Subspaces \(X_1,\ldots,X_J\) of a normed vector space \(X=\sum_{j=1}^JX_j\) are considered. Linear continuous mappings \(P_j:X\to X_j\), \(P_1+\cdots +P_J=\text{Id}\), are assumed. The goal is to minimize a functional \(f: X\to (-\infty,\infty]\) which can be decomposed as \(f=\phi +\psi \), where \(\phi\) is Fréchet-differentiable and \(D\phi\) is uniformly elliptic and Lipschitz continuous, and \(\psi\) is convex, lower semicontinuous and additive with respect to the partition of \(X\) introduced above. An algorithm based on the minimization of \(f\) on affine subspaces determined by \(X_j\) is presented. The author proves that a sequence generated by the algorithm converges to a minimizer of \(f\) in \(X\), and estimates the rate of convergence. It is a generalization of a result of \textit{P. L. Lions} where \(\psi =0\) [Domain decomposition methods for PDEs, 1st Int. Symp., Paris/France 1987, 1-42 (1988; Zbl 0658.65090)]. After an introduction to the theory of plasticity, where such functionals \(f\) arise, applications to domain decomposition in plasticity with hardening are given.