Controlled Block Proximal Linearized Method for Nonconvex and Nonsmooth Optimization

We consider multiblock nonconvex and nonsmooth optimization without assuming bounded iterates or coercivity of the original objective. We propose the Controlled Block Proximal Linearized (CBPL) method, which incorporates a control potential into block updates while retaining the original objective, variables, and constraints. Under explicit control design conditions and the KL property, the control yields boundedness, finite length, whole-sequence convergence of the original variables, and convergence of objective values. At the limit, one limiting subgradient of the original objective vanishes on all effective comparison directions. We apply CBPL to graph-regularized nonnegative matrix factorization (NMF), CP decomposition, and independent-core matrix tri-factorization (NMTF) with $\ell_0$ norm sparsity constraints. We characterize their reconstruction-preserving scale degeneracies, proving that scale criticality is equivalent to graph collapse and that full criticality forces such collapse. Explicit controls provide compact sublevel sets and a uniform positive lower bound on graph action, including with an independent Tucker core, while retaining genuine reconstruction-changing directions. Each block admits an explicit projection and a computable safe step parameter. For ordinary NMF and CP models, whole-sequence convergence to a full critical point follows. Experiments with direct safe updates demonstrate convergence, sustained graph action, and controlled factor and core scales.