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原文摘要

In this paper, 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, whose control potentials establish boundedness while retaining the original objective, constraints and variables. We prove finite length and whole-sequence convergence to an effective critical point of the original objective, where one limiting subgradient annihilates all effective feasible directions. We establish finite total variation of the original objective and block functions, summable original effective residuals, and finite termination and linear or sublinear convergence rates for the original objective values. Scale orthogonality further gives full original criticality and, for homogeneous balancing controls, summable absolute objective errors with an $o(1/K)$ best-iterate bound. We establish a complete degeneracy theory for graph-regularized nonnegative factorizations: scale criticality is equivalent to graph collapse, and full criticality eliminates graph action; our controls prevent this collapse while retaining genuine reconstruction changes. Furthermore, our proposed method provides the practical, simple, and effective schemes for these applications, combining guaranteed convergence with explicit closed-form updates that can be applied directly to solve the original problems. We apply CBPL to graph-regularized sparse nonnegative matrix factorization with $\ell_0$-constraints ($\ell_0$-GSNMF), graph-regularized sparse nonnegative canonical polyadic (CP) decomposition with $\ell_0$-constraints ($\ell_0$-GSNCP), and graph-regularized sparse nonnegative independent-core matrix tri-factorization with $\ell_0$-constraints ($\ell_0$-GSNMTF). Numerical experiments demonstrate effectiveness, convergence and structural control.