Applications of a splitting algorithm to decomposition in convex programming and variational inequalities. (English) Zbl 0737.90048

Tseng, Paul

SIAM J. Control Optimization 29, No. 1, 119-138 (1991).

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.

Reviewer: J.-E.Martínez-Legaz (Barcelona)

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

Keywords:

augmented Lagrangian; alternating minimization; real Hilbert spaces; continuous linear operators; maximal monotone operators