proximalbackward

Proximalbackward is a term used in optimization to describe a family of iterative schemes that blend proximal operators with backward (implicit) updates to solve composite objective functions. The construction draws on the proximal point framework for handling nonsmooth terms and on implicit discretizations common in numerical analysis. Proximalbackward methods are designed to improve stability and robustness, particularly for stiff or ill-conditioned problems, or when large time steps are advantageous.

In a typical setting, one aims to minimize a function f(x) = g(x) + h(x), where g is proper closed convex and possibly nonsmooth, and h is smooth with a Lipschitz continuous gradient. A proximalbackward update computes the next iterate x_{k+1} by solving an implicit subproblem that couples the proximal handling of g with a backward, or implicit, treatment of h. A representative variant can be written schematically as x_{k+1} ∈ argmin_x { g(x) + h(x_{k+1}) + (1/2t)||x - x_k||^2 }, which requires solving for x_{k+1} inside h. In practice, inner iterations or linearizations of h are often employed to make the subproblem tractable. When h is linearized at x_k, the scheme reduces to a familiar proximal-gradient-like step with improved stability.

Convergence properties depend on problem structure. Under convexity and Lipschitz-gradient assumptions, the objective values along the generated sequence are nonincreasing and converge to a minimizer; if f is strongly convex, convergence can be linear, otherwise sublinear rates like those of other proximal methods are typical. The backward component tends to enhance stability with larger step sizes relative to explicit schemes.

Variants of proximalbackward integrate with other splitting strategies, such as Douglas-Rachford or alternating direction methods, and adapt step sizes, proximal operators, or inner solver accuracy. Applications span image and signal processing, machine learning, compressed sensing, and numerical analysis of differential inclusions or stiff differential equations.

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