We first study its convergence property for a constant step-size rule. The analysis indicates that the proposed algorithm with a small constant step size approximates a solution to the problem. We next consider the case of a diminishing step-size sequence and prove that there exists a subsequence of the sequence generated by the algorithm which weakly converges to a solution to the problem.
We also give numerical examples to support the convergence analyses. Convex optimization theory has been widely used to solve practical convex minimization problems over complicated constraints, e. It enables us to consider constrained optimization problems in which the explicit form of the metric projection onto the constraint set is not always known; i.
The motivations behind studying the problem are to devise optimization algorithms which have a wider range of application compared with the previous algorithms for smooth convex optimization see, e. Many algorithms have been presented for solving nonsmooth convex optimization. To our knowledge, there are no references on parallel algorithms for nonsmooth convex optimization with fixed point constraints.
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In this paper, we propose a parallel subgradient algorithm for nonsmooth convex optimization with fixed point constraints. Our algorithm is founded on the ideas behind the two useful algorithms. It ensures that our algorithm converges to a point in the intersection of the fixed point sets of nonexpansive mappings.
Since the operator can communicate with all users, our parallel algorithm enables the operator to find a solution to the main problem by using information transmitted from all users. This paper has three contributions in relation to other work on convex optimization. The first is that our algorithm does not use any proximity operators, in contrast to the algorithms presented in [ 16 , 18 , 21 — 23 ]. Our algorithm can use subgradients, which are well defined for any nonsmooth, convex functions.
The second contribution is that our parallel algorithm can be applied to nonsmooth convex optimization problems over the fixed point sets of nonexpansive mappings, while the previous algorithms work in nonsmooth convex optimization over simple constraint sets [ 15 ], Subchapter 5. The third contribution is to present convergence analyses for different step-size rules. We show that our algorithm with a small constant step size approximates a solution to the problem of minimizing the sum of nonsmooth, convex functions over the fixed point sets of nonexpansive mappings.
We also show that there exists a subsequence of the sequence generated by our algorithm with a diminishing step size which weakly converges to a solution to the problem. This paper is organized as follows. It is clear that firm nonexpansivity implies nonexpansivity. The metric projection [ 15 ], Subchapter 4. Then :. This paper deals with a networked system with an operator denoted by user 0 and I users.
Moreover, we define. The next section describes a sufficient condition for satisfying A5. This section presents a parallel subgradient algorithm for solving Problem 2. In this case, user i can use.
Moreover, user i can compute. We can prove that Algorithm 3. The following lemma yields some properties of Algorithm 3. Suppose that Assumptions A1 - A5 and 3. Then the following properties hold :. Let us perform a convergence analysis on Algorithm 3. Suppose that Assumptions A1 - A5 , 3. Let us compare Algorithm 3.
Algorithm 3. Algorithm 7 satisfies. We can see that the previous algorithms 5 and 7 can be applied to the case where the projections onto constraint sets can easily be computed, whereas Algorithm 3. Assume that 8 does not hold.
Since the right side of the above inequality approaches minus infinity when n diverges, we have a contradiction. Therefore, 8 holds. Thus, 9 guarantees that. Assume that 10 does not hold. Since the above inequality does not hold for large enough n , we have arrived at a contradiction.
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Therefore, 10 holds. This completes the proof. The following broadcast gradient method [ 2 ], Algorithm 4. Meanwhile, Algorithm 3. Accordingly, we find that. Hence, 15 holds. Therefore, a similar discussion to the one for obtaining 14 ensures that. Let us look at some numerical examples to see how Algorithm 3.
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The experiment used a We used. We performed samplings, each starting from different random initial points given by MATLAB, and averaged their results. We can see that the sequences generated by Algorithm 3. This paper discussed the problem of minimizing the sum of nondifferentiable, convex functions over the intersection of the fixed point sets of firmly nonexpansive mappings in a real Hilbert space. It presented a parallel algorithm for solving the problem. The parallel algorithm does not use any proximity operators, in contrast to conventional parallel algorithms.
Moreover, the parallel algorithm can work in nonsmooth convex optimization over constraint sets onto which projections cannot be always implemented, while the conventional incremental subgradient method can only be applied when the constraint set is simple in the sense that the projection onto it can easily be implemented. Between these two pages, the complex of unavailable consequences minimally were as a tile of possible onset revolution, c. If new, usually the download Inherently Parallel Algorithms in its continuous Reference. You can Look a family capital and explain your jobs.
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