A Rockafellar-type theorem for non-traditional costs
aa r X i v : . [ m a t h . M G ] D ec A ROCKAFELLAR-TYPE THEOREM FOR NON-TRADITIONAL COSTS
S. ARTSTEIN-AVIDAN, S. SADOVSKY, K. WYCZESANY
Abstract.
We present a new approach to the problem of existence of a potential for theoptimal transport problem with respect to non-traditional cost functions, that is, costs thatmay assume infinite values. We define a notion of c-path-boundedness, and prove that thisproperty characterizes sets that are contained in the c-subgradient of a c-class function.We provide an example of a c-cyclically monotone set which does not admit a potential,and on the other hand, we present cases where c-path-boundedness is implied by c-cyclicmonotonicity. Our method also gives a new proof for the Rockafellar-R¨uschendorf theorem. Introduction
Mass transport problems have been widely studied in mathematics, and have a vast arrayof applications, for an overview see [1, 25]. The case when the cost c , with respect to whichoptimal transport is considered, is real-valued (namely does not attain the value ±∞ ) is wellstudied and understood. A key ingredient in the classical theory is the result of Rockafellar[21] for quadratic cost c ( x, y ) = | x − y | /
2, and its generalization to real valued costs byR¨uschendorf [23]. These results state that a set is cyclically monotone if and only if it lies inthe subgradient of a convex function, and more generally, it is c -cyclically monotone if andonly if it lies in the c -subgradient of a c -class function (see definitions in Sections 2.3 and2.4).The above results do not apply to the case of non-traditional costs, that is, cost functionswhich can assume the value + ∞ , or, in other words, when one prohibits certain pairs ofpoints to be mapped to one another. In particular, the case of the so-called polar cost ,connected to the polarity transform, which has received much attention lately [4, 6, 7], is animportant example to which the Rockafellar-R¨uschendorf theorem does not apply. Indeed,when c is the polar cost, one may find c -cyclically monotone sets which are not contained inthe c -subgradient of any c -class function (see Section 2.7).In this note we characterize sets that are c -subgradients of c -class functions. These setsmust satisfy a property slightly stronger than c -cyclic monotonicity, which we call c -path-boundedness. Our methods also give a new and simple proof of the Rockafellar–R¨uschendorftheorem.Non-traditional costs were studied for instance by Pratelli [20], who showed the existence ofoptimal transport plans with respect to continuous, but possibly + ∞ valued, costs. Bertrandand Puel [12], and Bertrand, Pratelli and Puel [11] considered relativistic cost functions ofthe form c ( x, y ) = h ( x − y ), where h is a strictly convex and differentiable function, restricted to a strictly convex set K , and infinite outside K , see also [16]. A special case of such a costis the relativistic heat cost, linked with the relativistic heat equation considered by Brenier[13]. Further, a non-traditional cost function on the sphere is used in [10] for a proof ofAlexandrov’s theorem about prescribing the Gauss curvature of convex sets in the Euclideanspace, following Oliker [19].Preceding results were usually connected with a specific cost, or a special family of costs. Inthis note, we develop the theory of mass transport with respect to a general non-traditionalcost. In particular, they apply to the polar cost on R n × R n , which is linked with the polaritytransform, see [5–7]. Transportation with respect to polar cost was first considered in [9],and the results obtained there are to appear in [3]. A constructive proof for geometric convexpotentials will appear in [8].Our main theorem is the following non-traditional analogue of the Rockafellar–R¨uschendorftheorem (restated as Theorem 3.2 in Section 3, after the relevant notions are defined). Theorem.
Let
X, Y be two arbitrary sets and c : X × Y → ( −∞ , ∞ ] an arbitrary costfunction. For a given subset G ⊂ X × Y there exists a c -class function ϕ : X → [ −∞ , ∞ ] such that G ⊂ ∂ c ϕ if and only if G is c -path-bounded. Structure of the paper : In Section 2 we provide the background for the problem, and de-fine the notions of c -transform, c -subgradient, and c -cyclic monotonicity. We connect thesenotions to the classical example of the Legendre transform and the more recent polaritytransform. We then explain the Rockafellar–R¨uschendorf theorem and show that it does nothold for non-traditional costs. In Section 3 we define c -path-bounded sets, prove our maintheorem, and show that it implies the Rockafellar–R¨uschendorf theorem for traditional costs.In Section 4 we show some consequences of our theorem under further restrictions, in partic-ular that it implies many of the previous results. Finally, in Section 5, we provide anotherproof of our main result in a slightly more restrictive setting, exposing the combinatorialflavour behind it. Acknowledgement : The first named author would like to thank her favourite mathemati-cian for useful conversations. The authors received funding from the European ResearchCouncil (ERC) under the European Union’s Horizon 2020 research and innovation programme(grant agreement No 770127). The second named author is grateful to the Azrieli foundationfor the award of an Azrieli fellowship.2.
Background
Mass transport.
The initial data of a mass transport problem is ( X, Σ X ) , ( Y, Σ Y ),where X, Y are sets and Σ X , Σ Y are σ -algebras, and a cost function c : X × Y → ( −∞ , ∞ ],which is assumed to be measurable (with respect to the product σ -algebra). One usuallyconsiders X and Y which are separable metric spaces endowed with the Borel σ -algebra, andin particular X = Y = R n serve as a good model space. ROCKAFELLAR-TYPE THEOREM FOR NON-TRADITIONAL COSTS 3
Given µ ∈ P ( X ) a probability measure on X , and ν ∈ P ( Y ), we say that a measurable map T : X → Y is a transport map between µ and ν if µ ( T − ( B )) = ν ( B )for all measurable B ∈ Σ Y .We say γ ∈ P ( X × Y ) is a transport plan between µ and ν , and denote γ ∈ Π( µ, ν ), if itsmarginals on X and Y are µ and ν respectively. In particular, to any transport map T therecorresponds (in the obvious way) a transport plan concentrated on its graph, denoted usuallyby ( Id, T ) µ .The total cost of transporting µ to ν is defined by C ( µ, ν ) = inf (cid:26)Z X × Y c ( x, y ) dγ : γ ∈ Π( µ, ν ) (cid:27) . A central theorem of Kantorovich [17, 18] states that in the case of a lower semi-continuouscost function, the above infimum is attained and is given by(1) C ( µ, ν ) = sup (cid:26)Z X ϕdµ + Z Y ψdν : ϕ ∈ L ( X, dµ ) , ψ ∈ L ( Y, dν ) , ( ϕ, ψ ) admissible (cid:27) , where a pair of functions is called admissible if ϕ ( x ) + ψ ( y ) ≤ c ( x, y ) for all x ∈ X, y ∈ Y .2.2. The c -class and basic functions. Given a cost c : X × Y → ( −∞ , ∞ ], and a function ϕ : X → [ −∞ , ∞ ], we define its c -transform ϕ c : Y → [ −∞ , ∞ ] by ϕ c ( y ) = inf x ∈ X ( c ( x, y ) − ϕ ( x )) . Similarly, when ψ : Y → [ −∞ , ∞ ], we use the same notation to define ψ c ( x ) = inf y ∈ Y ( c ( x, y ) − ψ ( y )) . In the case where the expression considered on the right hand side is + ∞− (+ ∞ ), we stipulatethat this quantity is equal to + ∞ . This corresponds to the fact that for ϕ ( x ) + ψ ( y ) ≤ ∞ tohold true, no condition on ϕ ( x ) or ψ ( y ) is needed. To avoid arithmetic manipulations withinfinite numbers, one may instead consider the infimum in the definition of ϕ c to be takenonly over those points x for which c ( x, y ) < ∞ , and similarly for ψ c .It is easy to check that if ( ϕ, ψ ) is an admissible pair then ψ ≤ ϕ c and ϕ ≤ ψ c , namelythe c -transform maps a function ϕ to the (point-wise) largest ψ such that the pair ( ϕ, ψ ) isadmissible. In particular, ϕ cc ≥ ϕ , since ( ϕ, ϕ c ) is an admissible pair. As a result of thisobservation we get that ϕ ccc = ϕ c , since in addition the mapping ϕ ϕ c is order reversing.In light of this discussion, we define the c -class associated with a cost function c : X × Y → ( −∞ , ∞ ] as the image of the c -transform, namely { ϕ c : ϕ : X → [ −∞ , ∞ ] } or { ψ c : ψ : Y → [ −∞ , ∞ ] } . When X = Y and c ( x, y ) = c ( y, x ) the two transforms coincide, as do the two c -classes. Weslightly abuse notation, since throughout this note we will be considering symmetric cost S. ARTSTEIN-AVIDAN, S. SADOVSKY, K. WYCZESANY functions, by referring to “the c -class” where in fact we should formally make a distinctionbetween the c -class of functions on X and the c -class of functions on Y .Within the c -class, we define the sub-class of basic functions to be functions of the form ϕ ( x ) = c ( x, y ) + β (and ψ ( y ) = c ( x , y ) + β , respectively) for some x ∈ X, y ∈ Y and β ∈ R . By definition, every c -class function is an infimum of basic functions, and it is nothard to check that the c -class is closed under infimum.2.3. The c -subgradient. Given a cost c : X × Y → ( −∞ , ∞ ], and a c -class function ϕ , wedefine its c -subgradient by ∂ c ϕ = { ( x, y ) ∈ X × Y : ϕ ( x ) + ϕ c ( y ) = c ( x, y ) < ∞} . Clearly ( x, y ) ∈ ∂ c ϕ if and only if ( y, x ) ∈ ∂ c ϕ c .As a motivation for this definition one may go back to Kantorovich’s theorem, recalled in (1),and note that a one sided inequality is trivial, since for any γ ∈ Π( µ, ν ), and any admissiblepair ϕ, ψ , we have Z X ϕdµ + Z Y ψdν = Z X × Y ( ϕ ( x ) + ψ ( y )) dγ ≤ Z X × Y c ( x, y ) dγ, and for equality to hold one needs that γ -almost everywhere in X × Y , we will have ϕ ( x ) + ψ ( y ) = c ( x, y ). Since replacing ψ by ϕ c only increases the integral, we see that in fact forequality to hold we need the plan γ to be concentrated on the c -subgradient of some c -classfunction ϕ .2.4. c -cyclic monotonicity. Given a cost c : X × Y → ( −∞ , ∞ ], a subset G ⊂ X × Y iscalled c -cyclically monotone if c ( x, y ) < ∞ for all ( x, y ) ∈ G , and for any m , any { ( x i , y i ) :1 ≤ i ≤ m } ⊂ G , and any permutation σ of [ m ] = { , . . . , m } it holds that(2) m X i =1 c ( x i , y i ) ≤ m X i =1 c ( x i , y σ ( i ) ) . To our best knowledge, this definition was first introduced by Knott and Smith [24], as ageneralization of cyclic monotonicity considered by Rockafellar [21] in the case of quadraticcost. When G ⊂ ∂ c ϕ , we have by definition that c ( x i , y i ) = ϕ ( x i ) + ϕ c ( y i ) and c ( x i , y σ ( i ) ) ≥ ϕ ( x i ) + ϕ c ( y σ ( i ) ) , and summing these inequalities over i ∈ [ m ] gives (2). In other words, c -cyclic monotonicityis a necessary condition for a set G ⊂ X × Y to have a potential ϕ , that is, a c -class functionfor which G ⊂ ∂ c ϕ .2.5. Example: The quadratic cost.
The most well studied example is that of quadraticcost, namely X = Y = R n and the Euclidean distance squared k x − y k /
2. From thetransportation perspective, since k x − y k / k x k / − h x, y i + k y k /
2, only the mixed termis important. We will thus work with c ( x, y ) = −h x, y i , bearing in mind that in this settingthe cost is no longer positive, and is unbounded both from above and below. It is, however,traditional, namely does not assume the value + ∞ . ROCKAFELLAR-TYPE THEOREM FOR NON-TRADITIONAL COSTS 5
The c -transform is given by ϕ c ( y ) = inf x ∈ R n ( −h x, y i − ϕ ( x )) , which can be written as − ϕ c = L ( − ϕ ) , where L ψ ( y ) = sup x ( h x, y i − ψ ( x )) is the Legendre transform (for an overview see e.g. [22]).It is easy to check that the associated c -class is that of all concave functions on R n which areupper semi-continuous.2.6. The Rockafellar-R¨uschendorf Theorem.
For costs c : X × Y → R , namely tra-ditional costs, the Rockafellar-R¨uschendorf Theorem states that the condition of c -cyclicmonotonicity for a set G ⊂ X × Y is equivalent to the condition that there exists a c -classfunction such that G ⊂ ∂ c ϕ [21, 23].This fundamental theorem has a beautiful constructive proof. Indeed, one fixes some element( x , y ) ∈ G and sets ϕ ( x ) = inf x, ( x i ,y i ) mi =1 ⊂ G c ( x, y m ) − c ( x , y ) + m X i =1 ( c ( x i , y i − ) − c ( x i , y i )) ! . Interestingly, examining the proof (which can be found in many places, see for example [25]),there is only one step where the c -cyclic monotonicity is used, and it is to prove that at( x , y ) the function ϕ defined above is finite. For non-traditional costs the proof fails (asdoes the theorem, as we shall shortly see), and the function ϕ defined above may be infiniteon points x with ( x, y ) ∈ G , in which case ( x, y ) cannot belong to ∂ c ϕ .Below we present a corresponding theorem for non-traditional costs. Applied to the case oftraditional costs, our method gives a new proof for the Rockafellar-R¨uschendorf Theorem,which is, in our view, more intuitive.2.7. An example of a c -cyclically monotone set with no potential. To end this sec-tion, and before moving to the proof of our main result, let us present an example of a c -cyclically monotone set which is not a subset of ∂ c ϕ for any c -class ϕ . In other words, G has no potential (no offense, G ). The cost we shall use is the polar cost in one dimension,namely c ( x, y ) = − ln( xy −
1) on R × R (where the logarithm of a non-positive number isdefined to be −∞ ). For this cost, c -cyclic monotonicity is equivalent to the set lying on thegraph of a decreasing function. More precisely we prove the following lemma. Lemma 2.1.
Let c ( x, y ) = − ln( xy − . A set G ⊂ R + × R + is c -cyclically monotone if andonly if for any ( x , y ) , ( x , y ) ∈ G one has ( x − x )( y − y ) ≤ .Proof. In one direction, assume G is c -cyclically monotone and let ( x , y ) , ( x , y ) ∈ G . Inparticular x y > x y >
1. The c -cyclic monotonicity implies c ( x , y ) + c ( x , y ) ≤ c ( x , y ) + c ( x , y ) , S. ARTSTEIN-AVIDAN, S. SADOVSKY, K. WYCZESANY which we rewrite asln( x y −
1) + ln( x y − ≤ ln( x y −
1) + ln( x y − . We may assume that x y > x y >
1, or else there is nothing to prove. We thus knowthat ( x y − x y − ≤ ( x y − x y − x − x )( y − y ) ≤
0, as claimed.For the other direction, given some G which satisfies that for any ( x , y ) , ( x , y ) ∈ G one has ( x − x )( y − y ) ≤
0, pick some m -tuple in G . It is enough to show that thesubset { ( x i , y i ) : i ∈ [ m ] } is c -cyclically monotone. However, as this is a finite subset, onemay consider all the sums C ( σ ) = P mi =1 c ( x i , y σ ( i ) ) for some permutation σ ∈ S m of theset [ m ]. The permutation with minimal cost C ( σ ) (not necessarily unique) will satisfy, bydefinition, that the set { ( x i , y σ ( i ) ) : i ∈ [ m ] } is c -cyclically monotone. In particular, by thefirst direction, we will have ( x i − x j )( y σ ( i ) − y σ ( j ) ) ≤
0. However, there is only one decreasingrearrangement of a set { y i : i ∈ [ m ] } , and we know already that the identity permutation Id is such a rearrangement, so in particular C ( Id ) is minimal. We conclude that G is c -cyclicallymonotone. (cid:3) Our example set G will be the following G = { ( x, y ) : ≤ x < , y = 3 − x } ∪ { ( , ) } . It is easy to check that G satisfies the condition in Lemma 2.1, namely it is c -cyclicallymonotone for the cost c ( x, y ) = − ln( xy − x, y ) = ( , ) and( z, w ) = ( , ), we may always add a third point ( x , y ) ∈ G such that the followingexpression c ( x, y ) − c ( x , y ) + c ( x , y ) − c ( z, y )is arbitrarily large.Indeed, picking ( x , y ) = ( t, − t ), compute c ( x, y ) − c ( x , y ) + c ( x , y ) − c ( z, y )= ln 8 + ln( 3 t − − ln(( t − − t )) + ln( 72 − t ) ≥ ln 8 − ln 2 + ln 52 − ln(( t − − t )) . As t → − , we have that ( t − − t ) → + . In particular the term depending on t in thelower bound for the expression tends to + ∞ , and there can be no upper bound for it whichdoes not depend on t , but only on the endpoints ( , ) and ( , ).This excludes the possibility for the existence of a potential, since if there existed some ϕ such that G ⊂ ∂ c ϕ , it would necessarily satisfy (as ( x, y ) ∈ G and ( x , y ) ∈ G ) that c ( x, y ) − c ( x , y ) ≤ ϕ ( x ) − ϕ ( x ) and c ( x , y ) − c ( z, y ) ≤ ϕ ( x ) − ϕ ( z ) , so that in particular c ( x, y ) − c ( x , y ) + c ( x , y ) − c ( z, y ) ≤ ϕ ( x ) − ϕ ( z ) , ROCKAFELLAR-TYPE THEOREM FOR NON-TRADITIONAL COSTS 7 getting a bound for the aforementioned expression which does not depend on x . We havethus shown that G is a c -cyclically monotone set which does not admit a potential. Remark 2.2.
From the transportational point of view it should be mentioned that c -cyclicmonotonicity is also not sufficient for optimality of the plan in the most general setting.Indeed, in [2] an example is given of a transport plan with respect to a non-traditional cost,which is not optimal yet it is supported on a c -cyclically monotone set. A Rockafellar-R¨uschendorf result for non-traditional costs
As we have demonstrated in the previous section, for non-traditional costs it may happenthat a set is c -cyclically monotone but fails to have a potential ϕ in the c -class, even when c is continuous and the spaces considered are simply R n . The example in Section 2.7 gives aclue as to what condition to consider. We call it c -path-boundedness. Definition 3.1.
Fix sets
X, Y and c : X × Y → ( −∞ , ∞ ] . A subset G ⊂ X × Y willbe called c -path-bounded if c ( x, y ) < ∞ for any ( x, y ) ∈ G , and for any ( x, y ) ∈ G and ( z, w ) ∈ G , there exists a constant M = M (( x, y ) , ( z, w )) ∈ R such that the following holds:For any m ∈ N and any { ( x i , y i ) : 2 ≤ i ≤ m − } ⊂ G , denoting ( x , y ) = ( x, y ) and ( x m , y m ) = ( z, w ) , we have m − X i =1 (cid:0) c ( x i , y i ) − c ( x i +1 , y i ) (cid:1) ≤ M. It is not hard to see that a c -path-bounded set must be c -cyclically monotone (indeed, if( x, y ) = ( z, w ) then if there is some path for which the sum is positive, one can duplicateit many times to get paths with arbitrarily large sums). It is also not hard to check (usingthe same reasoning we used in the example above) that c -path-boundedness is a necessarycondition for the existence of a potential (we do this in Section 3.3). Our main theorem isthat the condition of c -path-boundedness is in fact equivalent to the existence of a potential. Theorem 3.2.
Let
X, Y be two arbitrary sets and c : X × Y → ( −∞ , ∞ ] an arbitrary costfunction. For a given subset G ⊂ X × Y there exists a c -class function ϕ : X → [ −∞ , ∞ ] such that G ⊂ ∂ c ϕ if and only if G is c -path-bounded. Reformulation of the problem.
One can reformulate the problem of finding a po-tential for a given set G ⊂ X × Y as a question regarding the existence of a solution to alinear system of inequalities (possibly infinitely many of them). Theorem 3.3.
Let c : X × Y → ( −∞ , ∞ ] be a cost function and let G ⊂ X × Y . Then thereexists a potential for G , namely a c -class function ϕ : X → [ −∞ , ∞ ] such that G ⊂ ∂ c ϕ ifand only if the following system of inequalities, (3) c ( x, y ) − c ( z, y ) ≤ ϕ ( x ) − ϕ ( z ) , indexed by ( x, y ) , ( z, w ) ∈ G , has a solution ϕ : P X G → R , where P X G = { x ∈ X : ∃ y ∈ Y, ( x, y ) ∈ G } . S. ARTSTEIN-AVIDAN, S. SADOVSKY, K. WYCZESANY
Proof.
Assume that there exists a potential ϕ : X → [ −∞ , ∞ ] such that G ⊂ ∂ c ϕ . We mayrestrict ϕ to P X G , on which it must attain only finite values, because G ⊂ ∂ c ϕ means inparticular that ϕ ( x ) + ϕ c ( y ) = c ( x, y ) < ∞ . For every z ∈ P X G we have ϕ ( z ) = inf w ∈ Y ( c ( z, w ) − ϕ c ( w )) ≤ c ( z, y ) − ϕ c ( y ) . At the same time, since ( x, y ) ∈ ∂ c ϕ , ϕ ( x ) = inf w ∈ Y ( c ( x, w ) − ϕ c ( w )) = c ( x, y ) − ϕ c ( y ) . Taking the difference of the two equations we get c ( x, y ) − c ( z, y ) ≤ ϕ ( x ) − ϕ ( z ) . For the other direction, assume we have a solution to the system of inequalities. We wouldlike to extend it to some c -class function defined on X . To this end let˜ ϕ ( z ) = inf ( x,y ) ∈ G { c ( z, y ) − c ( x, y ) + ϕ ( x ) } . We need to show that the function ˜ ϕ , which is clearly in the c -class, satisfies that it is anextension of the original function ϕ : P X G → R , and that it is a potential, namely G ⊂ ∂ c ˜ ϕ .The assumption (3) implies that for z ∈ P X G we have ϕ ( z ) ≤ c ( z, y ) − c ( x, y ) + ϕ ( x )and so the infimum in the definition of the extended ˜ ϕ is attained at z itself. In particular,we get that ˜ ϕ is indeed an extension of the original function ϕ . This means that if ( x, y ) ∈ G then ˜ ϕ c ( y ) = inf z ∈ X ( c ( z, y ) − ˜ ϕ ( z ))= inf z ∈ X sup ( x ′ ,y ′ ) ∈ G ( c ( z, y ) − c ( z, y ′ ) + c ( x ′ , y ′ ) − ϕ ( x ′ )) ≥ inf z ∈ X ( c ( z, y ) − c ( z, y ) + c ( x, y ) − ϕ ( x ))= c ( x, y ) − ϕ ( x ) = c ( x, y ) − ˜ ϕ ( x ) . As the opposite inequality is trivial, we get ( x, y ) ∈ ∂ c ˜ ϕ , as required. (cid:3) The above theorem, while very simple in nature, reduces the question of finding a potential tothe question of determining when a set of linear inequalities has a solution. The index set forthe inequalities are pairs (( x, y ) , z ) ∈ G × P X G (or, equivalently, pairs (( x, y ) , ( z, w )) ∈ G × G ,where we ignore w as it does not appear in the inequalities). The solution vector we arelooking for is indexed by P X G , and we denote it ( ϕ ( x )) x ∈ P X G . In fact, formally, we shouldbe using ( ϕ ( x, y )) ( x,y ) ∈ G , which seems to allow multi-valued ϕ . However, note that if ( x, y )and ( x, y ′ ) are both in G then c ( x, y ) − c ( x, y ) ≤ ϕ ( x, y ) − ϕ ( x, y ′ )and c ( x, y ′ ) − c ( x, y ′ ) ≤ ϕ ( x, y ′ ) − ϕ ( x, y ) ROCKAFELLAR-TYPE THEOREM FOR NON-TRADITIONAL COSTS 9 which means ϕ ( x, y ) = ϕ ( x, y ′ ) . In other words, even if we do index the vector by ( x, y ) ∈ G instead of x ∈ P X G , the solutionvector depends only on the first coordinate.3.2. Solutions for families of linear inequalities.
Our main theorem will follow fromthe next theorem regarding systems of linear inequalities.
Theorem 3.4.
Let { α i,j } i,j ∈ I ∈ [ −∞ , ∞ ) , where I is some arbitrary index set, and with α i,i = 0 . The system of inequalities (4) α i,j ≤ x i − x j , i, j ∈ I has a solution if and only if for any i, j ∈ I there exists some constant M ( i, j ) such that forany m and any i , · · · , i m − , letting i = i and j = i m one has that P m − k =1 α i k ,i k +1 ≤ M ( i, j ) . Instead of proving the theorem directly, we shall prove the following theorem, which at firstglance might seem weaker.
Theorem 3.5.
Let { a i,j } i,j ∈ I ∈ [ −∞ , ∞ ) , where I is some arbitrary index set. Assume thatfor any m ≥ and any i , i , · · · , i m it holds that a i ,i m ≥ P m − k =1 a i k ,i k +1 . Then the system ofinequalities a i,j ≤ x i − x j , i, j ∈ I has a solution. Clearly, Theorem 3.4 implies Theorem 3.5. In fact, the reverse implication holds as well. Wewill show this by proving Theorem 3.4 under the assumption of Theorem 3.5.
Proof that Theorem 3.5 implies Theorem 3.4.
The “only if” part of Theorem 3.4 is easy anddoes not require Theorem 3.5. Indeed, let { α i,j } i,j ∈ I ∈ [ −∞ , ∞ ), where I is some arbitraryindex set, and with α i,i = 0. Assume that the system of inequalities α i,j ≤ x i − x j , i, j ∈ I has a solution, ( x i ) i ∈ I . Summing the relevant inequalities we see that M ( i, j ) = x i − x j provides the required bound.For the opposite direction, we will use Theorem 3.5. Assume that for any i, j ∈ I there existssome M ( i, j ) such that for any m and any { i k } m − k =2 , letting i = i and i m = j it holds that m − X k =1 α i k ,i k +1 ≤ M ( i, j ) . Define new constants a i,j ∈ [ −∞ , ∞ ) as follows: a i,j = sup { m − X k =1 α i k ,i k +1 : m ∈ N , m ≥ , i , . . . , i m − ∈ I } . By the above condition, the right hand side is bounded from above and so the supremum isnot + ∞ .We first claim that the system of equations a i,j ≤ x i − x j , satisfies the conditions of Theorem3.5. Assume we are given i , i , · · · , i m − , i m , and we want to prove that a i ,i m ≥ P m − k =1 a i k ,i k +1 .Fix ε >
0. For each k ∈ [ m ] use the definition of a i k ,i k +1 to pick some m k and i ( k )2 , . . . , i ( k ) m k − such that, letting i ( k )1 = i k and i ( k ) m k = i k +1 , we have a i k ,i k +1 ≤ m k − X l =1 α i ( k ) l ,i ( k ) l +1 + ε/m. We have thus identified some finite set of indices in I , the set J = { i k , i ( k )2 , . . . , i ( k ) m k − : k ∈ [ m − } ∪ { i m } , which is naturally arranged as a path from i to i m . Using again the definition of a i,j , thepath thus defined participates in the supremum, and we have that a i ,i m ≥ m X k =1 ( a i k ,i k +1 − ε/m ) = (cid:0) m X k =1 a i k ,i k +1 (cid:1) − ε. As this holds for any ε , we get the inequality in the condition of Theorem 3.5.Applying Theorem 3.5, we see that the system of inequalities(5) a i,j ≤ x i − x j , admits a solution. Moreover, since a i,j ≥ α i,j by definition, the resulting vector x is also asolution of the original system of inequalities. (cid:3) Having made the reduction from Theorem 3.4 to Theorem 3.5, we proceed by proving thelatter.
Proof of Theorem 3.5.
We use Zorn’s Lemma. Consider the partially ordered set of pairs(
J, f J ) where J ⊂ I and f J : J → R are such that for any i, j ∈ J we have f J ( i ) − f J ( j ) ≥ a i,j .We know the set is non-empty because it contains pairs ( { i } , c ). The partial order we consideris ( J, f J ) ≤ ( K, f K ) if J ⊂ K and f K | J = f J .First let us notice that every chain has an upper bound. Assume ( J α , f J α ) α ∈ A is a chain(namely any two elements are comparable). Consider J = ∪ α J α and f J = ∪ α f J α . Thisfunction is well defined because of the chain properties (at a point i ∈ J it is defined as f J α ( i )for any α with i ∈ J α ). The pair ( J, f | J ) is in our set because if i, j ∈ J then for some α wehave i, j ∈ J α , so f | J α satisfies the inequality on f J ( i ) − f J ( j ) ≥ α i,j and so does f J . Finally,( J, f J ) is clearly an upper bound for the chain. So, we have shown that every chain has anupper bound, and we may use Zorn’s lemma to find a maximal element. Denote the maximalelement by ( J , f J ).Assume towards a contradiction that J = I , that is, there exists some element i ∈ I suchthat i J . If we are able to extend f J to be defined on { i } in such a way that allinequalities with indices of the form ( i , j ) and ( j, i ) with j ∈ J still hold, we will contradict ROCKAFELLAR-TYPE THEOREM FOR NON-TRADITIONAL COSTS 11 maximality and complete the proof. Note that the inequalities that need to be satisfied inorder to extend the function are a i ,j ≤ f ( i ) − f ( j ) and a j,i ≤ f ( j ) − f ( i ) . That is, f ( i ) is to be defined in such a way thatsup j ∈ J ( a i ,j + f J ( j )) ≤ f ( i ) ≤ inf j ∈ J ( f J ( j ) − a j,i ) . For there to exist such an element, f J must satisfy that(6) sup j ∈ J ( a i ,j + f J ( j )) ≤ inf j ∈ J ( f J ( j ) − a j,i ) , or, in other words, that for any j, k ∈ J a i ,j + f J ( j ) ≤ f J ( k ) − a k,i . We can rewrite the condition as a i ,j + a k,i ≤ f J ( k ) − f J ( j ). Recall that under our assump-tions a k,j ≥ a k,i + a i ,j . Since f J already satisfies the inequality a k,j ≤ f J ( k ) − f J ( j ), weknow the above inequality holds for any j, k , and so the inequality (6) holds and we mayextend the function f J . This is a contradiction to the maximality, and we conclude J = I ,so that we have found a solution to the full system of inequalities. (cid:3) Summary.
The reader will have noticed that the above arguments already provide aproof for the “if” part of Theorem 3.2. The other direction is simple, and so we present thecomplete proof below.
Proof of Theorem 3.2.
Assume G ⊂ X × Y satisfies that for some c -class ϕ : X → [ −∞ , ∞ ], G ⊂ ∂ c ϕ , then c ( x, y ) − c ( z, y ) ≤ ϕ ( x ) − ϕ ( z )holds for all ( x, y ) , ( z, w ) ∈ G . So, given a pair ( x, y ) , ( z, w ) ∈ G we set M = M (( x, y ) , ( z, w )) = ϕ ( x ) − ϕ ( z ). Any sum, as in the definition of c -path-boundedness, will be bounded by thecorresponding sum of differences ϕ ( x i ) − ϕ ( x i +1 ), which make for a telescopic sum adding upto M . So we conclude that G is c -path-bounded.Assume, in the other direction, that G ⊂ X × Y is c -path-bounded. By Theorem 3.3 weneeded to show that the family of inequalities c ( x, y ) − c ( z, y ) ≤ ϕ ( x ) − ϕ ( z ) , where ( x, y ) , ( z, w ) ∈ G , has a solution. By Theorem 3.4, as G is c -path-bounded, a solutionexists. (cid:3) Rockafellar-R¨uschendorf theorem.
As a corollary we have a new and simple prooffor the Rockafellar-R¨uschendorf theorem for traditional costs.
Corollary 3.6 (Rockafellar-R¨uschendorf) . Let c : X × Y → R be a traditional (i.e. finitelyvalued) cost function. Assume we are given a set G ⊂ X × Y which is c -cyclically monotone.Then there exists a c -class function ϕ : X → [ −∞ , ∞ ] such that G ⊂ ∂ c ϕ .Proof. In order to apply Theorem 3.2 we proceed to show that G is c -path-bounded. Let( x, y ) , ( z, w ) ∈ G , and let M := M (( x, y ) , ( z, w )) = c ( x, w ) − c ( w, z ). To show this indeedsatisfies the condition, namely that there is an upper bound on the total cost of any path,we use the c -cyclic monotonicity. Let m ≥ x , y ) , . . . , ( x m − , y m − ) ∈ G . Denote( x, y ) = ( x , y ) and ( z, w ) = ( x m , y m ). The condition of c -cyclic monotonicity implies that m − X i =1 ( c ( x i , y i ) − c ( x i +1 , y i )) + c ( x m , y m ) − c ( x , y m ) ≤ . In particular, m − X i =1 ( c ( x i , y i ) − c ( x i +1 , y i )) ≤ c ( x, w ) − c ( z, w ) . (cid:3) It is important to note that we relied heavily on the fact that c ( x, w ) < ∞ , otherwise thisupper bound might be infinite, and therefore meaningless.4. Special cases
We now show that under certain assumptions, c -path-boundedness is implied by c -cyclicmonotonicity, even for non-traditional costs. This is motivated by the fact that usually, whenconsidering optimal transport plans, they are concentrated on sets which are automatically c -cyclically monotone (e.g. using a result of Pratelli [20], under the assumption that c iscontinuous). So, it is useful to indicate cases in which the c -cyclical monotonicity propertyimplies that the set is also c -path-bounded.In this section we collect several such results. An important ingredient in these results is thefollowing observation, which is very similar to the one we used to prove Corollary 3.6.Consider a c -cyclically monotone set G . With it, we may associate a directed graph witha vertex set G in which there is an edge pointing from ( x, y ) to ( z, w ) if c ( z, y ) < ∞ . Onthe vertex set of this graph (namely on G ) we define an equivalence relation ∼ , where twopoints are equivalent, ( x, y ) ∼ ( z, w ), if there exists a directed cycle passing through both(or, equivalently, if there is a directed path from each of the points to the other). This isclearly an equivalence relation. If the points ( x s , y s ) and ( x f , y f ) are in the same equivalenceclass, then there is a constant M as required in Theorem 3.2. Indeed, fix an arbitrary path ROCKAFELLAR-TYPE THEOREM FOR NON-TRADITIONAL COSTS 13 from ( x f , y f ) to ( x s , y s ), say ( z , w ) , . . . , ( z k , w k ), the edges of which are graph edges, and let M = − " c ( x f , y f ) − c ( z , y f ) + k − X j =1 (cid:0) c ( z i , w i ) − c ( z i +1 , w i ) (cid:1) + c ( z k , w k ) − c ( x s , w k ) < ∞ . Any path from ( x s , y s ) to ( x f , y f ) (paths not on the graph have total cost −∞ ) is completedto a cycle using the above path, and using c -cyclic monotonicity we have for any m and any( x i , y i ) mi =2 ⊂ Gc ( x s , y s ) − c ( x , y s ) + m − X i =2 (cid:0) c ( x i , y i ) − c ( x i +1 , y i ) (cid:1) + c ( x m , y m ) − c ( x f , y m ) ≤ M. Summarizing, we have proved the following proposition.
Proposition 4.1.
Let c : X × Y → ( −∞ , ∞ ] be some cost function and let G ⊂ X × Y bea c -cyclically monotone set. Assume the equivalence relation ∼ defined above had just oneequivalence class. Then G is c -path-bounded. Corollary 4.2.
Let c : X × Y → ( −∞ , ∞ ] be a continuous cost function on separable metricspaces X, Y and let G ⊂ X × Y be c -cyclically monotone and path connected. Let T > andassume G satisfies that c ( x, y ) ≤ T for all ( x, y ) ∈ G . Then G is c -path-bounded.Proof. It is enough to show that any two points ( x, y ) , ( z, w ) ∈ G satisfy ( x, y ) ∼ ( z, w ). Bypath connectivity we may find a continuous γ : [0 , → G with γ (0) = ( x, y ) and γ (1) =( z, w ). The compact set γ ([0 , c ) set { ( x, y ) : c ( x, y ) = ∞} . In particular, there exists some δ > | ( η, ξ ) − γ ( t ) | < δ then, denoting γ ( t ) = ( x t , y t ), we have c ( x t , ξ ) < ∞ . Since γ is uniformly continuous, wemay find m and t = 0 < t < · · · < t m = 1 such that | γ ( t j ) − γ ( t j − ) | < δ for j = 1 , . . . , m .Denote γ ( t j ) = ( x j , y j ). Since | ( x j +1 , y j ) − ( x j , y j ) | ≤ | ( x j +1 , y j +1 ) − ( x j , y j ) | < δ , we get that c ( x j +1 , y j ) < ∞ , which means that the path ( γ ( t j )) mj =1 connects the points ( x, y ) and ( z, w )in the graph. By symmetry, we get that any two points are connected and there is only oneequivalence class for the relation ∼ . Applying Proposition 4.1, the proof is complete. (cid:3) As may be apparent from the proof of the corollary, the connectedness of G is not elemental,and we may replace it with other assumptions, so long as these imply that there is onlyone equivalence class for ∼ . Another useful variant, in which there may be more then oneequivalence class, is the following. Proposition 4.3.
Let c : X × Y → ( −∞ , ∞ ] be a continuous cost function on separablemetric spaces X, Y and let G ⊂ X × Y be c -cyclically monotone and bounded. Let T > and assume that for every ( x, y ) ∈ G we have that c ( x, y ) ≤ T . Then G is c -path-bounded.Proof. By the argument given at the beginning of this section, we only need to address pairswhich lie in different equivalence classes of the relation ∼ , that is, show that for such pairs abound on the total cost of a path between them exists. We first observe that under the assumptions we have made, there are only finitely manyequivalence classes for ∼ . Indeed, let S T = { ( x, y ) : c ( x, y ) ≤ T } , then by continuity of c on the compact set G ⊂ S T , we can find some δ > | ( z, w ) − ( x, y ) | < δ and( x, y ) , ( z, w ) ∈ G then max( c ( x, w ) , c ( z, y )) < T + 1. In particular, any two points in G whosedistance is less than δ belong to the same equivalence class. Using compactness of G again,we may cover it with a finite number of δ -balls, so there can be no more than a finite numberof different equivalence classes. Denote the number of equivalence classes of ∼ by k ∈ N .Next, we claim that for each equivalence class ( denoted by [ v ]) there exists a bound M = M ([ v ]) (depending only on the equivalence class) such that for any two points ( x s , y s ) and( x f , y f ) in [ v ], the total cost of any path between then is bounded by M . To this end definethe function F : [ v ] × [ v ] → R to be the supremum over the total cost on any path from thefirst given point to the second one, namely let F (( x s , y s ) , ( x f , y f )) be given bysup ( c ( x s , y s ) − c ( x , y s ) + m − X k =1 ( c ( x k , y k ) − c ( x k +1 , y k )) + c ( x m , y m ) − c ( x f , y m ) ) where the supremum runs over all m ∈ N and any ( x i , y i ) mi =1 ∈ G . We have already shown(using c -cyclic monotonicity and the definition of the relation ∼ ) that F is finite. To seethat it is bounded, it suffices to show that F is uniformly continuous (since the domain isbounded as well).Let ( x s , y s ) , ( x f , y f ) ∈ [ v ]. Given some path ( x i , y i ) mi =1 joining ( x ′ s , y ′ s ) and ( x ′ f , y ′ f ), we mayadd to it the two points ( x s , y s ) and ( x f , y f ) as the first and last points, getting a new pathbetween ( x s , y s ) and ( x f , y f ). We thus see that F (( x ′ s , y ′ s ) , ( x ′ f , y ′ f )) + c ( x s , y s ) − c ( x ′ s , y s ) + c ( x ′ f , y ′ f ) − c ( x f , y ′ f ) ≤ F (( x s , y s ) , ( x f , y f )) . However, as c is continuous on the compact set G ⊂ S T , it is uniformly continuous, and forany ε > δ = δ ( ε ) such that if | ( x s , y s ) − ( x ′ s , y ′ s ) | < δ and | ( x f , y f ) − ( x ′ f , y ′ f ) | < δ then | c ( x s , y s ) − c ( x ′ s , y s ) | < ε/ (cid:12)(cid:12) c ( x ′ f , y ′ f ) − c ( x f , y ′ f ) (cid:12)(cid:12) < ε/ F (( x ′ s , y ′ s ) , ( x ′ f , y ′ f )) − F (( x s , y s ) , ( x f , y f )) ≤ ε. By symmetry of F in its arguments, we get that F is indeed uniformly continuous on [ v ] × [ v ],and in particular bounded. Denote this bound by M ([ v ]).Finally, by the definition of our equivalence relation, any path joining two points in G canbe split into at most k paths, each one within one of the equivalence classes, and at most k − z, w ) ∈ [ v ] and ( z ′ , w ′ ) ∈ [ v ], notice first that the cost c is bounded from below on P X G × P Y G by continuity. Therefore, c ( z, w ) − c ( z ′ , w ) ≤ T − inf { c ( x, w ) : ( x, y ) ∈ G, ( z, w ) ∈ G } =: M . To sum up, denoting the k equivalence classes by ([ v i ]) k i =1 , the total cost for any path in G ⊂ S T is bounded from above by P k i =1 M ([ v i ]) + ( k − M . As a result we have a ROCKAFELLAR-TYPE THEOREM FOR NON-TRADITIONAL COSTS 15 uniform bound for any path with any beginning and end point in G , that is, G as a whole is c -path-bounded (in fact, uniformly, which is a much stronger statement). (cid:3) Summing up, we have seen that under additional geometric or topological conditions, c -cyclicmonotonicity is in fact enough, and does imply c -path-boundedness (and in particular, theexistence of a potential). For example, in the results of [15], where the assumptions on thecost are that it is continuous, and infinite only on the diagonal, it is easy to check that acompact c -cyclically monotone set satisfies the conditions in Proposition 4.3.5. An alternative route
The course of proving our main theorem was not as linear as presented in this note. One ofthe methods we developed was more hands on, but worked only for countable sets G . Upontrying to generalize it to uncountable ones, we discovered the current proof. However, theprevious proof we had, which entails much more combinatorics, is interesting in its own right,and the methods we found may be useful in other scenarios, so we present it here as well.5.1. Graph theoretical component of the proof.
We shall make use of a decompositionof a nearly balanced weighted acyclic directed graph into weighted paths. A directed graphis called acyclic if there are no directed cycles in the graph. We call a weighted graph nearlybalanced if we have some control over the difference between the in-coming and out-comingtotal weight in each vertex. We assume that all the weights are non-negative.
Proposition 5.1.
Let
Γ = (
V, E, ( w e ) e ∈ E ) be a finite directed weighted acyclic graph. Assumethat it is almost balanced in the following sense: for some fixed vector ( ε v ) v ∈ V with non-negative entries, we have for every vertex v ∈ V that X ( x,v ) ∈ E w ( x,v ) − X ( v,y ) ∈ E w ( v,y ) ∈ [ − ε v , ε v ] . Then there exists a weighted decomposition of Γ into paths P k = v ( k )1 → v ( k )2 → · · · → v ( k ) m k with equal weights µ k for each edge in P k , such that (7) w e = X k : e ∈ P k µ k and X k µ k < X ε i . Moreover, it holds individually for each v ∈ V that (8) X { k : v = s k or v = f k } µ k < ε v . Here s k = v ( k )1 denotes the starting vertex of the path P k and f k = v ( k ) m k denotes its end point.In fact, if P ( x,v ) ∈ E w ( x,v ) ≤ P ( v,y ) ∈ E w ( v,y ) then v = f k for any k and if P ( x,v ) ∈ E w ( x,v ) ≥ P ( v,y ) ∈ E w ( v,y ) then v = s k for any k . Remark 5.2.
Note that (8) implies (7) , which can be seen by summing the inequalities in (8) over all v ∈ V . Then the right hand side becomes P v ε v , while the left hand side is X v X { k : s k = v or f k = v } µ k , and since every path has precisely one starting point s k and one final point f k , we get thateach µ k was summed twice, that is we get exactly P µ k .Proof of Proposition 5.1. Note that if | V | = 2 then the claim is trivial as we can use justone path. We then have that ε = ε = w (1 , = µ , where we used 1 and 2 as labels of thevertices, and assumed the only edge is (1 , V is any set and | E | = 1 then the situationis exactly as in the first case we considered and there is nothing to prove.Assume we know the claim for | E | < k and we are given a graph Γ with | E | = k . Consideran edge e ∗ = ( x, y ) with minimal weight w ∗ and pick a maximal path P which includes it(maximal in the sense that it cannot be extended to a longer path), say s = v → v → · · · → v m = f . Maximality implies that there is no outgoing edge from its end vertex f = v m , andno edge going into its start vertex s = v . In particular, the “almost balanced” restrictionon s reads P ( s,y ) ∈ E w ( s,y ) < ε s and on f reads P ( x,f ) ∈ E w ( x,f ) < ε f . Moreover, since w ∗ wasa minimal weight in the whole graph, it follows that ε s > w ∗ and ε f > w ∗ .Define Γ ′ to be a graph with the same vertices V and edges e ∈ E , whose weights are definedas w ′ e = ( w e − w ∗ if e ∈ P , w e otherwise.Since w ∗ was chosen as the minimal weight, we see that all new weights remain non-negative.Note that the edge e ∗ = ( x, y ) now has weight zero and thus can be omitted. Therefore,the graph Γ ′ is a directed weighted acyclic graph with at most k − ε ′ v given by ε ′ s = ε s − w ∗ , ε ′ f = ε f − w ∗ , and ε ′ v = ε v for v ∈ V \ { s, f } . Note that P v ∈ V ε ′ v = P v ∈ V ε v − w ∗ .By the induction assumption, the new graph Γ ′ has a weighted decomposition: that is, wecan find paths ( P k ) k ∈ S and weights µ k such that P { k : e ∈ P k } µ k = w ′ e and for every v ∈ V , wehave X { k : s k = v or f k = v } µ k ≤ ε ′ v . We add to the collection the path P with a weight w ∗ on each edge. We claim that thisconstitutes the desired weighted decomposition of Γ.Indeed, if we compute P { k : s k = v or f k = v } µ k for a vertex which is neither s nor f , i.e. not anend point of P , we get the same result as in Γ ′ and hence it is at most ε ′ v = ε v . If we compute ROCKAFELLAR-TYPE THEOREM FOR NON-TRADITIONAL COSTS 17 the sum for v = s or v = f , we get the sum in Γ ′ with added w ∗ , which is thus bounded by ε ′ v + w ∗ = ε v , as needed.Finally, by construction, a vertex can be chosen as a starting vertex s k for some path P k onlyif, after equal weights were removed from its inwards and outwards pointing edges, therewas no weight left in the inwards pointing edges. In other words, only if P ( x,v ) ∈ E w ( x,v ) < P ( v,y ) ∈ E w ( v,y ) . Similarly, a vertex can be chosen as f k for some path P k only if P ( x,v ) ∈ E w ( x,v ) > P ( v,y ) ∈ E w ( v,y ) , which completes the proof. (cid:3) A result of Ky Fan.
For the proof, we use the following result of Ky Fan [14]:
Theorem 5.3 (Ky Fan) . Let E be a locally convex, real Hausdorff vector space. Let x ν ∈ E bean indexed set of vectors with indices ν ∈ I , and let α ν ∈ R . Then the system of inequalities f ( x ν ) ≥ α ν for ν ∈ I has a solution f ∈ E ∗ (that is, f a continuous linear functional on E )is and only if the point (0 , ∈ E × R does not belong to the closed convex cone C ∈ E × R spanned by the elements { ( x ν , α ν ) : ν ∈ I } . Since the proof is a direct application of Hahn Banach theorem, we include it here forcompleteness.
Proof of Theorem 5.3.
Denote by C the closed convex cone spanned by ( x ν , α ν ). Assume thesystem has a solution f , then f × ( − id R ) ≥ C . On the other hand f × ( − id R )(0 ,
1) = − , C .For the other direction, we use the Hahn-Banach theorem to separate (0 ,
1) from the closedconvex cone C . That is, we find a linear functional ( h, a ) ∈ E ∗ × R such that for any( x, α ) ∈ C one has h ( x ) + aα ≥ b > h (0) + a = a . Since C is a cone, (0 , ∈ C and so0 ≥ b > a . In particular we see that h ( x ) + aα ≥ b can be written as f ( x ) − α ≥ b/ ( − a ) ≥ f = h/ ( − a ) ∈ E ∗ . That is, we have found a solution f satisfying f ( x ν ) ≥ α ν for all ν . (cid:3) Completing the (alternative) proof.
We are ready to provide our alternative prooffor Theorem 3.5, in the case where the family of inequalities is countable. This in turn impliesa countable version of Theorem 3.4 and thus a countable version for Theorem 3.2.
Proof of Theorem 3.5.
We will use Theorem 5.3 for the space R I with the box topology.It is clearly Hausdorff and locally convex. The convex cone is generated by the vectors( e i − e j , a i,j ), and so we must show that the point (0 ,
1) has a neighborhood separated fromthis cone.The neighborhood we pick is of the form Q i ∈ I ( − ε i , ε i ) × (1 / , ∞ ), where the sequence ε i willbe chosen in a way which depends only on { a i,j } i,j ∈ I .To define the neighborhood, using that I is countable we fix an ordering ≤ of it, and forevery i ∈ I we define ε i = 15 2 − i { a k,j : k ≤ i, j ≤ i } + 1 . Note that ε i > i . Then, towards a contradiction, assume that the set Q i ∈ I ( − ε i , ε i ) × (1 / , ∞ ), which is anopen neighborhood of (0 , ∈ R I × R in the box topology, intersects with the closed convexcone generated by the vectors ( e i − e j , a i,j ). This means that there is some finite J ⊂ I ,and a positive combination P i,j ∈ J λ i,j ( e i − e j , a i,j ) which is inside this neighborhood. Thiscondition amounts to(9) X j ∈ J λ i,j − λ j,i ∈ ( − ε i , ε i ) ∀ i ∈ J and X i,j ∈ J λ i,j a i,j ≥ / . Let Λ be the matrix with entries λ i,j . Note that subtracting any positive multiple of apermutation matrix from the matrix Λ has no effect on the sum on the left and only increases(by cyclic monotonicity) the sum on the right. Thus we may assume without loss of generalitythat the matrix Λ is not larger (entry-wise) than any positive multiple of a permutationmatrix.Consider the elements of J as vertices of a weighted directed graph Γ, where we definethe weights to be w ( i,j ) = λ i,j . The assumption that Λ contains no positive multiple of apermutation matrix implies that the graph Γ is acyclic. Moreover, the first condition in (9)means that the graph Γ is almost balanced, up to the weights ( ε i ), in the sense of Proposition5.1. Using Proposition 5.1 we find paths P k = v ( k ) i → v ( k ) i → · · · → v ( k ) i mk and weights µ k thatcover the graph Γ and satisfy for every vertex v i that X { k : s k = v i or f k = v i } µ k ≤ ε i (where we let s k = v ( k ) i and f k = v ( k ) i mk as before).Denote by A k the adjacency matrix of the path P k , so that Λ = P µ k A k . Then X i,j ∈ J λ i,j a i,j = X k µ k X i,j ∈ J ( A k ) i,j a i,j . Moreover, by the assumption on a i,j in the statement of the theorem we are proving, andsince ( A k ) i,j ∈ { , } and are indicating a path, we get that X i,j ∈ J ( A k ) i,j a i,j ≤ a s k ,f k . Therefore, X i,j ∈ J λ i,j a i,j ≤ X k µ k a s k ,f k =: S. Let us now decompose the sum S according to the start and end vector of the path P k in thefollowing way. Fix an ordering of the (finite number of) elements in J , and write S = | J | X l =1 X { k : max( s k ,f k )= l } µ k a s k ,f k . Indeed, each path is summed exactly once, according to the quantity l = max( s k , f k ). ROCKAFELLAR-TYPE THEOREM FOR NON-TRADITIONAL COSTS 19
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