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[[Proximal Gradient Methods|Proximal gradient methods]] are applicable in a wide variety of scenarios for solving [[convex optimization]] problems of the form
:<math> \min_{x\in \mathcal{H}} F(x)+R(x),</math>
where <math>F</math> is [[Convex_function|convex]] and differentiable with [[Lipschitz_continuity|Lipschitz continuous]] [[Gradient|gradient]], and <math> R</math> is a [[Convex_function|convex]], [[Semicontinuous_function|lower semicontinuous]] function which is possibly nondifferentiable, and <math>\mathcal{H}</math> is some set, typically a [[Hilbert_space|Hilbert space]]. The usual criterion of <math> x</math> minimizes <math> F(x)+R(x)</math> if and only if <math> \nabla (F+R)(x) = 0</math> in the convex, differentiable setting is now replaced by
:<math> 0\in \partial (F+R)(x), </math>
where <math>\partial \varphi</math> denotes the [[subdifferential]] of a real-valued, convex function <math> \varphi</math>.
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