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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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