In this paper, we consider a class of composite multiobjective optimization problems, subject to a closed convex constraint set, defined on Riemannian manifolds. To tackle this problem, we propose the generalized cond...
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In this paper, we consider a class of composite multiobjective optimization problems, subject to a closed convex constraint set, defined on Riemannian manifolds. To tackle this problem, we propose the generalized conditional gradient method with two step size strategies, including Armijo step size and the nonmonotone line search step size. Under some reasonable conditions, the global convergence result is established, and the iteration-complexity bound for composite multiobjective optimization problems is presented on Riemannian manifolds.
Iterative soft thresholding algorithm (ISTA) has a simple formulation and it can easily be implemented. Nevertheless, ISTA is limited to well-conditioned problems, e.g. compressive sensing. In this paper, we present a...
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Iterative soft thresholding algorithm (ISTA) has a simple formulation and it can easily be implemented. Nevertheless, ISTA is limited to well-conditioned problems, e.g. compressive sensing. In this paper, we present an ISTA type algorithm based on the generalized conditional gradient method (GCGM) to solve elastic-net regularization which is commonly adopted in ill-conditioned problems. Furthermore, we propose a projected gradient (PG) method to accelerate the ISTA type algorithm. In addition, we discuss the existence of the radius R and we give a strategy to determine the radius R of the l1-ball constraint in the PG method by Morozov's discrepancy principle (MDP). Numerical results are reported to illustrate the efficiency of the proposed approach. (C) 2021 Elsevier B.V. All rights reserved.
The ill-posed problem of solving linear equations in the space of vector-valued finite Radon measures with Hilbert space data is considered. Approximate solutions are obtained by minimizing the Tikhonov functional wit...
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The ill-posed problem of solving linear equations in the space of vector-valued finite Radon measures with Hilbert space data is considered. Approximate solutions are obtained by minimizing the Tikhonov functional with a total variation penalty. The well-posedness of this regularization method and further regularization properties are mentioned. Furthermore, a flexible numerical minimization algorithm is proposed which converges subsequentially in the weak* sense and with rate O(n(-1)) in terms of the functional values. Finally, numerical results for sparse deconvolution demonstrate the applicability for a finite-dimensional discrete data space and infinite-dimensional solution space.
A new iterative algorithm for the solution of minimization problems in infinite-dimensional Hilbert spaces which involve sparsity constraints in form of l(p)-penalties is proposed. In contrast to the well-known algori...
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A new iterative algorithm for the solution of minimization problems in infinite-dimensional Hilbert spaces which involve sparsity constraints in form of l(p)-penalties is proposed. In contrast to the well-known algorithm considered by Daubechies, Defrise, and De Mol, it uses hard instead of soft shrinkage. It is shown that the hard shrinkage algorithm is a special case of the generalized conditional gradient method. Convergence properties of the generalized conditional gradient method with quadratic discrepancy term are analyzed. This leads to strong convergence of the iterates with convergence rates O(n(-1/2)) and O(lambda(n)) for p = 1 and 1 < p <= 2, respectively. Numerical experiments on image deblurring, backwards heat conduction, and inverse integration are given.
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