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Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. (English) Zbl 0737.90048
The problem considered is: Find $$p^*\in{\mathcal H}$$ satisfying $$b\in A\Phi(A'p^*)+B\Gamma B'p^*)$$, with $$b\in {\mathcal H}$$, $$A: {\mathcal V}\to {\mathcal H}$$, $$B: {\mathcal W}\to {\mathcal H}$$, $$\Phi: {\mathcal V}\to {\mathcal V}$$ and $$\Gamma: {\mathcal W}\to {\mathcal W}$$, $${\mathcal H}$$, $${\mathcal V}$$ and $${\mathcal W}$$ being real Hilbert spaces and the superscript $$'$$ denoting adjoint. It is assumed that $$A$$ and $$B$$ are continuous linear operators and $$\Phi$$ and $$\Gamma$$ are maximal monotone operators. To solve this problem, an algorithm is proposed which, starting with some $$p(0)\in {\mathcal H}$$, generates three sequences $$\{p(t)\}$$, $$\{x(t)\}$$, $$\{z(t)\}$$ according to $$x(t)\in\Phi(A'p(t))$$, $$z(t)\in\Gamma(B'[p(t)-c(t)(Ax(t)+Bz(t)-b])$$, $$p(t+1)=p(t)+c(t)(b-Ax(t)-Bz(t))$$, $$c(t)$$ being a positive stepsize. Under suitable assumptions these sequences are well defined and $$\{p(t)\}$$ converges, at least linearly, to a solution of the problem in the weak topology. Applications of this algorithm to variational inequalities, separable convex programming and linear complementarity problems are studied.

##### MSC:
 90C25 Convex programming 49J40 Variational inequalities 47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 90C48 Programming in abstract spaces 90-08 Computational methods for problems pertaining to operations research and mathematical programming
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