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Algorithms for the split variational inequality problem. (English) Zbl 1239.65041

The authors present a new problem, the split variational inequality problem (SVIP). SVIP is quite general and should enable split minimization between two spaces so that the image of a solution point of one minimization problem, under a given bounded linear operator, is a solution point of another minimization problem. Here two approaches to the solution of the SVIP, are considered. The first approach is to look at the product space \(H_1 \times H_2\) and to transform the SVIP:
Find a point \(x^* \in C\) such that \(\langle f (x^*), x-x^* \rangle \geq0\) for all \(x \in C\) (and given operators \(f: H_1 \rightarrow H_1\) and \(g: H_2 \rightarrow H_2\), where \(H_1\) and \(H_2\) are two Hilbert spaces) and such that the point \(y^* = Ax^* \in Q\) and solves \(\langle g (y^*), y-y^* \rangle \geq0\) for all \(y \in Q\). (\(A: H_1 \rightarrow H_2\) is a bounded linear operator, and \(C\subseteq H_1, Q \subseteq H_2\) are nonempty, closed and convex subsets) into an equivalent constrained variational inequality problem in the product space. A prototypical split inverse problem concerns a model in which there are two spaces \(X\) and \(Y\) and there is given a bounded linear operator \(A: X \rightarrow Y\). Additionally, there are two inverse problems involved, one inverse problem \(IP_1\) formulated in the space \(X\) and another inverse problem \(IP_2\) formulated in the space \(Y\). the split inverse problem is the following:
Find a point \(x^* \in X\) that solves \(IP_1\) such that the point \(y^* = Ax^* \in Y\) solves \(IP_2\).
The algorithm for the constrained variational inequality problem (VIP) is presented. The authors propose their own method for solving the SVIP which does not rely on any product space formulation, and then prove convergence: For \(f= \nabla F\) and \(g = \nabla G\) in the SVIP (\(F:{\mathbb{R}}^n \rightarrow {\mathbb{R}}^n, G: {\mathbb{R}}^m \rightarrow {\mathbb{R}}^m\) are continuously differentiable convex functions on closed and convex subsets \(C \subseteq {\mathbb{R}}^n\) and \(Q \subseteq {\mathbb{R}}^m\)) the split minimization problem:
Find a point \(x^* \in C\) such that \(x^* = \arg \min \{f(x) | x\in C \}\) and such that, the point \(y^* = Ax^*\) is in \(Q\) and solves \(y^* = \arg\min \{g(y)|y \in Q\}\).
Finally the split zeros problem (SZP), newly introduced in this paper, is defined as follows. For given operators \(B_1: H_1 \rightarrow H_1\) and \(B_2: H_2 \rightarrow H_2\) (\(H_1, H_2\) are Hilbert spaces) and a bounded linear operator \(A: H_1 \rightarrow H_2\) the authors formulate SZP as follows: find a point \(x^* \in H_1\) such that \(B_1(x^*) = 0\) and \(B_2(Ax^*) = 0\). (This problem is a special case of the SVIP if \(A\) is a surjective operator.) The authors construct iterative algorithms that solve such problems under reasonable conditions in Hilbert space and show when the unique solution of an SVIP is a solution of an SZP. A similar result concerning the relationship between the (un-split) zero finding problem and the VIP is proposed.

MSC:

65K15 Numerical methods for variational inequalities and related problems
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