Proximal gradient method: Difference between revisions

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== Projection onto convex sets (POCS) ==

One of the widely used convex optimization algorithms is [[projections onto convex sets]] (POCS). This algorithm is employed to recover/synthesize a signal satisfying simultaneously several convex constraints. Let <math>f_i</math> be the indicator function of non-empty closed convex set <math>C_i</math> modeling a constraint. This reduces to convex feasibility problem, which require us to find a solution such that it lies in the intersection of all convex sets <math>C_i</math>. In POCS method each set <math>C_i</math> is incorporated by its [[projection operator]] <math>P_{C_i}</math>. So in each [[iteration]] <math>x</math> is updated as

One of the widely used convex optimization algorithms is POCS ([[Projections onto convex sets|Projection Onto Convex Sets]]).

This algorithm is employed to recover/synthesize a signal satisfying simultaneously several convex constraints.

Let <math>f_i</math> be the indicator function of non-empty closed convex set <math>C_i</math> modeling a constraint.

This reduces to convex feasibility problem, which require us to find a solution such that it lies in the intersection

of all convex sets <math>C_i</math>. In POCS method each set <math>C_i</math> is incorporated by its [[projection operator]]

<math>P_{C_i}</math>. So in each [[iteration]] <math>x</math> is updated as

: <math>

x_{k+1} = P_{C_1} P_{C_2} \cdots P_{C_n} x_k

</math>

However beyond such problems [[projection operator]]s are not appropriate and more general operators are required to tackle them. Among the various generalizations of the notion of a convex projection operator that exist, proximity operators are best suited for other purposes.

However beyond such problems [[projection operator]]s are not appropriate and more general operators

are required to tackle them. Among the various generalizations of the notion of a convex projection

operator that exist, proximity operators are best suited for other purposes.

== Definition ==