An inexact method of partial inverses and a parallel bundle method.

*(English)* Zbl 1091.49006
Summary: For a maximal monotone operator $T$ on a Hilbert space $H$ and a closed subspace $A$ of $H$, we consider the problem of finding $(x, y \in T(x))$ satisfying $x \in A$ and $y \in A^{\bot}$. An equivalent formulation of this problem makes use of the partial inverse operator of Spingarn. The resulting generalized equation can be solved by using the proximal point algorithm. We consider instead the use of hybrid proximal methods. Hybrid methods use enlargements of operators, close in spirit to the concept of $\varepsilon$-subdifferentials. We characterize the enlargement of the partial inverse operator in terms of the enlargement of $T$ itself. We present a new algorithm of resolution that combines Spingarn and hybrid methods, we prove for this method global convergence only assuming existence of solutions and maximal monotonicity of T. We also show that, under standard assumptions, the method has a linear rate of convergence. For the important problem of finding a zero of a sum of maximal monotone operators $T_{1}, \cdots , T_{m}$, we present a highly parallelizable scheme. Finally, we derive a parallel bundle method for minimizing the sum of polyhedral functions.

##### MSC:

49J40 | Variational methods including variational inequalities |

49J52 | Nonsmooth analysis (other weak concepts of optimality) |

47N10 | Applications of operator theory in optimization, convex analysis, programming, economics |

65J15 | Equations with nonlinear operators (numerical methods) |

47H05 | Monotone operators (with respect to duality) and generalizations |