Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFinitely Convergent Iterative Methods with Overrelaxations Revisited
361
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study the finite convergence of iterative methods for solving convex feasibility problems. Our key assumptions are that the interior of the solution set is nonempty and that certain overrelaxation parameters converge to zero, but with a rate slower than any geometric sequence. Unlike other works in this area, which require divergent series of overrelaxations, our approach allows us to consider some summable series. By employing quasi-Fej{e}rian analysis in the latter case, we obtain additional asymptotic convergence guarantees, even when the interior of the solution set is empty.
rate research
Read More
We propose finitely convergent methods for solving convex feasibility problems defined over a possibly infinite pool of constraints. Following other works in this area, we assume that the interior of the solution set is nonempty and that certain overrelaxation parameters form a divergent series. We combine our methods with a very general class of deterministic control sequences where, roughly speaking, we require that sooner or later we encounter a violated constraint if one exists. This requirement is satisfied, in particular, by the cyclic, repetitive and remotest set controls. Moreover, it is almost surely satisfied for random controls.
Nesterovs well-known scheme for accelerating gradient descent in convex optimization problems is adapted to accelerating stationary iterative solvers for linear systems. Compared with classical Krylov subspace acceleration methods, the proposed scheme requires more iterations, but it is trivial to implement and retains essentially the same computational cost as the unaccelerated method. An explicit formula for a fixed optimal parameter is derived in the case where the stationary iteration matrix has only real eigenvalues, based only on the smallest and largest eigenvalues. The fixed parameter, and corresponding convergence factor, are shown to maintain their optimality when the iteration matrix also has complex eigenvalues that are contained within an explicitly defined disk in the complex plane. A comparison to Chebyshev acceleration based on the same information of the smallest and largest real eigenvalues (dubbed Restricted Information Chebyshev acceleration) demonstrates that Nesterovs scheme is more robust in the sense that it remains optimal over a larger domain when the iteration matrix does have some complex eigenvalues. Numerical tests validate the efficiency of the proposed scheme. This work generalizes and extends the results of [1, Lemmas 3.1 and 3.2 and Theorem 3.3].
We present two globally convergent Levenberg-Marquardt methods for finding zeros of H{o}lder metrically subregular mappings that may have non-isolated zeros. The first method unifies the Levenberg- Marquardt direction and an Armijo-type line search, while the second incorporates this direction with a nonmonotone trust-region technique. For both methods, we prove the global convergence to a first-order stationary point of the associated merit function. Furthermore, the worst-case global complexity of these methods are provided, indicating that an approximate stationary point can be computed in at most $mathcal{O}(varepsilon^{-2})$ function and gradient evaluations, for an accuracy parameter $varepsilon>0$. We also study the conditions for the proposed methods to converge to a zero of the associated mappings. Computing a moiety conserved steady state for biochemical reaction networks can be cast as the problem of finding a zero of a H{o}lder metrically subregular mapping. We report encouraging numerical results for finding a zero of such mappings derived from real-world biological data, which supports our theoretical foundations.
We apply novel inner-iteration preconditioned Krylov subspace methods to the interior-point algorithm for linear programming (LP). Inner-iteration preconditioners recently proposed by Morikuni and Hayami enable us to overcome the severe ill-conditioning of linear equations solved in the final phase of interior-point iterations. The Krylov subspace methods do not suffer from rank-deficiency and therefore no preprocessing is necessary even if rows of the constraint matrix are not linearly independent. By means of these methods, a new interior-point recurrence is proposed in order to omit one matrix-vector product at each step. Extensive numerical experiments are conducted over diverse instances of 138 LP problems including the Netlib, QAPLIB, Mittelmann and Atomizer Basis Pursuit collections. The largest problem has 434,580 unknowns. It turns out that our implementation is more robust than the standard public domain solvers SeDuMi (Self-Dual Minimization), SDPT3 (Semidefinite Programming Toh-Todd-Tutuncu) and the LSMR iterative solver in PDCO (Primal-Dual Barrier Method for Convex Objectives) without increasing CPU time. The proposed interior-point method based on iterative solvers succeeds in solving a fairly large number of LP instances from benchmark libraries under the standard stopping criteria. The work also presents a fairly extensive benchmark test for several renowned solvers including direct and iterative solvers.
Model-free learning-based control methods have seen great success recently. However, such methods typically suffer from poor sample complexity and limited convergence guarantees. This is in sharp contrast to classical model-based control, which has a rich theory but typically requires strong modeling assumptions. In this paper, we combine the two approaches to achieve the best of both worlds. We consider a dynamical system with both linear and non-linear components and develop a novel approach to use the linear model to define a warm start for a model-free, policy gradient method. We show this hybrid approach outperforms the model-based controller while avoiding the convergence issues associated with model-free approaches via both numerical experiments and theoretical analyses, in which we derive sufficient conditions on the non-linear component such that our approach is guaranteed to converge to the (nearly) global optimal controller.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
