)1 xT(cid:19)(cid:29)+(cid:10)A ,xxT −X(cid:11)≥(cid:10)A ,xxT −X(cid:11),
i x X i i
(cid:28) (cid:18)1 xT (cid:19)(cid:29) (cid:28) (cid:18)1 xT(cid:19)(cid:29) (cid:28) (cid:18)0 0 (cid:19)(cid:29)
g (x)= Q , = Q , + Q ,
j j xxxT j x X j 0xxT −X
=(cid:28)Q ,(cid:18)1 xT(cid:19)(cid:29)+(cid:10)B ,xxT −X(cid:11)>(cid:10)B ,xxT −X(cid:11).
j x X j j
Since f and g are all CQ, A (cid:22) 0 and B (cid:22) 0. Moreover, X − xxT (cid:23) 0, i.e.,
i j i j
xxT −X (cid:22)0.Thus,(cid:10)A ,xxT −X(cid:11)≥0and(cid:10)B ,xxT −X(cid:11)≥0.Hence,wehave
i j
f (x)≥0andg (x)>0,sox∈K,thatisK ⊆K. (cid:116)(cid:117)
i j 1
3.2 Motzkin’stheoreminMatrixForm
(cid:28) (cid:18)1 xT(cid:19)(cid:29)
If P, is seen as a linear expression in x,X, then Corollary 1 can be
x X
reformulatedas:
Corollary2. Let x be a column vector variable of dimension n and X be a n×n
symmetricmatrixvariable.SupposeP ,P ,...,P andQ ,...,Q are(n+1)×(n+1)
0 1 r 1 s
symmetricmatrices.Let
(cid:28) (cid:18)1 xT(cid:19)(cid:29) (cid:28) (cid:18)1 xT(cid:19)(cid:29)
W=ˆ{(x,X)|∧r P , ≥0,∧s Q , >0},
i=1 i x X i=1 j x X
thenW =∅iffthereexistλ ,λ ,...,λ ≥0andη ,η ,...,η ≥0suchthat
0 1 r 0 1 s
(cid:88)r (cid:28) (cid:18)1 xT(cid:19)(cid:29) (cid:88)s (cid:28) (cid:18)1 xT(cid:19)(cid:29)
λ P , + η Q , +η ≡0, and
i i x X j j x X 0
i=0 j=1
η +η +...+η >0.
0 1 s
4 AlgorithmforgeneratinginterpolantsforConcaveQuadraticPoly-
nomialinequalities
Problem1. Giventwoformulasφandψonnvariableswithφ∧ψ |=⊥,where
φ=f ≥0∧...∧f ≥0∧g >0∧...∧g >0,
1 r1 1 s1
ψ =f ≥0∧...∧f ≥0∧g >0∧...∧g >0,
r1+1 r s1+1 s
inwhichf ,...,f ,g ,...,g areallCQ,developanalgorithmtogeneratea(reverse)
1 r 1 s
CraiginterpolantI forφandψ,onthecommonvariablesofφandψ,suchthatφ|=I
andI∧ψ |=⊥.Forconvenience,wepartitionthevariablesappearinginthepolynomials
above into three disjoint subsets x = (x ,...,x ) to stand for the common variables
1 d
appearinginbothφandψ,y=(y ,...,y )tostandforthevariablesappearingonlyin
1 u
φandz=(z ,...,z )tostandforthevariablesappearingonlyinψ,whered+u+v =
1 v
n.
Sincelinearinequalitiesaretriviallyconcavequadraticpolynomials,ouralgorithm
(Algorithm IGFQC in Section 4.4) can deal with the linear case too. In fact, it is a
generalizationofthealgorithmforlinearinequalities.
Theproposedalgorithmisrecursive:thebasecaseiswhennosumofsquares(SOS)
polynomial can be generated by a nonpositive constant combination of nonstrict in-
equalitiesinφ∧ψ.Whenthisconditionisnotsatisfied,i.e.,anSOSpolynomialcanbe
generatedbyanonpositiveconstantcombinationofnonstrictinequalitiesinφ∧ψ,then
itispossibletoidentifyvariableswhichcanbeeliminatedbyreplacingthembylinear
expressionsintermsofothervariablesandthusgenerateequisatisfiableproblemwith
fewervariablesonwhichthealgorithmcanberecursivelyinvoked.
(cid:28) (cid:18)1 xT(cid:19)(cid:29)
Lemma1. LetU ∈R(n+1)×(n+1) beamatrix.If U, ≤0foranyx∈Rn
x X
(cid:18)1 xT(cid:19)
andsymmetricmatrixX ∈Rn×nwith (cid:23)0,thenU (cid:22)0.
x X
Proof. AssumethatU (cid:54)(cid:22)0.Thenthereexistsacolumnvectory=(y ,y ,...,y )T ∈
0 1 n
Rn+1suchthatc:=yTUy=(cid:10)U,yyT(cid:11)>0.DenoteM =yyT,thenM (cid:23)0.
(cid:18)1 xT(cid:19) (cid:18)1 xT (cid:19)
Ify (cid:54)=0,thenletx=(y1,...,yn)T,andX =xxT.Thus, = =
0 y0 y0 x X xxxT
(cid:28) (cid:18)1 xT(cid:19)(cid:29) (cid:68) (cid:69) (cid:28) (cid:18)1 xT(cid:19)(cid:29)
1 M (cid:23),and U, = U, 1 M = c >0,whichcontradictswith U, ≤
y2 x X y2 y2 x X
0 0 0
0.
If y = 0, then M = 0. Let M(cid:48) = |U(1,1)|+1M, then M(cid:48) (cid:23) 0. Further, let
0 (1,1) c
 
10···0
00···0 (cid:18)1 xT(cid:19)
M(cid:48)(cid:48) =M(cid:48)+... ... ... ....ThenM(cid:48)(cid:48) (cid:23)0andM(cid:48)(cid:48)(1,1) =1.Let x X =M(cid:48)(cid:48),then
00···0
 
10···0
(cid:28) (cid:18)1 xT(cid:19)(cid:29) (cid:42) 00···0(cid:43)
U, x X =(cid:104)U,M(cid:48)(cid:48)(cid:105)= U,M(cid:48)+... ... ... ...
00···0
 
10···0
(cid:42) (cid:43)
|U |+1 00···0
= U, (1,1c) M +... ... ... ...
00···0
|U |+1
(1,1)
= (cid:104)U,M(cid:105)+U
c (1,1)
=|U |+1+U >0,
(1,1) (1,1)
(cid:28) (cid:18)1 xT(cid:19)(cid:29)
whichalsocontradictswith U, ≤ 0.Thus,theassumptiondoesnothold,
x X
thatisU (cid:22)0. (cid:116)(cid:117)
Lemma2. LetA={y∈Rm |A y−α ≥0,B y−β >0, fori=1,...,r,j =
i i j j
1,...,}beanonemptysetandB ⊆Rmbeannonemptyconvexclosedset.IfA∩B =∅
andtheredoesnotexistalinearformL(y)suchthat
∀y∈A,L(y)>0, and ∀y∈B,L(y)≤0, (5)
thenthereisalinearformL (y)(cid:54)≡0andδ ,...,δ ≥0suchthat
0 1 r
r
(cid:88)
L (y)= δ (A y−α )and ∀y∈B,L (y)≤0. (6)
0 i i i 0
i=1
Proof. Since A is defined by a set of linear inequalities, A is a convex set. Using
the separation theorem on disjoint convex sets, cf. e.g. [1], there exists a linear form
L (y)(cid:54)≡0suchthat
0
∀y∈A,L (y)≥0, and ∀y∈B,L (y)≤0. (7)
0 0
From(5)wehavethat
∃y ∈A, L (y )=0. (8)
0 0 0
Since
∀y∈A,L (y)≥0, (9)
0
then
A y−α ≥0∧...∧A y−α ≥0∧
1 1 r r
B y−β >0∧...∧B y−β >0∧−L (y)>0
1 1 s s 0
hasnosolutionw.r.t.y.UsingCorollary1,thereexistλ ,...,λ ≥ 0,η ,...,η ≥ 0
1 r 0 s
andη ≥0suchthat
r s
(cid:88) (cid:88)
λ (A y−α )+ η (B y−β )+η(−L (y))+η ≡0, (10)
i i i j j j 0 0
i=1 j=1
s
(cid:88)
η +η >0. (11)
j
j=0
Applyingy in(8)to(10)and(11),itfollows
0
η =η =...=η =0, η >0.
0 1 s
Fori=1,...,r,letδi = ληi ≥0,then
r
(cid:88)
L (y)= δ (A y−α )and ∀y∈B,L (y)≤0. (cid:116)(cid:117)
0 i i i 0
i=1
The lemma below asserts the existence of a strict linear inequality separating A
and B defined above, for the case when any nonnegative constant combination of the
linearizationoff sispositive.
i
Lemma3. LetA={y∈Rm |A y−α ≥0,B y−β >0, fori=1,...,r,j =
i i j j
1,...,}beanonemptysetandB ⊆Rmbeannonemptyconvexclosedset,A∩B =∅.
ThereexistsalinearformL(x,X)suchthat
∀(x,X)∈A,L(x,X)>0, and ∀(x,X)∈B,L(x,X)≤0,
whenevertheredoesnotexistλ ≥0,s.t.,(cid:80)r λ P (cid:22)0.
i i=1 i i
Proof. Proof is by contradiction. Given that A is defined by a set of linear inequal-
ities and B is a closed convex nonempty set, by Lemma 2, there exist a linear form
L (x,X)(cid:54)≡0andδ ,...,δ ≥0suchthat
0 1 r
(cid:88)r (cid:28) (cid:18)1 xT(cid:19)(cid:29)
L (x,X)= δ P , and ∀(x,X)∈B,L (x,X)≤0.
0 i i x X 0
i=1
I.e.thereexistsansymmetricalmatrixL(cid:54)≡0suchthat
(cid:28) (cid:18)1 xT(cid:19)(cid:29) (cid:88)r (cid:28) (cid:18)1 xT(cid:19)(cid:29)
L, ≡ δ P , , (12)
x X i i x X
i=1
(cid:28) (cid:18)1 xT(cid:19)(cid:29)
∀(x,X)∈B, L, ≤0. (13)
x X