Circumcentered reflections method for wavelet feasibility problems

We introduce a two-stage global-then-local search method for solving feasibility problems. The approach pairs the advantageous global tendency of the Douglas–Rachford method to find a basin of attraction for a fixed point, together with the local tendency of the circumcentered reflections method to perform faster within such a basin. We experimentally demonstrate the success of the method for solving nonconvex problems in the context of wavelet construction formulated as a feasibility problem. References F. J. Aragón Artacho, R. Campoy, and M. K. Tam. The Douglas–Rachford algorithm for convex and nonconvex feasibility problems. Math. Meth. Oper. Res. 91 (2020), pp. 201–240. doi: 10.1007/s00186-019-00691-9 R. Behling, J. Y. Bello Cruz, and L.-R. Santos. Circumcentering the Douglas–Rachford method. Numer. Algor. 78.3 (2018), pp. 759–776. doi: 10.1007/s11075-017-0399-5 R. Behling, J. Y. Bello-Cruz, and L.-R. Santos. On the linear convergence of the circumcentered-reflection method. Oper. Res. Lett. 46.2 (2018), pp. 159–162. issn: 0167-6377. doi: 10.1016/j.orl.2017.11.018 J. M. Borwein, S. B. Lindstrom, B. Sims, A. Schneider, and M. P. Skerritt. Dynamics of the Douglas–Rachford method for ellipses and p-spheres. Set-Val. Var. Anal. 26 (2018), pp. 385–403. doi: 10.1007/s11228-017-0457-0 J. M. Borwein and B. Sims. The Douglas–Rachford algorithm in the absence of convexity. Fixed-point algorithms for inverse problems in science and engineering. Springer, 2011, pp. 93–109. doi: 10.1007/978-1-4419-9569-8_6 I. Daubechies. Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41.7 (1988), pp. 909–996. doi: 10.1002/cpa.3160410705 N. D. Dizon, J. A. Hogan, and J. D. Lakey. Optimization in the construction of nearly cardinal and nearly symmetric wavelets. In: 13th International conference on Sampling Theory and Applications (SampTA). 2019, pp. 1–4. doi: 10.1109/SampTA45681.2019.9030889 N. D. Dizon, J. A. Hogan, and S. B. Lindstrom. Circumcentering reflection methods for nonconvex feasibility problems. arXiv preprint arXiv:1910.04384 (2019). url: https://arxiv.org/abs/1910.04384 D. J. Franklin. Projection algorithms for non-separable wavelets and Clifford Fourier analysis. PhD thesis. University of Newcastle, 2018. doi: 1959.13/1395028. D. J. Franklin, J. A. Hogan, and M. K. Tam. A Douglas–Rachford construction of non-separable continuous compactly supported multidimensional wavelets. arXiv preprint arXiv:2006.03302 (2020). url: https://arxiv.org/abs/2006.03302 D. J. Franklin, J. A. Hogan, and M. K. Tam. Higher-dimensional wavelets and the Douglas–Rachford algorithm. 13th International conference on Sampling Theory and Applications (SampTA). 2019, pp. 1–4. doi: 10.1109/SampTA45681.2019.9030823 B. P. Lamichhane, S. B. Lindstrom, and B. Sims. Application of projection algorithms to differential equations: Boundary value problems. ANZIAM J. 61.1 (2019), pp. 23–46. doi: 10.1017/S1446181118000391 S. B. Lindstrom and B. Sims. Survey: Sixty years of Douglas–Rachford. J. Aust. Math. Soc. 110 (2020), 1–38. doi: 10.1017/S1446788719000570 S. B. Lindstrom, B. Sims, and M. P. Skerritt. Computing intersections of implicitly specified plane curves. J. Nonlin. Convex Anal. 18.3 (2017), pp. 347–359. url: http://www.yokohamapublishers.jp/online2/jncav18-3 S. G. Mallat. Multiresolution approximations and wavelet orthonormal bases of L2(R). Trans. Amer. Math. Soc. 315.1 (1989), pp. 69–87. doi: 10.1090/S0002-9947-1989-1008470-5 Y. Meyer. Wavelets and operators. Cambridge University Press, 1993. doi: 10.1017/CBO9780511623820 G. Pierra. Decomposition through formalization in a product space. Math. Program. 28 (1984), pp. 96–115. doi: 10.1007/BF02612715

Circumcentering approximate reflections for solving the convex feasibility problem

The circumcentered-reflection method (CRM) has been applied for solving convex feasibility problems. CRM iterates by computing a circumcenter upon a composition of reflections with respect to convex sets. Since reflections are based on exact projections, their computation might be costly. In this regard, we introduce the circumcentered approximate-reflection method (CARM), whose reflections rely on outer-approximate projections. The appeal of CARM is that, in rather general situations, the approximate projections we employ are available under low computational cost. We derive convergence of CARM and linear convergence under an error bound condition. We also present successful theoretical and numerical comparisons of CARM to the original CRM, to the classical method of alternating projections (MAP), and to a correspondent outer-approximate version of MAP, referred to as MAAP. Along with our results and numerical experiments, we present a couple of illustrative examples.

On the complexity of equational decision problems for finite height complemented and orthocomplemented modular lattices

We study the computational complexity of the satisfiability problem and the complement of the equivalence problem for complemented (orthocomplemented) modular lattices L and classes thereof. Concerning a simple L of finite height, $$\mathcal {NP}$$ NP -hardness is shown for both problems. Moreover, both problems are shown to be polynomial-time equivalent to the same feasibility problem over the division ring D whenever L is the subspace lattice of a D-vector space of finite dimension at least 3. Considering the class of all finite dimensional Hilbert spaces, the equivalence problem for the class of subspace ortholattices is shown to be polynomial-time equivalent to that for the class of endomorphism $$*$$ ∗ -rings with pseudo-inversion; moreover, we derive completeness for the complement of the Boolean part of the nondeterministic Blum-Shub-Smale model of real computation without constants. This result extends to the additive category of finite dimensional Hilbert spaces, enriched by adjunction and pseudo-inversion.

Approximation of Fixed Points for Enriched Suzuki Nonexpansive Operators with an Application in Hilbert Spaces

In this article, we introduce the class of enriched Suzuki nonexpansive (ESN) mappings. We show that this new class of mappings properly contains the class of Suzuki nonexpansive as well as the class of enriched nonexpansive mappings. We establish existence of fixed point and convergence of fixed point in a Hilbert space setting under the Krasnoselskii iteration process. One of the our main results is applied to solve a split feasibility problem (SFP) in this new setting of mappings. Our main results are a significant improvement of the corresponding results of the literature.

The inertial relaxed algorithm with Armijo-type line search for solving multiple-sets split feasibility problem

The multiple-sets split feasibility problem is the generalization of split feasibility problem, which has been widely used in fuzzy image reconstruction and sparse signal processing systems. In this paper, we present an inertial relaxed algorithm to solve the multiple-sets split feasibility problem by using an alternating inertial step. The advantage of this algorithm is that the choice of stepsize is determined by Armijo-type line search, which avoids calculating the norms of operators. The weak convergence of the sequence obtained by our algorithm is proved under mild conditions. In addition, the numerical experiments are given to verify the convergence and validity of the algorithm.

Inertial accelerated algorithms for solving split feasibility with multiple output sets in Hilbert spaces

In this paper, we propose two inertial accelerated algorithms which do not require prior knowledge of operator norm for solving split feasibility problem with multiple output sets in real Hilbert spaces. We prove weak and strong convergence results for approximating the solution of the considered problem under certain mild conditions. We also give some numerical examples to demonstrate the performance and efficiency of our proposed algorithms over some existing related algorithms in the literature.

Iterative approximation of split feasibility problem in real Hilbert spaces

In this paper, we propose an iterative algorithm for finding solution of split feasibility problem involving a λ−strictly pseudo-nonspreading map and asymptotically nonexpansive semigroups in two real Hilbert spaces. We prove weak and strong convergence theorems using the sequence obtained from the proposed algorithm. Finally, we applied our result to solve a monotone inclusion problem and present a numerical example to support our result.

WEAK AND STRONG CONVERGENCE THEOREMS OF ALGORITHMIC SCHEMES FOR THE MULTIPLE-SETS SPLIT FEASIBILITY PROBLEM

Trong bài báo này, để giải bài toán chấp nhận tách đa tập (MSSFP) trong không gian Hilbert, chúng tôi trình bày một cách tiếp cận tổng quát để xây dựng các phương pháp lặp. Chúng tôi đề xuất một lược đồ thuật toán xâu trung bình với sự hội tụ yếu và một lược đồ thuật toán xâu trung bình với sự hội tụ mạnh. Lược đồ thuật toán xâu trung bình với sự hội tụ mạnh được xây dựng dựa trên phương pháp lặp tổng quát cho ánh xạ không giãn, trong đó cỡ bước được tính toán trực tiếp trong mỗi bước lặp mà không cần sử dụng chuẩn của toán tử. Những lược đồ thuật toán này không chỉ bao hàm những cải tiến của phương pháp lặp vòng và lặp đồng thời đã biết như những trường hợp riêng mà còn bao hàm cả những phương pháp lặp mới

An Inertial Accelerated Algorithm for Solving Split Feasibility Problem with Multiple Output Sets

The paper proposes an inertial accelerated algorithm for solving split feasibility problem with multiple output sets. To improve the feasibility, the algorithm involves computing of projections onto relaxed sets (half spaces) instead of computing onto the closed convex sets, and it does not require calculating matrix inverse. To accelerate the convergence, the algorithm adopts self-adaptive rules and incorporates inertial technique. The strong convergence is shown under some suitable conditions. In addition, some newly derived results are presented for solving the split feasibility problem and split feasibility problem with multiple output sets. Finally, numerical experiments illustrate that the algorithm converges more quickly than some existing algorithms. Our results extend and improve some methods in the literature.