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\iteman{io-port 05351710}
\itemau{Attouch, Hedy; Bolte, J\'er\^ome; Redont, Patrick; Soubeyran, Antoine}
\itemti{Alternating proximal algorithms for weakly coupled convex minimization problems. Applications to dynamical games and PDE's.}
\itemso{J. Convex Anal. 15, No. 3, 485-506 (2008).}
\itemab
The authors consider structured convex optimization problems of the following type $$\min\left\{L(x,y)= f(x)+ g(x)+ \tfrac\mu 2\,Q(x,y): x\in X,\ y\in Y\right\},$$ where $X$, $Y$ are real Hilbert spaces, $f$ and $g$ are closed convex proper functions, $Q$ is a continuous nonnegative quadratic form and $\mu$ is a positive parameter. For this problem, a proximal-like algorithm of the following type is introduced and studied: $(x_0, y_0)\in X\times Y$, $\alpha$, $\mu$, $\nu> 0$ given $$(x_k,y_k)\ge (x_{k+1}, y_k)\to (x_{k+1},y_{k+1})$$ as follows: $$\gather x_{k+1}= \text{arg\,min}\left\{f(\xi)+ \tfrac\mu 2\, Q(\xi, y_k)+ \tfrac \alpha 2\,\Vert\xi- x_k\Vert^2: \xi\in X\right\},\\ y_{k+1}= \text{arg\,min}\left\{g(\eta)+ \tfrac\mu 2\, Q(x_{k_1},\eta)+ \tfrac \nu 2\,\Vert \eta- y_k\Vert^2: \eta\in Y\right\}.\endgather$$ The authors show that the generated sequence $(x_k,y_k)$ converges weakly to a minimum point of $L(x,y)$.  Applications are given in game theory, variational problems and partial differential equations (PDEs).
\itemrv{Hans Benker (Merseburg)}
\itemcc{}
\itemut{convex optimization; alternating minimization; splitting methods; proximal algorithm; weak coupling; quadratic coupling; costs to change; anchoring effect; dynamical games; best response; domain decomposition for PDE's; monotone inclusions}
\itemli{http://www.heldermann.de/JCA/JCA15/JCA153/jca15035.htm}
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