A simple algorithm for computing projection onto intersection of finite level sets
© He et al.; licensee Springer. 2014
Received: 21 August 2013
Accepted: 30 July 2014
Published: 21 August 2014
We consider the problem of computing the projection , where u is chosen in a real Hilbert space H arbitrarily and the closed convex subset C of H is the intersection of finite level sets of convex functions given as follows: , where m is a positive integer and is a convex function for . A relaxed Halpern-type algorithm is proposed for computing the projection in this paper, which is defined by , , where the initial guess is chosen arbitrarily, the sequence is in and is a sequence of half-spaces containing for . Since calculations of the projections onto half-spaces (; ) are easy in practice, this algorithm is quite implementable. Strong convergence of our algorithm is proved under some ordinary conditions. Some numerical experiments are provided which show advantages of our algorithm.
MSC:58E35, 47H09, 65J15.
Keywordsprojection Hilbert space half-space strong convergence
converges in norm to when and are two closed subspaces of H. In 1965, Bregman  showed that the iterates generated by (1.2) converge weakly to for any pair of closed convex subsets and . Gubin et al.  proved that the iterates will converge linearly to if and are ‘boundedly regular’. Actually, they proved this result for alternating projections between any finite collection of closed convex sets. Strong convergence also holds when the sets are symmetric , Theorem 2.2; , Corollary 2.6]. However, in 2004, Hundal  proved that the sequence of iterates generated by (1.2) does not always converge in norm to by providing an explicit counterexample.
Since the computation of a projection onto a closed convex subset is generally difficult, to overcome this difficulty, Fukushima  suggested a way to calculate the projection onto a level set of a convex function by computing a sequence of projections onto half-spaces containing the original level set. This idea is followed by Yang  and Lopez et al. , respectively, who introduced the relaxed CQ algorithms for solving the split feasibility problem in the setting of finite-dimensional and infinite-dimensional Hilbert spaces, respectively. The idea is also used by Gibali et al.  and He et al.  for solving variational inequalities in a Hilbert space.
where m is a positive integer and is a convex function for .
where the initial guess is chosen arbitrarily, the sequence is in and is a sequence of half-spaces containing for (the specific structure of the half-spaces will be described in Section 3). Since calculations of the projections onto half-spaces (; ) are easy in practice, this algorithm is quite implementable. Moreover, strong convergence of our algorithm can be proved under some ordinary conditions.
The rest of this paper is organized as follows. Some useful lemmas are given in Section 2. In Section 3, the strong convergence of our algorithm is proved. Some numerical experiments are given in Section 4 which show advantages of our algorithm.
Throughout the rest of this paper, we denote by H a real Hilbert space. We will use the notations:
→ denotes strong convergence.
⇀ denotes weak convergence.
denotes the weak ω-limit set of .
A function is said to be subdifferentiable at x, if it has at least one subgradient at x. The set of subgradients of f at the point x, denoted by , is called the subdifferential of f at x. The last relation above is called the subdifferential inequality of f at x. A function f is called subdifferentiable, if it is subdifferentiable at all .
This inequality is trivial but in common use.
The following lemma is the key to the proofs of strong convergence of our algorithms. In fact, it can be used as a new fundamental tool for solving some nonlinear problems, particularly, some problems related to projection operator.
Lemma 2.2 
implies for any subsequence .
3 Iterative algorithms
where and are given by (3.2) and the sequence is in .
Theorem 3.2 Assume that () and . Then the sequence generated by Algorithm 3.1 converges strongly to the point .
which means that is bounded.
where M is some positive constant such that (noting that is bounded).
for any subsequence . From the Lemma 2.2, we get , which means . □
where . By the subdifferential inequality, it is easy to see that holds for all and .
where are given by (3.15) and the sequence is in .
By an argument very similar to the proof of Theorem 3.2, it is not difficult to see that our result of Theorem 3.2 can be extended easily to the general case.
Theorem 3.4 Assume that () and . Then the sequence generated by Algorithm 3.3 converges strongly to the point .
Finally, we point out that if the computation of the projection operator is easy for all (for example, is a closed ball or a half-space for all ), then we have no need to adopt the relaxation technique in the algorithm designs, that is, one can use the following algorithm to compute the projection for a given point . Moreover, the strong convergence of this algorithm can be proved by an argument similar to the proof of Theorem 3.4 (in fact, its proof is much simpler than that of Theorem 3.4).
where the sequence is in .
Theorem 3.6 Assume that () and . Then the sequence generated by Algorithm 3.5 converges strongly to the point .
4 Numerical experiments
In this section, in order to show advantages of our algorithms, we present some numerical results via implementing Algorithm 3.1 and Algorithm 3.5 for two examples, respectively, in the setting of finite-dimensional Hilbert space. The codes were written in Matlab 2013a and run on an Amd Liano APU A4-3300M Core4 CPU k43t (CPU 1.9 GHz) personal computer. In the following two examples, we always take and for . The n th step iterate is denoted by . Since we do not know the exact projection , we use to measure the error of the n th step iteration.
Use Algorithm 3.1 to calculate the projection .
Use Algorithm 3.5 to calculate the projection .
Numerical results as regards Example 4.1
1.33E − 01
4.00E − 03
1.82E − 03
1.32E − 03
1.28E − 03
8.78E − 04
7.15E − 04
7.01E − 04
6.77E − 04
5.53E − 04
5.27E − 04
5.10E − 04
Numerical results as regards Example 4.2
1.16E − 01
9.69E − 03
3.94E − 03
3.45E − 03
2.98E − 03
2.63E − 03
2.02E − 03
1.77E − 03
1.67E − 03
1.52E − 03
1.47E − 03
1.37E − 03
1.20E − 03
1.07E − 03
This work was supported by National Natural Science Foundation of China (Grant No. 11201476) and the Fundamental Research Funds for the Central Universities (3122014K010).
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