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2022. Т. 41. С. 96—106
Онлайн-доступ к журналу: http://mathizv.isu.ru
Серия «Математика»
Research article УДК 517.9
MSC 49Q22, 28C15, 28A33, 46E27
DOI https://doi.org/10.26516/1997-7670.2022.41.96
Nonlinear Kantorovich Problems with a Parameter
Vladimir I. Bogachev1'2'3'4®, Ilyal. Malofeev2'3
1 Lomonosov Moscow State University, Moscow, Russian Federation
2 National Research University Higher School of Economics, Moscow, Russian Federation
3 Saint Tikhon's Orthodox University, Moscow, Russian Federation
4 Moscow Center for Fundamental and Applied Mathematics, Moscow, Russian Federation
El vibogach@mail.ru
Abstract. We consider nonlinear Kantorovich problems with marginal distributions and cost functions depending measurably on a parameter and prove that there exist optimal transportation plans that are also measurable with respect to the parameter. Unlike the classical linear Kantorovich problem of minimization of the integrals of a given cost function with respect to transportation plans, we deal with nonlinear cost functionals in which integrands depend on transportation plans. Dependence of cost functions on conditional measures of transportation plans is also allowed.
Keywords: Kantorovich problem, optimal plan, measurability with respect to a parameter
Acknowledgements: This research is supported by the Saint Tiknon's Orthodox University, the Foundation "Living tradition" and the Russian Foundation for Basic Research Grant 20-01-00432. The results of Section 2 (Theorem 1) have been supported by the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement N 075-15-2022-284. The results of Section 3 (Theorem 2) have been supported by the RSF grant N 19-71-30020.
For citation: Bogachev V. I., Malofeevl.I. Nonlinear Kantorovich Problems with a Parameter. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 41, pp. 96-106.
https://doi.org/10.26516/1997-7670.2022.41.96
Научная статья
Нелинейные задачи Канторовича с параметром В. И. Богачев1'2'3'4^, И. И. Малофеев2'3
1 Московский государственный университет им. М. В. Ломоносова , Москва, Российская Федерация
2 Национальный исследовательский университет «Высшая школа экономики», Москва, Российская Федерация,
3 Православный Свято-Тихоновский гуманитарный университет, Москва, Российская Федерация,
4 Московский центр фундаментальной и прикладной математики, Москва, Российская Федерация
KI vibogach@mail.ru
Аннотация. Рассматриваются нелинейные задачи Канторовича с маргинальными распределениями и функциями стоимости, измеримо зависящими от параметра. Доказывается существование оптимальных планов транспортировки, также измеримо зависящих от параметра. В отличие от классической линейной задачи Канторовича минимизации интегралов заданной функции стоимости по транспортным планам мы имеем дело с нелинейными функционалами стоимости, в которых подынтегральные функции зависят от транспортных планов. Допускается также зависимость от условных мер транспортных планов.
Ключевые слова: задача Канторовича, оптимальный план, измеримость по параметру
Благодарности: Работа выполнена при поддержке Православного Свято-Тихоновского университета, фонда «Живая традиция» и гранта Российского фонда фундаментальных исследований 20-01-00432. Результаты раздела 2 (Теорема 1) поддержаны Минобрнауки России в рамках реализации программы Московского центра фундаментальной и прикладной математики по соглашению N 075-15-2022-284. Результаты раздела 3 (Теорема 2) поддержаны грантом РНФ 19-71-3002.
Ссылка для цитирования: Bogachev V. I., MalofeevI.I. Nonlinear Kantorovich Problems with a Parameter // Известия Иркутского государственного университета. Серия Математика. 2022. Т. 41. C. 96-106. https://doi.org/10.26516/1997-7670.2022.41.96
1. Introduction
In this paper we study nonlinear Kantorovich problems with marginal distributions and cost functions depending measurably on a parameter. We consider measures on completely regular Luzin spaces, i.e., images of complete separable metric spaces under continuous injective mappings.
This important class of spaces contains all Borel subspaces of complete
separable metric spaces (see [5, Corollary 6.8.5]), but also many nonmetriz-able spaces encountered in applications. The parameter takes values in
a Souslin space (Souslin spaces are images of complete separable metric spaces under continuous mappings). For the classical linear Kantorovich problem the existence of optimal plans measurable with respect to the parameter has recently been proved in [9] under broad conditions. In particular, if X, Y are complete separable metric spaces or, more generally, completely regular Luzin spaces, a parameter set T is a Souslin space, a cost function h: T x X x Y ^ [0, is Borel measurable and for every fixed t £ T the function (x, у) ^ h(t, x, y) is lower semicontinuous (which means that the sets {(x,y): h(t,x,y) < c} are closed), then, for any given Borel mappings t ^ ^t and t ^ щ with values in the spaces V(X) and V(Y) of Borel probability measures on X and Y, where the spaces of measures are equipped with their weak topologies, one can find optimal Kantorovich plans at that are Borel measurable in t.
Recall (see [2], [8], [17], [18], and [19] for more details) that for a single Borel measurable cost function h: X x Y ^ [0, and two given measures ^ £ V(X) and v £ V(Y) an optimal Kantorovich plan for the triple (ц,, v, h) is a Borel probability measure on X x Y belonging to the set П(^, v) of measures with projections ^ and v on the factors and minimizing the integral of h with respect to measures from П(^, v). Such an optimal plan exists if h is lower semicontinuous. In other words, the value
Kh(p,v) := inf / hda стеП(p,v) JxxY
is attained.
A nonlinear Kantorovich problem, investigated recently by several authors (see [1], [3], [4], and [14]), deals with minimization of more general integrals of the form
Jh(&)= h(x,y,a) a(dxdy), (1.1)
Jx xY
where the cost function h: X x Y x V(X x Y) ^ [0, can now depend on the measure a with respect to which it is integrated. The term "weak transport cost" used in some of these works does not look appropriate. So we call problems of this new type "nonlinear" to emphasize that the cost functional is not linear with respect to the plan.
In the parametric version the cost function also depends on a parameter t from a Souslin space T. Our main result states that this more general problem has solutions measurably dependent on the parameter. There is an interesting special case of dependence of the cost function on the plan (actually, the cost functions considered in [1], [3], [4], and [14] are of this type):
h(x,y,a)= H (x,ax), (1.2)
where H is defined on X x V(Y) and ax are conditional measures for a with respect to its projection ax on X, that is, x ^ ax is a Borel mapping
from X to V(Y) such that
XxY
f (x, y) a(dx dy) = f (x, y) ax(dy) ax (dx)
for every bounded Borel function f on X x Y. Such cost functions have worse continuity properties, because conditional measures depend on a measurably, but not always continuously. It is also possible to define conditional measures ax as measures on X x Y concentrated on the sets {x} x Y, in which case the function H is defined on X x V(X x Y) and can be identified with h. Our second main result states that for cost functions of type (1.2) with a parameter there are also solutions depending measurably on this parameter, but here the convexity of h in a is additionally required as in the cited papers. For the classical Kantorovich problem measurable dependence on parameters has been studied in [7], [9], [10], [16], and [19], including the case of problems with pointwise constraints in the spirit of [15] (see also [12]).
Before giving exact formulations, we recall that the weak topology on the set V(X) of Borel probability measures on a completely regular topological space X is induced by the weak topology on the whole space M.(X) of signed Borel measures on X generated by all seminorms of the form
where f belongs to the space of bounded continuous functions on X. If X is a complete metric space, then V(X) with the weak topology is metriz-able with a complete metric, for example, one can use the Kantorovich-Rubinshtein norm
where Lip1(X) is the space of 1-Lipschitz functions. Recall that on a Souslin space every measure ^ e V(X) is Radon: for all Borel sets B one has
If X is a Luzin completely regular space, then V(X) is also Luzin in the weak topology and its compact subsets are metrizable (see [5, Theorem 8.9.6] or [6, Theorem 5.1.8]). In particular, every uniformly tight set is metrizable, that is, a set M c V(X) such that for every e > 0 there is a
2. Measurable plans for general nonlinear costs
ß(B) = sup{^(K) : K C B, K is compact}.
compact set K C X with /i(X\K) < e for all ^ £ M. Uniformly tight sets have compact closures in the weak topology by the Prohorov theorem. For more details on this background material see [5] and [6].
Given a nonnegative Borel function h on X x Y xV(X x Y), we set
We need a simple observation: the function Jh with values in [0, +rc>] is lower semicontinuous on uniformly tight subsets of V (X x Y) provided that h is lower semicontinuous on every set of the form S x M, where S is compact in X x Y and M C V (X x Y) is compact and uniformly tight. Indeed, since the values Jmax(h,n)(a) increase to Jh(a), the assertion reduces to the case h < 1. Next, Jh can be uniformly approximated on M by functions Jg with g lower semicontinuous on X x Y x M, because for each e £ (0,1) there is a compact set K C X x Y with a(K) > 1 — e for all a £ M. On the set K x M the function h can be represented as supra hn for a sequence of continuous functions hn > 0 (see [13, 1.7.15(c)]), but the same expression defines a lower semicontinuous function g on all of X xY x M and |Jh(a) — Jg(a)| < 2e for all a £ M. Finally, dealing with continuous h, we obtain yet another uniform approximation of Jh by Jf with f of the form f(x,y,a) = ^r[=l pi(x,y)ipi(a), where pi is a bounded continuous function on X xY and ^ is a bounded continuous function on M. For such f the function Jf is continuous. In order to construct f we apply the Stone-Weierstrass theorem and find f of the indicated form such that lh(x, y, a) — f (x, y,a)l < e for all (x, y) £ K, a £ M. Finally, the functions (pi can be redefined in such a way that <pi = 0 outside a suitable neighborhood U of K, the values on K remain the same and f takes values in [0, 2] (the original function f takes values in [0,2] only on K x M).
This implies that the sets {a £ M\: Jh(/j.) < c} are closed. In addition, the function Jh attains a minimum on every closed uniformly tight set on which it is finite. If X and Y are complete separable metric spaces and the function h is lower semicontinuous, then the functional Jh is lower semicontinuous on all of X xY xV (X xY), which is verified shorter (see
Theorem 1. Suppose that ^ V(X) and t^ vt, T ^ V(Y) are
Borel mappings and
is a Borel function such that for every t £T the function
ht: (x,y,a) ^ h(t,x,y,a)
is lower semicontinuous on sets of the form K x n(^, vt) with compact K C X xY and Kht (¡it, vt) is finite for every t £ T. Then the function
[3]).
h :T xX xY x V (X xY) ^ [0, +то)
t ^ Kht (^t, vt) is Borel measurable and there is a Borel mapping t ^ at from T to V(X x Y) such that at is optimal for (^t,^t,ht) for each t. Moreover, there exists a sequence of Borel mappings from T to V(X x Y) such that for each t the sequence of measures £n(t) is everywhere dense in the set of optimal plans for the triple (^t, ^t, ht).
Proof. It is worth noting that we can assume from the very beginning that the spaces X and Y are complete separable metric. Indeed, a Luzin space is obtained from such a space by weakening the topology, but the Borel aalgebra remains the same. In addition, Borel measures on the original space remain Radon measures with respect to the metric (because on complete separable metric spaces all Borel measures are Radon) and every set n(^, v) is uniformly tight and compact in the weak topology on the space V(X x Y) corresponding to the metrics on X and Y, hence on such sets we have coincidence of the weak topologies on V(X x Y) generated by the original Luzin topology and the stronger metrizable topology. On the whole space V (X x Y) these two topologies are different if X x Y is not a Polish space in the original topology. Finally, the lower continuity is also preserved in the stronger metrizable topology.
is Borel, because the inclusion a E n(/j,t, vt) can be written as a countable family of equalities
where {fj } and {gj } are some sequences of bounded continuous functions on X and Y separating measures (such sequences exist on all completely regular Souslin spaces, see [5, Corollary 6.7.5, Theorem 6.7.7, Lemma 8.10.38]). The sections
are compact.
Let us recall the following classical result ("Measurable Choice Theorem"), see [11, p. 224, 225]. Let T be a Souslin space, E a Luzin space, and let B c T x E be a Borel set such that for all t e T the sections Bx are ^-compact (countable unions of compact sets). Then B admits a Borel uniformization, which means that the projection kt(B) of B on T is a Borel set and there is a Borel mapping f: (B) ^ E whose graph is contained in B.
The set
5 = {(t, a) E T x V(X x Y) : a E U(ßt, vt)}
St = {a : (t,a) E S} =n(vt,vt)
By this theorem applied to E = V(XxY), in which case the projection of M on T is T, there exists a sequence of Borel mappings (n: T ^V(X x Y) such that for each t the sequence {(ra(t)} is everywhere dense in n(^t, ut).
The function (t, a) ^ Jht (a) with values in [0, +rc>] is Borel measurable on T x V(X x Y). Indeed, this function is the result of substitution of a for ( in the function
(t,a,( h(t,x,y,() a(dxdy),
JX xY
which is Borel measurable. This is shown in [9, Lemma 3.1] for bounded h, but for unbounded h the result follows by considering max(h, N) and letting N ^
Next, if for every fixed t the function a ^ Jht(a) is continuous on n(^t,ut), then
Kht (fa, n) = inf Jht ((n(t)).
n
Each function Jht((n(t)) is Borel in t, because (n(t) is Borel in t and (t, a) ^ Jht (a) is jointly Borel measurable. Hence the left-hand side is Borel measurable in t. But the functions Jht can be only lower semicontinuous, when the previous argument does not work. In this case we apply the measurable selection theorem to verify that Kht(pt,^t) is Borel measurable. With the aid of this theorem we shall construct a sequence of Borel functions Jn on T x V(X x Y) such that Jn(t, a) increases to Jt(&) for each a e n(^t,vt) and is continuous in the second argument. It suffices to find a countable family of functions Jn of this kind such that J = supra Jn and pass to the functions max(Ji,..., Jn). The required family is constructed as follows. For each rational number r the set
Zr = {(t,a): a e n(^t,vt), Jht(a) < r}
is Borel in TxV(XxY) and its sections Sr,t = {a: (t, a) e Zr} are compact by the compactness of n(^t, vt) and the lower semicontinuity of Jht. Hence the projection Ar of this set on T is Borel and there is a sequence of Borel mappings Tr,j: Ar ^ V(X x Y) such that {rrj(i)} is dense in Sr,t for each t e Ar. The function 5r(t,a) = infj ||a — Tr,j(t)\\KR, i.e., the KE-distance from a to Sr,t, is Borel measurable on T x V(X x Y). For each n, let dr,n be the function on R defined by dr,n(s) = 0 if s < 1/n, dr,n(s) = r if s > 2/n, and dr,n(s) = nr(s — 1/n) if 1/n < s < 2/n. Set Jr,n(t,a) = dr,n(5r(t,a)) if t e Ar and Jr,n(t,a) = r otherwise. It is readily seen that Jr,n is Borel measurable and continuous in a. In addition, Jr,n(t,a) < Jht (a) whenever a e n(ft, vt), since Jr,n < r and Jht (a) > r if a e Sr,t, while Jr,n(t, a) = 0 if a e Sr,t. Finally,
Jht(a) = sup Jr,n(t,a) for all a e n(/j,t,vt).
r,n
Indeed, otherwise there is a rational number with
Jht (a) > r> sup Jr,n(t, a).
r,n
By the lower semicontinuity of Jht there is a number n with
\\a — rr,3(t)\\KR > 2/n
for all j. Then Jr,n(t,a) = r, a contradiction. It remains to use the following simple fact: if continuous functions fn on a compact space K increase pointwise to a bounded function , then their minima increase to the minimum of , which exists since is lower semicontinuous. Indeed, let m be the minimum of f. The functions fn Am increase to m and by Dini's theorem convergence is uniform. Hence their minima increase to m, which yields our claim. In our situation, having Borel functions Jn(t,a) increasing to Jht (a) and continuous in a, we obtain that their minima on n(/j,t, vt) are Borel measurable in t and increase to Kht (¡it, vt)-Our next step is to repeat the same reasoning for the set
M = {(t ,a) £ S: Jht (a)=Kht (lit, vt)},
which is also Borel in the space T x V(X x Y). The sections Mt are compact by the lower continuity of the function Jh and the fact that the set of points of minimum of a lower continuous function on a compact set is compact. Now we get a sequence of mappings such that {£n(t)} is dense in Mt. □
Remark 1. It is clear from the proof that the same assertion is true if h takes values in [0, +rc>]. Moreover, if we agree that the minimum is attained in the case where Jht (a) = for all a £ n(/it, vt), then the assumption that Kht (¡it, vt) < can be removed. It follows from the proof that the sets {t £ T: Kht(pt,vt) < C} are Borel, so we can apply the theorem to such parametric sets.
3. Nonlinear costs with conditional measures
We now turn to cost functions of the form (1.2). In this case, an additional restriction is imposed on cost functions.
Theorem 2. Suppose that in Theorem 1 the cost function h is of the form (1.2), where H is defined onT x X x V(Y), the functions
Ht: (x,p) ^ H(t,x,p)
are lower semicontinuous and, in addition, the functions p ^ H(t, x, p) are convex for all t,x. Then the conclusion of Theorem 1 is true.
Proof. In order to obtain a Borel function when substituting ax for p in H(t,x,p), we use the fact established in [9] that conditional measures ax (which are unique only up to a redefinition on a measure zero set) can be selected in such a way that the functions (x, a) ^ ax(B) will be Borel measurable for each Borel set B CY .It follows from [3] that the function a ^ Jht (a) is lower semicontinuous on , vt). To be more precise, this is shown in [3] in the case of metric spaces, but as explained at the beginning of the proof of Theorem 1 this is sufficient for our purposes. Actually, the reasoning in [3] directly applies in the case of Luzin spaces (however, it is important to consider Jht on uniformly tight sets). Therefore, the sets
Mt = j(i,a): a £ U(pt,vt), j H(t,x,ax) ^t(clx) = Kht (jit, ft)j
are compact. Hence the same reasoning as in Theorem 1 applies. □
What we have said in Remark 1 is also valid in the present situation.
It would be interesting to consider more general cost functions in (1.2) of the form H(t, x, y, ax) or H(t, x, y, ax, ay).
Note that in our paper [9] the case of Souslin spaces X and Y was also considered with appropriate concepts of measurability. This leads to some complications in the proof, because the measurable choice theorem applied above is not valid for such spaces. However, it is likely that the constructions from [9] extend to Souslin spaces in the present more general setting, which will be the subject of another paper.
References
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Список источников
1. Alibert J.-J., Bouchitté G., Champion T. A new class of costs for optimal transport planning // European J. Appl. Math. 2019. Vol. 30, N 6. P. 1229-1263. https://doi.org/10.1017/S0956792518000669
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3. Backhoff-Veraguas J., Beiglbock M., Pammer G. Existence, duality, and cyclical monotonicity for weak transport costs. Calc. Var. Partial Differ. Equ. 2019. Vol. 58, N 203. P. 1-28. https://doi.org/10.1007/s00526-019-1624-y
4. Backhoff-Veraguas J., Pammer G. Applications of weak transport theory. Bernoulli. 2022. Vol. 28, N 1. P. 370-394. https://doi.org/10.3150/21-BEJ1346
5. Bogachev V.I. Measure Theory. Vols. 1, 2. Berlin : Springer, 2007. Vol. 1. 500 p.; Vol. 2. 575 p. https://doi.org/10.1007/978-3-540-34514-5
6. Bogachev V. I. Weak Convergence of Measures / Amer. Math. Soc., Providence. Rhode Island, 2018., 286 p. https://doi.org/10.1090/surv/234
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10. Dedecker J., Prieur C., Raynaud De Fitte P. Parametrized Kantorovich-Rubinstein theorem and application to the coupling of random variables // Dependence in probability and statistics. Lect. Notes Stat. New York : Springer, 2006. Vol. 187. P. 105—121. https://doi.org/10.1007/0-387-36062-X_5
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Об авторах
Богачев Владимир Игоревич,
д-р физ.-мат. наук, проф., Московский государственный университет, Российская Федерация, 119991, г. Москва, vibogach@mail.ru
About the authors Vladimir I. Bogachev, Dr. Sci.
(Phys.-Math.), Prof., Moscow State University, Moscow, 119991, Russian Federation, vibogach@mail.ru
Малофеев Илья Игоревич, канд. физ.-мат. наук, н.с., Национальный исследовательский университет «Высшая школа экономики» , Российская Федерация, 101000, г. Москва, ilmalofeev@yandex.ru, https://orcid.org/0000-0002-4091-1242
Поступила в 'редакцию / Received 17.03.2022 Поступила после рецензирования / Revised 12.07.2022 Принята к публикации / Accepted 19.07.2022
Ilya I. Malofeev, Cand. Sci. (Phys.Math.), Researcher, National Research University Higher School of Economics, Moscow, 101000, Russian Federation, ilmalofeev@yandex.ru, https://orcid.org/0000-0002-4091-1242
