Strong Duality of Linear Optimisation Problems over Measure Spaces 5 1 Raphael Hauser∗1 and Sergey Shahverdyan†2 0 2 1Mathematical Institute, University of Oxford n a 2Mathematical Institute, University of Oxford J 7 1 ] C O . 1 Introduction h t a m In [2] the authors introduce a general duality result for linear optimisation prob- [ lemsoversignedmeasureswithinfinitelymanyconstraintsintheformofintegrals of functions with respect to the decision variables (the measure in question). In 1 v this work we present two particular cases of the general duality result for which 3 strongdualityholds. Inthefirstcasetheoptimisationproblemsareovermeasures 4 2 withLp densityfunctionswith1< p<¥ . Inthesecondcaseweconsiderasemi- 4 infiniteoptimisationproblemwherefinitelymanyconstraintsaregiveninformof 0 . boundson integrals. Thelattercase has aparticularimportanceinpractice where 1 0 the model can be applied in robust risk management and model-free option pric- 5 ing, e.g. [2, 3]. 1 In the next section we present the general duality result first introduced in [2]. In : v Section 3 we introduce results on conic linear optimisation problems from [8]. i X In Sections 4 and 5 we use the results from Section 3 to prove duality for two ar cases described above: measures that have Lp density functions for 1 < p < ¥ and semi-infiniteproblemswithspecial structure. ∗AndrewWilesBuilding,RadcliffeObservatoryQuarter,WoodstockRoad,OxfordOX26GG, UnitedKingdom,hauser@maths.ox.ac.uk †AndrewWilesBuilding,RadcliffeObservatoryQuarter,WoodstockRoad,OxfordOX26GG, UnitedKingdom,shahverdyan@maths.ox.ac.uk 1 2 Problem Formulation Let(F ,F),(G ,G)and(S ,S)becompletemeasurespaces,andletA: G ×F →R, a : G → R, B : S ×F → R, b : S → R, and c : F → R be bounded measurable functions on these spaces and the corresponding product spaces. Let M , M F G and M be the set of signed measures with finite variation on (F ,F), (G ,G) and S (S ,S)respectively. Wenowconsiderthefollowingpairofoptimisationproblems over M and M ×M respectively, which authors show to be duals of each F G S other, (P’) sup c(x)dF(x) F∈MF ZF s.t. A(y,x)dF(x)≤a(y), (y∈G ), F Z B(z,x)dF(x)=b(z), (z∈S ), F Z F ≥0, and (D’) inf a(y)dG(y)+ b(z)dS(z), (G,S)∈MG×MS ZG ZS s.t. A(y,x)dG(y)+ B(z,x)dS(z)≥c(x), (x∈F ), G S Z Z G ≥0. Theorem 2.1(WeakDuality) Forevery(P’)-feasiblemeasureF andevery(D’)- feasiblepair(G,S)we have c(x)dF(x)≤ a(y)dG(y)+ b(z)dS(z). F G S Z Z Z Proof. UsingFubini’sTheorem, wehave c(x)dF(x)≤ A(y,x)d(G ×F)(y,x)+ B(z,x)d(S ×F)(z,x) F G ×F S ×F Z Z Z ≤ a(y)dG(y)+ b(z)dS(z). G S Z Z We are interested in finding conditions on measures that imply strong duality be- tweentheprimalanddualproblems,i.e.Val(P′)=Val(D′),wherebyVal(P′)and Val(D′)wedenotetheoptimalvaluesoftheproblems(P′) and (D′)respectively. 2 3 General Results for Conic Optimisation Problems As mentioned above, in this section we introduce results on conic linear optimi- sationproblemsfrom [8]. Consideraconiclinearoptimisationproblemofthefollowingform (P) minhc, fi subjectto A f +h∈K, (1) f∈C where X and Y are linear spaces, C ⊂ X, K ⊂Y are convex cones, h ∈Y and A : X →Y is a linear map. Assume that X and Y are paired with some linear spaces X′ and Y′ respectively, so bilinear forms h·,·,i : X′×X → R and h·,·,i:Y′×Y →R aredefined. We calltheproblem(1)thePrimalproblem. The results of Shapiro are based on conjugate duality first introduced by Rocke- fellar[6], [7]. Define thepositivedualconeofC as C∗ :={f∗ ∈X∗ :hf∗, fi≥0, ∀f ∈C}, (2) and similarlyfortheconeK K∗ :={g∗ ∈Y∗ :hg∗,gi≥0, ∀g∈K}, (3) We alsoneed an assumptionforX′ sothattheadjointmappingofA exists. Assumption 3.1 Foranyg∗∈Y′thereexistsaunique f∗∈X′suchthathg∗,A fi= hf∗, fiforall f ∈X. Based onthisassumptionwecan definetheadjointmappingA∗ :Y′ →X′ bythe equation hg∗,A fi=hA∗g∗, fi, ∀f ∈X. (4) NowconsidertheLagrangianfunctionoftheprimalproblem(1) L(f,g∗):=hc, fi+hg∗,Af +hi (5) and thefollowingoptimisationproblem min y (f):= max {L(f,g∗)} (6) f∈C g∗∈−K∗ (cid:26) (cid:27) By changingtheminand maxoperators weget theLagrangian Dual problem max f (g∗):=min{L(f,g∗)} (7) g∗∈−K∗ f∈C (cid:26) (cid:27) 3 which isequivalenttothefollowingoptimisationproblem (D) max hg∗,hi subjectto A∗g∗+c∈C∗. (8) g∗∈−K∗ which wecall thedual problem. TheaimofthissectionistofindconditionsunderwhichVal(D)=Val(P),where byVal(D) andVal(P) we denote the objective values of the Dual (8) and Primal (1)problemsrespectively. Note that the dual problem is also a conic linear problem, and that it is easy to showtheweak duality,i.e.,Val(D)≤Val(P). Further, we associatewiththeprimalproblemtheoptimalvaluefunction v(g):=inf{hc, fi: f ∈C,A f +g∈K}. (9) We define v(g) to be +¥ if the set {f ∈C:A f +g∈K} is empty. We have Val(P)=v(h). From [7] we know that the extended optimal value function v(g) is convex and positivelyhomogeneousofdegree1, i.e., ∀t >0and g∈Y v(tg)=tv(g). Theconjugateofv(g)isdefined as v∗(g∗):=sup{hg∗,gi−v(g)}. (10) g∈Y Evaluatingtheformulaeabovewe get v∗(g∗)=sup{hg∗,gi−hc, fi:(f,g∗)∈X×Y∗, f ∈C,A f +g∈K} =sup sup {hg∗,gi−hc, fi} f∈CA f+g∈K =supsup{hg∗,g−A fi−hc, fi} f∈Cg∈K =supsup{hg∗,gi−hA∗g∗+c, fi} f∈Cg∈K Itiseasytoshowthatv∗(g∗)istheindicatorfunctionofthefeasiblesetofthedual problem, since from g∗ ∈ −K∗ and A∗g∗+c ∈C∗ follows that v∗(g∗) = 0, and v∗(g∗)=+¥ otherwise. So wecan writethedualproblemas max {hg∗,hi−v∗(g∗)} (11) g∗∈Y∗ Taking thebiconjugateofv(y) v∗∗(g):= sup {hg∗,gi−v∗(g∗)}, (12) g∗∈Y∗ 4 we see thatVal(D)=v∗∗(h), hence we get, that if v(h)=v∗∗(h), then there is no dualitygapbetween Lagrangian primaland dualproblems. Nowwe aimtofind conditionssuch thatv(h)=v∗∗(h). We described the main approach to the proof of the strong duality in this frame- work, and now, without going into details, we will introduce the results given by Shapiro. Moreinterestedreader can refer to[8]formoredetails. We makean assumptionwhich willbe considered to behold throughoutthis section. Assumption 3.2 The spacesY andY′ are paired locally convex topologicalvec- tor spaces. Denoteby lscvthelowersemicontinoushullofthefunctionv, i.e. lscv(g)=min v(g),liminfv(z) , (13) z→g (cid:26) (cid:27) and by clvtheclosureofthefunctionv: lscv(·), if lscv(g)>−¥ forall g∈Y, clv(·):= (14) (−¥ , if lscv(g)=−¥ forat least oneg∈Y. We say that the problem (P) is sub-consistent if lscv(h) < +¥ (if the problem (P) is consistent, i.e. it’s feasible set is nonempty, then it is also sub-consistent). Moreover, the Fenchel-Moreau theorem implies that v∗∗ = clv. Taking into ac- count the fact that if lscv(h) < +¥ then clv(h) = lscv(h) we get the following proposition: Proposition3.3 (Proposition2.2,[8]) Thefollowingholds: 1. Val(D)=clv(h). 2. If (P)issub-consistent,thenVal(D)=lscv(h). Theabovepropositionshowsthat ifP issub-consistentthenstrong dualityholds, i.e., Val(D) =Val(P) iff v(h) is lower semicontinous at g = h. But it may be difficulttoverifythesemicontinuitydirectly,soweseekmoretractableconditions in thesubsequentanalysis. Define thesub-differentialofthefunctionvat apointg (where visfinite)as ¶ v(g):= g∗ ∈Y′ :v(z)−v(g)≥hg∗,z−gi, ∀z∈Y . (15) We say that v is sub(cid:8)-differentiable at a point g if v(g) is finite(cid:9)and ¶ v(g) is nonempty. Further, we know that if v is sub-differentiable at g = h, then v∗∗ = v(h), andconversely;ifv(b)isfiniteand v∗∗ =v(h),then¶ v(h)=¶ v∗∗(h)[7]. We nowgetthefollowingproposition. 5 Proposition3.4 (Proposition2.5,[8]) If the optimal value function v(g) is sub- differentiable at the point g = h, then Val(P) =Val(D) and the set of optimal solutions of (D) is ¶ v(h). Conversely, if Val(P) = Val(D) and is finite, then Sol(D)=¶ v(h). BySol(D)wedenotetheset ofsolutionsoftheproblem(D). However, checking the sub-differentiability for the optimal value function may stillbedifficult. Considertheset M :={(g,a )∈Y ×R:g=k−A f,a ≥hc, fi, f ∈C,k∈K}. (16) Itiseasytoshowthattheoptimalvalueoftheproblem(P)isequaltotheoptimal valueofthefollowingproblem mina s.t. (h,a )∈M. Proposition3.5 (Proposition2.6,[8]) SupposethatVal(P)isfiniteandthecone M is closed in the product topology of Y ×R. Then Val(P) =Val(D) and the primalproblem(P) hasanoptimalsolution. Fromconvexanalysisweknowthatifv(g)<¥ andcontinuousath,Y isaBanach space and Y∗ is its standard dual, then ¶ v(h) is closed and bounded in the dual topologyofY∗ ([4], p. 84). Weget thefollowingproposition. Proposition3.6 (Proposition2.7,[8]) If the optimal value function v(g) is con- tinuousatg=handifY isaBanachspacepairedwithitsstandarddualY∗,then theset ofoptimalsolutionsof (D)isboundedinthedualnormtopologyofY∗. From[5]weknowthatifX andY areBanachspacesequippedwithstrongtopolo- gies, the conesC and K are closed and hc,·i and A :X →Y are continuous, then v(g) iscontinuousat g=h ifand onlyif h∈int(domv). (17) Since domv=−A(C)+K,wecan writethecondition(17)as −h∈int(A(C)−K). (18) Hencewegetthefinalstoneoftheframeworkweneedtoproveourstrongduality. 6 Proposition3.7 (Proposition2.9,[8]) SupposethatX andY areBanachspaces, theconesC andK areclosed,hc,·iandA :X →Y arecontinuous,andthatCon- dition(18)holds. ThenVal(P)=Val(D)andSol(D)is nonemptyandbounded. If the cone K has a non-empty interior, then Condition (18) is equivalent to ([1], Proposition2.106) ∃f¯∈C such thatA f¯+h∈int(K). (19) If the later condition holds, then it is said that the generalized Slater condition is satisfied forProblem(1). In manyapplicationswehaveequalitytypeconstraintsforoptimisationproblems of the form (1). In this case the cone K has obviously a single element 0, and hence, theinterioris empty. If theconstraintin theproblem(1)is givenbytheequality A f +h=0, then theregularitycondition(18)is equivalentto([1], section2.3.4) A(X)=Y, ∃f¯∈int(C)s. t. A f¯+h=0. After having introduced themathematicalframework for general coniclinearop- timisationproblems,weareready to usetheseresultsto getstrongdualityresults fortwoparticularcasesofourgeneraldualitytheory,whichwediscussinthenext two sections. 4 Measures with Lp Densities Consider a special case of the primal problem (P′) from Section 2, where the optimisationis overmeasuresthat haveadensityfunctionthatbelongsto Lp(F ). Let F ⊆Rn, G ⊆Rm and S ⊆Rk. (PL) sup c(x)f(x)dx f∈Lp(F )ZF A(y,x)f(x)dx≤a(y), a.e. y∈G F Z B(z,x)f(x)dx=b(z), a.e. z∈S F Z f ≥0. 7 Let c∈Lq(F )and a∈Lp(Y ) forsome1< p<¥ and n,m,k. AssumeA:G ×F →R and B:S ×F →Rare such that A(y,·) ∈Lq(F ) A(·,x) ∈Lp(G ) B(z,·) ∈Lq(F ) B(·,x) ∈Lp(S ) ∀x∈F ,∀y∈G and∀z∈S . Hencethefunctionst (x)=kA(·,x)k ,t (x)=kB(·,x)k andr (y)=kA(y,·)k , A p B p A q r (z)=kB(z,·)k arewell defined. B q We alsomakethefollowingassumption. Assumption 4.1 FunctionsA andB aresuch,that A(y,x)f(x)dx∈Lp(G ), a.e. y∈G F Z B(z,x)f(x)dx∈Lp(S ), a.e. z∈S F Z Later we will prove a lemma which, under some conditions, guarantees that the aboveassumptionsaretrue, butfornowweconsiderthemas given. Denote X := Lp(F ), C := Lp(F ) + Y := Lp(G )×Lp(S ), K := Lp(G )×{0}. − In thiscase, since1< p<+¥ wehave X∗ := Lq(F ), C∗ := Lq (F ) + Y∗ := Lq(G )×Lq(S ), K∗ := Lq (G )×Lq(S ). − and X∗∗ =X, Y∗∗ =Y, C∗∗ =C, K∗∗ =K. With thesenotationswecan writetheproblem(PL) as minh−c, fi subjectto A f +h∈K, (20) f∈C 8 where thelinearoperatorA :X →Y isdefined as A(y,x)f(x)dx A f(y,z)= F B(z,x)f(x)dx F (cid:18) R (cid:19) and h isdefined as R −a(y) h(y,z)= −b(z) (cid:18) (cid:19) Construct theLagrangian oftheproblem(PL) L(f,l ∗)=−hc, fi+hl ∗,A f +hi, (21) where l ∗ ∈Y, i.e. it hasthefollowingform g∗ l ∗ = , g∗ ∈Lq(G ), s∗ ∈Lq(S ) s∗ (cid:18) (cid:19) TheLagrangian functioncan thusbewrittenas L(f,g∗,s∗) = − c(x)f(x)dx F Z + A(y,x)f(x)dx−a(y) g∗(y)dy (22) G F Z (cid:18)Z (cid:19) + B(z,x)f(x)dx−b(z) s∗(z)dz S F Z (cid:18)Z (cid:19) TheLagrangian primalproblemis min sup L(f,l ∗) (23) f∈Cl ∗∈−K∗ Interchangingtheminandmaxoperators,weobtainthedualLagrangian,problem sup minL(f,l ∗). (24) l ∗∈−K∗ f∈C In orderto evaluate(24) wechangetheorderofintegrationin (22). Wehave L(f,g∗,s∗)=−ha,g∗i−hb,s∗i+hf , fi, (25) where f (x)= A(y,x)g∗(y)dy+ B(z,x)s∗(z)dz−c(x) (26) G S Z Z Thus, −ha,g∗i−hb,s∗i, iff (x)≥0,(a.e. x∈f ), minL(f,g∗,s∗)= (27) f∈C (−¥ , otherwise 9 Thisleads totheequivalenceoftheLagrangian dualproblemto (DL) inf a(y)g(y)dy+ b(z)s(z)dz g∈Lq(G ),s∈Lq(S ) G S Z Z A(y,x)g(y)dy+ B(z,x)s(z)dz≥c(x), a.e. x∈F G S Z Z g≥0. Definition 4.2 We say that Slater condition holds for the problem (PL) if ∃f¯∈ Lp(F )suchthat A(y,x)f¯(x)dx

Strong Duality of Linear Optimisation Problems over Measure Spaces

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