In this article, a family of discrete-time, supertwisting-like algorithms is presented. The algorithms are naturally vector-valued and are described in an implicit fashion, reminiscent of backward-Euler discretization...
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In this article, a family of discrete-time, supertwisting-like algorithms is presented. The algorithms are naturally vector-valued and are described in an implicit fashion, reminiscent of backward-Euler discretization schemes. The well-posedness of the closed-loop is established and the robust stability, against a family of external disturbances, is thoroughly studied. Implementation strategies, involving splitting-algorithms from convex optimization, are also discussed and compared. Finally, numerical simulations show the performance of the proposed schemes.
A general algorithmic scheme for solving inclusions in a Banach space is investigated in respect to its local convergence behavior. Particular emphasis is placed on applications to certain proximal-point-type algorith...
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A general algorithmic scheme for solving inclusions in a Banach space is investigated in respect to its local convergence behavior. Particular emphasis is placed on applications to certain proximal-point-type algorithms in Hilbert spaces. The assumptions do not necessarily require that a solution be isolated. In this way, results existing for the case of a locally unique solution can be extended to cases with nonisolated solutions. Besides the convergence rates for the distance of the iterates to the solution set, strong convergence to a sole solution is shown as well. As one particular application of the framework, an improved convergence rate for an important case of the inexact proximal-point algorithm is derived.
We introduce an iterative sequence for finding the solution to 0 is an element of T(v), where T : E paired right arrows E* is a maximal monotone operator in a smooth and uniformly convex Banach space E. This iterative...
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We introduce an iterative sequence for finding the solution to 0 is an element of T(v), where T : E paired right arrows E* is a maximal monotone operator in a smooth and uniformly convex Banach space E. This iterative procedure is a combination of iterative algorithms proposed by Kohsaka and Takahashi (Abstr. Appl. Anal. 3:239-249, 2004) and Kamamura, Kohsaka and Takahashi (Set-Valued Anal. 12:417-429, 2004). We prove a strong convergence theorem and a weak convergence theorem under different conditions respectively and give an estimate of the convergence rate of the algorithm. An application to minimization problems is given.
Let A be a maximal monotone operator in a real Hilbert space H and let {u(n)} be the sequence in H given by the proximalpoint algorithm, defined by u (n) =(I+c(n) A)(-1)(u(n-1)-f(n) ), for all n >= 1, with u(0) = ...
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Let A be a maximal monotone operator in a real Hilbert space H and let {u(n)} be the sequence in H given by the proximalpoint algorithm, defined by u (n) =(I+c(n) A)(-1)(u(n-1)-f(n) ), for all n >= 1, with u(0) = z, where c(n) > 0 and f(n) is an element of H. We show, among other things, that under suitable conditions, u(n) converges weakly or strongly to a zero of A if and only if lim inf(n ->+infinity) vertical bar w(n)vertical bar +infinity, where w(n) = (Sigma(n)(k=1) c(k))(-1) Sigma(n)(k=1) c(k)u(k). Our results extend previous results by several authors who obtained similar results by assuming A(-1)(0) not equal phi.
In the context of convex analysis, macro-hybrid variational formulations of constrained boundary value problems are presented. Monotone mixed variational inclusions are macro-hybridized on the basis of nonoverlapping ...
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In the context of convex analysis, macro-hybrid variational formulations of constrained boundary value problems are presented. Monotone mixed variational inclusions are macro-hybridized on the basis of nonoverlapping domain decompositions, and corresponding three-field versions are derived. Then, for regularization purposes, augmented formulations are established via preconditioned exact penalizations and expressed in terms of proximation operators. Optimization interpretations are given for potential problems, recovering the classic two- and three field augmented Lagrangian formulations. Furthermore, associated parallel two and three-field proximal-point algorithms are discussed for numerical resolution of finite element discretizations. Applications to dual mixed variational formulations of problems from mechanics illustrate the theory.
We introduce a new barrier function to build new interior-pointalgorithms to solve optimization problems with bounded variables. First, we show that this function is a (3/2)n-self-concordant barrier for the unitary h...
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We introduce a new barrier function to build new interior-pointalgorithms to solve optimization problems with bounded variables. First, we show that this function is a (3/2)n-self-concordant barrier for the unitary hypercube [0,1]n, assuring thus the polynomial property of related algorithms. Second, using the Hessian metric of that barrier, we present new explicit algorithms from the point of view of Riemannian geometry applications. Third, we prove that the central path defined by the new barrier to solve a certain class of linearly constrained convex problems maintains most of the properties of the central path defined by the usual logarithmic barrier. We present also a primal long-step path-following algorithm with similar complexity to the classical barrier. Finally, we introduce a new proximal-point Bregman type algorithm to solve linear problems in [0,1]n and prove its convergence.
We apply the Banach contraction-mapping fixed-point principle for solving multivalued strongly monotone variational inequalities. Then, we couple this algorithm with the proximal-point method for solving monotone mult...
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We apply the Banach contraction-mapping fixed-point principle for solving multivalued strongly monotone variational inequalities. Then, we couple this algorithm with the proximal-point method for solving monotone multivalued variational inequalities. We prove the convergence rate of this algorithm and report some computational results.
Composition duality methods for mixed variational inclusions are studied in a functional framework of reflexive Banach spaces. On the basis of duality principles, the solvability of maximal monotone and subdifferentia...
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Composition duality methods for mixed variational inclusions are studied in a functional framework of reflexive Banach spaces. On the basis of duality principles, the solvability of maximal monotone and subdifferential mixed variational inclusions is established. For computational purposes, mass-preconditioned augmented formulations are introduced for regularization, as well as three-field and macro-hybrid variational versions. At a finite-dimensional level, corresponding discrete mixed and macro-hybrid internal approximations are discussed, as well as proximal-point iterative algorithms. Primal and dual mixed variational inclusions from contact mechanics illustrate the theory.
Iterative numerical algorithms for variational inequalities are systematically constructed from fixed-point problem characterizations in terms of resolvent operators. The applied method is the one introduced by Gabay ...
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Iterative numerical algorithms for variational inequalities are systematically constructed from fixed-point problem characterizations in terms of resolvent operators. The applied method is the one introduced by Gabay in [Ga], used here in the context of discrete variational inequalities, and with the emphasis on mixed finite element models. The algorithms apply to nonnecessarily potential problems, generalizing primal and mixed Uzawa and augmented Lagrangian-type algorithms. They are also identified with Euler and operator splitting methods for the time discretization of evolution first-order problems.
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