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Proximal Gradient Method
Proximal gradient methods are a generalized form of projection used to solve non-differentiable convex optimization problems. Many interesting problems can be formulated as convex optimization problems of the form \operatorname\limits_ \sum_^n f_i(x) where f_i: \mathbb^N \rightarrow \mathbb,\ i = 1, \dots, n are possibly non-differentiable convex functions. The lack of differentiability rules out conventional smooth optimization techniques like the steepest descent method and the conjugate gradient method, but proximal gradient methods can be used instead. Proximal gradient methods starts by a splitting step, in which the functions f_1, . . . , f_n are used individually so as to yield an easily implementable algorithm. They are called proximal because each non-differentiable function among f_1, . . . , f_n is involved via its proximity operator. Iterative shrinkage thresholding algorithm, projected Landweber, projected gradient, alternating projections, alternating ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Optimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard. Definition Abstract form A convex optimization problem is defined by two ingredients: * The ''objective function'', which is a real-valued convex function of ''n'' variables, f :\mathcal D \subseteq \mathbb^n \to \mathbb; * The ''feasible set'', which is a convex subset C\subseteq \mathbb^n. The goal of the problem is to find some \mathbf \in C attaining :\inf \. In general, there are three options regarding the existence of a solution: * If such a point ''x''* exists, it is referred to as an ''optimal point'' or ''solution''; the set of all optimal points is called the ''optimal set''; and the problem is called ''solvable''. * If f is unbou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
