20210620 Successive projection algorithm（连续投影法）

https://en.wikipedia.org/wiki/Projections_onto_convex_sets

Algorithm
The POCS algorithm solves the following problem:
find x∈Rnx \in \mathbb{R}^{n} \quadx∈Rn such that x∈C∩Dx \in C \cap Dx∈C∩D
where CCC and DDD are closed convex sets.
To use the POCS algorithm, one must know how to project onto the sets CCC and DDD separately. The algorithm starts with an arbitrary value for x0x_{0}x0 and then generates the sequence
xk+1=PC(PD(xk))x_{k+1}=\mathcal{P}_{C}\left(\mathcal{P}_{D}\left(x_{k}\right)\right) xk+1=PC(PD(xk))
The simplicity of the algorithm explains some of its popularity. If the intersection of CCC and DDD is non-empty, then the sequence generated by the algorithm will converge to some point in this intersection.
Unlike Dykstra’s projection algorithm, the solution need not be a projection onto the intersection CCC and DDD.

Related algorithms
The method of averaged projections is quite similar. For the case of two closed convex sets CCC and DDD, it proceeds by
xk+1=12(PC(xk)+PD(xk))x_{k+1}=\frac{1}{2}\left(\mathcal{P}_{C}\left(x_{k}\right)+\mathcal{P}_{D}\left(x_{k}\right)\right) xk+1=21(PC(xk)+PD(xk))
It has long been known to converge globally. [8]{ }^{[8]}[8] Furthermore, the method is easy to generalize to more than two sets; some convergence results for this case are in. [9]^{[9]}[9]

The averaged projections method can be reformulated as alternating projections method using a standard trick. Consider the set
E={(x,y):x∈C,y∈D}E=\{(x, y): x \in C, y \in D\} E={(x,y):x∈C,y∈D}
which is defined in the product space Rn×Rn\mathbb{R}^{n} \times \mathbb{R}^{n}Rn×Rn. Then define another set, also in the product
space:
F={(x,y):x∈Rn,y∈Rn,x=y}.F=\left\{(x, y): x \in \mathbb{R}^{n}, y \in \mathbb{R}^{n}, x=y\right\} . F={(x,y):x∈Rn,y∈Rn,x=y}.
Thus finding C∩DC \cap DC∩D is equivalent to finding E∩FE \cap FE∩F.

To find a point in E∩FE \cap FE∩F, use the alternating projection method. The projection of a vector (x,y)(x, y)(x,y) onto the set FFF is given by (x+y,x+y)/2(x+y, x+y) / 2(x+y,x+y)/2. Hence
(xk+1,yk+1)=PF(PE((xk,yk)))=PF((PCxk,PDyk))=12(PC(xk)+PD(yk),(PC(xk)+PD(yk))\left(x_{k+1}, y_{k+1}\right)=\mathcal{P}_{F}\left(\mathcal{P}_{E}\left(\left(x_{k}, y_{k}\right)\right)\right)=\mathcal{P}_{F}\left(\left(\mathcal{P}_{C} x_{k}, \mathcal{P}_{D} y_{k}\right)\right)=\frac{1}{2}\left(\mathcal{P}_{C}\left(x_{k}\right)+\mathcal{P}_{D}\left(y_{k}\right),\left(\mathcal{P}_{C}\left(x_{k}\right)+\mathcal{P}_{D}\left(y_{k}\right)\right)\right. (xk+1,yk+1)=PF(PE((xk,yk)))=PF((PCxk,PDyk))=21(PC(xk)+PD(yk),(PC(xk)+PD(yk))

Since xk+1=yk+1x_{k+1}=y_{k+1}xk+1=yk+1 and assuming x0=y0x_{0}=y_{0}x0=y0, then xj=yjx_{j}=y_{j}xj=yj for all j≥0j \geq 0j≥0, and hence we can simplify the iteration to xk+1=12(PC(xk)+PD(xk))x_{k+1}=\frac{1}{2}\left(\mathcal{P}_{C}\left(x_{k}\right)+\mathcal{P}_{D}\left(x_{k}\right)\right)xk+1=21(PC(xk)+PD(xk)).

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20210620 Successive projection algorithm（连续投影法）

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