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Distribution functions, extremal limits and optimal transport. (English) Zbl 1334.90145
Let \(c(x,y)\) be a cost function and \(C(x,y)\) be a copula. This paper is an expository paper on different types of optimizations, namely: (1) Find extremes of \(\frac{1}{N}\sum_{n=1}^Nc(x_n,y_n)\) over uniformly distributed sequences \(x_n\) and \(y_n\). (2) For \(\int_0^1\int_0^1c(x,y)dC(x,y)\) find optimal bounds with respect to \(C(x,y)\). (3) General theory of Monge-Kantorovich transport problem and problems from risk management.
In these directions the authors present some known theorems, we mention here three results:
(I) Let a cost function \(c(x,y)\) be a Riemann integrable and assume that \(d_x d_y c(x,y)>0\) for \((x,y)\in(0,1)^2\). Then \[ \max_{C(x,y)\text{-copula}}\int_0^1\int_0^1c(x,y)d_xd_yC(x,y) =\int_0^1c(x,x)d x, \] \[ \min_{C(x,y)\text{-copula}}\int_0^1\int_0^1c(x,y)d_xd_yC(x,y) =\int_0^1c(x,1-x)d x, \] where the \(\max\) is attained in \(C(x,y)=\min(x,y)\) and \(\min\) in \(C(x,y)=\max(x+y-1,0)\), uniquely.
(II) L. Uckelmann [in: Distributions with given marginals and moment problems. Proceedings of the 1996 conference, Prague, Czech Republic. Dordrecht: Kluwer Academic Publishers. 275–281 (1997; Zbl 0907.60022)] established: Let \(c(x,y)=\Phi(x+y)\) for \((x,y)\in[0,1]^2\); For \(0<k_1<k_2<2\) let \(\Phi(x)\) be a twice differentiable function such that \(\Phi(x)\) is strictly convex on \([0,k_1]\cup[k_2,2]\) and concave on \([k_1,k_2]\). If \(\alpha\) and \(\beta\) are the solutions of \[ \Phi(2\alpha)-\Phi(\alpha+\beta)+(\beta-\alpha)\Phi'(\alpha+\beta)=0, \] \[ \Phi(2\beta)-\Phi(\alpha+\beta)+(\alpha-\beta)\Phi'(\alpha+\beta)=0 \] such that \(0<\alpha<\beta<1\), then the optimal copula \(C(x,y)\) is attained at the shuffle of \(M\) with the support \(\Gamma(x)\), where \(\Gamma(x)=x\) for \(x\in[0,\alpha]\cup[\beta,1]\), and \(\Gamma(x)=\alpha+\beta-x\) for \(x\in(\alpha,\beta)\). The authors use this result for solving sine problem \(c(x,y)=\sin(\pi(x+y))\) in Problem 1.29 [https://math.boku.ac.at/udt/unsolvedproblems.pdf].
(III) Approximated the cost function \(c(x,y)\) by piecewise constant functions M. Hofer and M. R. Iacò [J. Optim. Theory Appl. 161, No. 3, 999–1011 (2014; Zbl 1293.90036)] proved: Let \((a_{i,j})\), \(i,j=1,2,\dots,n\) be a real-valued \(n\times n\) matrix. Let \(I_{i,j}=\big[\frac{i-1}{n},\frac{i}{n}\big]\times\big[\frac{j-1}{n},\frac{j}{n}\big]\), \(i,j=1,2,\dots,n\) and let the piecewise constant function \(c(x,y)\) be defined as \(c(x,y)=a_{i,j}\text{ if }(x,y)\in I_{i,j}, i,j=1,2,\dots,n.\) Then \[ \max_{C(x,y)\text{-copula}}\int_0^1\int_0^1c(x,y)d_xd_yC(x,y) =\frac{1}{n}\sum_{i=1}^na_{i,\pi^*(i)}. \] Here \(\pi^*(i)\) maximizes \(\sum_{i=1}^na_{i,\pi(i)}\), where \(\pi\) is a permutation of \((1,2,\dots,n)\). finally, the authors use this method, to approximate solution of \(c(x,y)=\sin(\pi x)\sin(\pi y)\), \(c(x,y)=\sin(\pi x)\cos(\pi y)\) and \(c(x,y)=\sin(\pi/x)\cos(2\pi y)\), respectively. The exact values of optimal \(C(x,y)\) are open.

MSC:
90C27 Combinatorial optimization
60E05 Probability distributions: general theory
90C46 Optimality conditions and duality in mathematical programming
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