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<button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.09927">arXiv:2407.09927</a> <span> [<a href="https://arxiv.org/pdf/2407.09927">pdf</a>, <a href="https://arxiv.org/ps/2407.09927">ps</a>, <a href="https://arxiv.org/format/2407.09927">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> An Adaptive Proximal ADMM for Nonconvex Linearly-Constrained Composite Programs </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Maia%2C+L+F">Leandro Farias Maia</a>, <a href="/search/math?searchtype=author&query=Gutman%2C+D+H">David H. Gutman</a>, <a href="/search/math?searchtype=author&query=Monteiro%2C+R+D+C">Renato D. C. Monteiro</a>, <a href="/search/math?searchtype=author&query=Silva%2C+G+N">Gilson N. Silva</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2407.09927v1-abstract-short" style="display: inline;"> This paper develops an adaptive Proximal Alternating Direction Method of Multipliers (P-ADMM) for solving linearly-constrained, weakly convex, composite optimization problems. This method is adaptive to all problem parameters, including smoothness and weak convexity constants. It is assumed that the smooth component of the objective is weakly convex and possibly nonseparable, while the non-smooth… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.09927v1-abstract-full').style.display = 'inline'; document.getElementById('2407.09927v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.09927v1-abstract-full" style="display: none;"> This paper develops an adaptive Proximal Alternating Direction Method of Multipliers (P-ADMM) for solving linearly-constrained, weakly convex, composite optimization problems. This method is adaptive to all problem parameters, including smoothness and weak convexity constants. It is assumed that the smooth component of the objective is weakly convex and possibly nonseparable, while the non-smooth component is convex and block-separable. The proposed method is tolerant to the inexact solution of its block proximal subproblem so it does not require that the non-smooth component has easily computable block proximal maps. Each iteration of our adaptive P-ADMM consists of two steps: (1) the sequential solution of each block proximal subproblem, and (2) adaptive tests to decide whether to perform a full Lagrange multiplier and/or penalty parameter update(s). Without any rank assumptions on the constraint matrices, it is shown that the adaptive P-ADMM obtains an approximate first-order stationary point of the constrained problem in a number of iterations that matches the state-of-the-art complexity for the class of P-ADMMs. The two proof-of-concept numerical experiments that conclude the paper suggest our adaptive P-ADMM enjoys significant computational benefits. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.09927v1-abstract-full').style.display = 'none'; document.getElementById('2407.09927v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 13 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.12464">arXiv:2402.12464</a> <span> [<a href="https://arxiv.org/pdf/2402.12464">pdf</a>, <a href="https://arxiv.org/format/2402.12464">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> An Adaptive Cubic Regularization quasi-Newton Method on Riemannian Manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Louzeiro%2C+M+S">Mauricio S. Louzeiro</a>, <a href="/search/math?searchtype=author&query=Silva%2C+G+N">Gilson N. Silva</a>, <a href="/search/math?searchtype=author&query=Yuan%2C+J">Jinyun Yuan</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+D">Daoping Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2402.12464v1-abstract-short" style="display: inline;"> A quasi-Newton method with cubic regularization is designed for solving Riemannian unconstrained nonconvex optimization problems. The proposed algorithm is fully adaptive with at most ${\cal O} (蔚_g^{-3/2})$ iterations to achieve a gradient smaller than $蔚_g$ for given $蔚_g$, and at most $\mathcal O(\max\{ 蔚_g^{-\frac{3}{2}}, 蔚_H^{-3} \})$ iterations to reach a second-order stationary point respec… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.12464v1-abstract-full').style.display = 'inline'; document.getElementById('2402.12464v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.12464v1-abstract-full" style="display: none;"> A quasi-Newton method with cubic regularization is designed for solving Riemannian unconstrained nonconvex optimization problems. The proposed algorithm is fully adaptive with at most ${\cal O} (蔚_g^{-3/2})$ iterations to achieve a gradient smaller than $蔚_g$ for given $蔚_g$, and at most $\mathcal O(\max\{ 蔚_g^{-\frac{3}{2}}, 蔚_H^{-3} \})$ iterations to reach a second-order stationary point respectively. Notably, the proposed algorithm remains applicable even in cases of the gradient and Hessian of the objective function unknown. Numerical experiments are performed with gradient and Hessian being approximated by forward finite-differences to illustrate the theoretical results and numerical comparison. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.12464v1-abstract-full').style.display = 'none'; document.getElementById('2402.12464v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2303.10554">arXiv:2303.10554</a> <span> [<a href="https://arxiv.org/pdf/2303.10554">pdf</a>, <a href="https://arxiv.org/format/2303.10554">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Inexact Newton Methods for Solving Generalized Equations on Riemannian Manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Louzeiro%2C+M+S">Mauricio S. Louzeiro</a>, <a href="/search/math?searchtype=author&query=Silva%2C+G+N">Gilson N. Silva</a>, <a href="/search/math?searchtype=author&query=Yuan%2C+J">Jinyun Yuan</a>, <a href="/search/math?searchtype=author&query=Zhang%2C+D">Daoping Zhang</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2303.10554v3-abstract-short" style="display: inline;"> The convergence of inexact Newton methods is studied for solving generalized equations on Riemannian manifolds by using the metric regularity property, which is also explored. Under appropriate conditions and without any additional geometric assumptions, local convergence results with linear and quadratic rates, as well as a semi-local convergence result, are obtained for the proposed method. Fina… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.10554v3-abstract-full').style.display = 'inline'; document.getElementById('2303.10554v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2303.10554v3-abstract-full" style="display: none;"> The convergence of inexact Newton methods is studied for solving generalized equations on Riemannian manifolds by using the metric regularity property, which is also explored. Under appropriate conditions and without any additional geometric assumptions, local convergence results with linear and quadratic rates, as well as a semi-local convergence result, are obtained for the proposed method. Finally, the theory is applied to the problem of finding a singularity for the sum of two vector fields. In particular, the KKT system for the constrained Riemannian center of mass on the sphere is explored numerically. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.10554v3-abstract-full').style.display = 'none'; document.getElementById('2303.10554v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">39 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2101.10863">arXiv:2101.10863</a> <span> [<a href="https://arxiv.org/pdf/2101.10863">pdf</a>, <a href="https://arxiv.org/ps/2101.10863">ps</a>, <a href="https://arxiv.org/format/2101.10863">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.sysconle.2006.08.011">10.1016/j.sysconle.2006.08.011 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the value function for nonautonomous optimal control problems with infinite horizon </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Baumeister%2C+J">J. Baumeister</a>, <a href="/search/math?searchtype=author&query=Leitao%2C+A">A. Leitao</a>, <a href="/search/math?searchtype=author&query=Silva%2C+G+N">G. N. Silva</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2101.10863v1-abstract-short" style="display: inline;"> In this paper we consider nonautonomous optimal control problems of infinite horizon type, whose control actions are given by $L^1$-functions. We verify that the value function is locally Lipschitz. The equivalence between dynamic programming inequalities and Hamilton-Jacobi-Bellman (HJB) inequalities for proximal sub (super) gradients is proven. Using this result we show that the value function i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.10863v1-abstract-full').style.display = 'inline'; document.getElementById('2101.10863v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2101.10863v1-abstract-full" style="display: none;"> In this paper we consider nonautonomous optimal control problems of infinite horizon type, whose control actions are given by $L^1$-functions. We verify that the value function is locally Lipschitz. The equivalence between dynamic programming inequalities and Hamilton-Jacobi-Bellman (HJB) inequalities for proximal sub (super) gradients is proven. Using this result we show that the value function is a Dini solution of the HJB equation. We obtain a verification result for the class of Dini sub-solutions of the HJB equation and also prove a minimax property of the value function with respect to the sets of Dini semi-solutions of the HJB equation. We introduce the concept of viscosity solutions of the HJB equation in infinite horizon and prove the equivalence between this and the concept of Dini solutions. In the appendix we provide an existence theorem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.10863v1-abstract-full').style.display = 'none'; document.getElementById('2101.10863v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">17 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 49L20; 49L25; 49J15 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Systems and Control Letters 56 (2007), no. 3, 188-196 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1605.03627">arXiv:1605.03627</a> <span> [<a href="https://arxiv.org/pdf/1605.03627">pdf</a>, <a href="https://arxiv.org/ps/1605.03627">ps</a>, <a href="https://arxiv.org/format/1605.03627">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Consistent Approximations to Impulsive Optimal Control Problems </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Porto%2C+D">Daniella Porto</a>, <a href="/search/math?searchtype=author&query=Silva%2C+G+N">Geraldo Nunes Silva</a>, <a href="/search/math?searchtype=author&query=Silva%2C+H+H+M">Helo铆sa Helena Marino Silva</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1605.03627v2-abstract-short" style="display: inline;"> We analyse the theory of consistent approximations given by Polak and we use it in an impulsive optimal control problem. We reparametrize the original system and build consistent approximations for this new reparametrized problem. So, we prove that if a sequence of solution of the consistent approximations is converging, it will converge to a solution of the reparametrized problem, and, finally, w… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.03627v2-abstract-full').style.display = 'inline'; document.getElementById('1605.03627v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1605.03627v2-abstract-full" style="display: none;"> We analyse the theory of consistent approximations given by Polak and we use it in an impulsive optimal control problem. We reparametrize the original system and build consistent approximations for this new reparametrized problem. So, we prove that if a sequence of solution of the consistent approximations is converging, it will converge to a solution of the reparametrized problem, and, finally, we show that from a solution of the reparametrized problem we can find a solution of the original one. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.03627v2-abstract-full').style.display = 'none'; document.getElementById('1605.03627v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 July, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 May, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2016. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1604.04569">arXiv:1604.04569</a> <span> [<a href="https://arxiv.org/pdf/1604.04569">pdf</a>, <a href="https://arxiv.org/ps/1604.04569">ps</a>, <a href="https://arxiv.org/format/1604.04569">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Kantorovich's theorem on Newton's method for solving strongly regular generalized equation </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ferreira%2C+O+P">O. P. Ferreira</a>, <a href="/search/math?searchtype=author&query=Silva%2C+G+N">G. N. Silva</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1604.04569v1-abstract-short" style="display: inline;"> In this paper we consider the Newton's method for solving the generalized equation of the form $ f(x) +F(x) \ni 0, $ where $f:惟\to Y$ is a continuously differentiable mapping, $X$ and $Y$ are Banach spaces, $惟\subseteq X$ an open set and $F:X \rightrightarrows Y$ be a set-valued mapping with nonempty closed graph. We show that, under strong regularity of the generalized equation, concept introduce… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1604.04569v1-abstract-full').style.display = 'inline'; document.getElementById('1604.04569v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1604.04569v1-abstract-full" style="display: none;"> In this paper we consider the Newton's method for solving the generalized equation of the form $ f(x) +F(x) \ni 0, $ where $f:惟\to Y$ is a continuously differentiable mapping, $X$ and $Y$ are Banach spaces, $惟\subseteq X$ an open set and $F:X \rightrightarrows Y$ be a set-valued mapping with nonempty closed graph. We show that, under strong regularity of the generalized equation, concept introduced by S.M.Robinson in [27], and starting point satisfying the Kantorovich's assumptions, the Newton's method is quadratically convergent to a solution, which is unique in a suitable neighborhood of the starting point. The analysis presented based on Banach Perturbation Lemma for generalized equation and the majorant technique, allow to unify some results pertaining the Newton's method theory. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1604.04569v1-abstract-full').style.display = 'none'; document.getElementById('1604.04569v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 April, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20 pages. arXiv admin note: substantial text overlap with arXiv: 1604.04568, arXiv:1603.04782</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1604.04568">arXiv:1604.04568</a> <span> [<a href="https://arxiv.org/pdf/1604.04568">pdf</a>, <a href="https://arxiv.org/ps/1604.04568">ps</a>, <a href="https://arxiv.org/format/1604.04568">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Optimization and Control">math.OC</span> </div> </div> <p class="title is-5 mathjax"> Local convergence analysis of Newton's method for solving strongly regular generalized equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ferreira%2C+O+P">O. P. Ferreira</a>, <a href="/search/math?searchtype=author&query=Silva%2C+G+N">G. N. Silva</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1604.04568v3-abstract-short" style="display: inline;"> In this paper we study Newton's method for solving generalized equations in Banach spaces. We show that under strong regularity of the generalized equation, the method is locally convergent to a solution with superlinear/quadratic rate. The presented analysis is based on Banach Perturbation Lemma for generalized equation and the classical Lipschitz condition on the derivative is relaxed by using a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1604.04568v3-abstract-full').style.display = 'inline'; document.getElementById('1604.04568v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1604.04568v3-abstract-full" style="display: none;"> In this paper we study Newton's method for solving generalized equations in Banach spaces. We show that under strong regularity of the generalized equation, the method is locally convergent to a solution with superlinear/quadratic rate. The presented analysis is based on Banach Perturbation Lemma for generalized equation and the classical Lipschitz condition on the derivative is relaxed by using a general majorant function, which enables obtaining the optimal convergence radius, uniqueness of solution as well as unifies earlier results pertaining to Newton's method theory. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1604.04568v3-abstract-full').style.display = 'none'; document.getElementById('1604.04568v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 September, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 April, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages. arXiv admin note: text overlap with arXiv:1604.04569</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1603.05280">arXiv:1603.05280</a> <span> [<a href="https://arxiv.org/pdf/1603.05280">pdf</a>, <a href="https://arxiv.org/ps/1603.05280">ps</a>, <a href="https://arxiv.org/format/1603.05280">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Local convergence of Newton's method for solving generalized equations with monotone operator </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Silva%2C+G+N">Gilson N. Silva</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1603.05280v3-abstract-short" style="display: inline;"> In this paper we study Newton's method for solving the generalized equation $F(x)+T(x)\ni 0$ in Hilbert spaces, where $F$ is a Fr茅chet differentiable function and $T$ is set-valued and maximal monotone. We show that this method is local quadratically convergent to a solution. Using the idea of majorant condition on the nonlinear function which is associated to the generalized equation, the converg… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1603.05280v3-abstract-full').style.display = 'inline'; document.getElementById('1603.05280v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1603.05280v3-abstract-full" style="display: none;"> In this paper we study Newton's method for solving the generalized equation $F(x)+T(x)\ni 0$ in Hilbert spaces, where $F$ is a Fr茅chet differentiable function and $T$ is set-valued and maximal monotone. We show that this method is local quadratically convergent to a solution. Using the idea of majorant condition on the nonlinear function which is associated to the generalized equation, the convergence of the method, the optimal convergence radius and results on the convergence rate are established. The advantage of working with a majorant condition rests in the fact that it allow to unify several convergence results pertaining to Newton's method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1603.05280v3-abstract-full').style.display = 'none'; document.getElementById('1603.05280v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 July, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 March, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">12 pages, 0 figure. arXiv admin note: substantial text overlap with arXiv:1603.04782, arXiv:1604.04568</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1603.04782">arXiv:1603.04782</a> <span> [<a href="https://arxiv.org/pdf/1603.04782">pdf</a>, <a href="https://arxiv.org/ps/1603.04782">ps</a>, <a href="https://arxiv.org/format/1603.04782">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> On the Kantorovich's theorem for Newton's method for solving generalized equations under the majorant condition </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Silva%2C+G+N">Gilson N. Silva</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1603.04782v2-abstract-short" style="display: inline;"> In this paper we consider a version of the Kantorovich's theorem for solving the generalized equation $F(x)+T(x)\ni 0$, where $F$ is a Fr茅chet derivative function and $T$ is a set-valued and maximal monotone acting between Hilbert spaces. We show that this method is quadratically convergent to a solution of $F(x)+T(x)\ni 0$. We have used the idea of majorant function, which relaxes the Lipschitz c… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1603.04782v2-abstract-full').style.display = 'inline'; document.getElementById('1603.04782v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1603.04782v2-abstract-full" style="display: none;"> In this paper we consider a version of the Kantorovich's theorem for solving the generalized equation $F(x)+T(x)\ni 0$, where $F$ is a Fr茅chet derivative function and $T$ is a set-valued and maximal monotone acting between Hilbert spaces. We show that this method is quadratically convergent to a solution of $F(x)+T(x)\ni 0$. We have used the idea of majorant function, which relaxes the Lipschitz continuity of the derivative $F'$. It allows us to obtain the optimal convergence radius, uniqueness of solution and also to solving generalized equations under Smale's condition. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1603.04782v2-abstract-full').style.display = 'none'; document.getElementById('1603.04782v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 May, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 15 March, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages, 0 figure</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1510.01947">arXiv:1510.01947</a> <span> [<a href="https://arxiv.org/pdf/1510.01947">pdf</a>, <a href="https://arxiv.org/ps/1510.01947">ps</a>, <a href="https://arxiv.org/format/1510.01947">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Numerical Analysis">math.NA</span> </div> </div> <p class="title is-5 mathjax"> Inexact Newton's method to nonlinear functions with values in a cone </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Ferreira%2C+O+P">O. P. Ferreira</a>, <a href="/search/math?searchtype=author&query=Silva%2C+G+N">G. N. Silva</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1510.01947v2-abstract-short" style="display: inline;"> The problem of finding a solution of nonlinear inclusion problems in Banach space is considered in this paper. Using convex optimization techniques introduced by Robinson (Numer. Math., Vol. 19, 1972, pp. 341-347), a robust convergence theorem for inexact Newton's method is proved. As an application, an affine invariant version of Kantorovich's theorem and Smale's 伪-theorem for inexact Newton's me… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1510.01947v2-abstract-full').style.display = 'inline'; document.getElementById('1510.01947v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1510.01947v2-abstract-full" style="display: none;"> The problem of finding a solution of nonlinear inclusion problems in Banach space is considered in this paper. Using convex optimization techniques introduced by Robinson (Numer. Math., Vol. 19, 1972, pp. 341-347), a robust convergence theorem for inexact Newton's method is proved. As an application, an affine invariant version of Kantorovich's theorem and Smale's 伪-theorem for inexact Newton's method is obtained. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1510.01947v2-abstract-full').style.display = 'none'; document.getElementById('1510.01947v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 April, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 October, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">15 pages, 0 figure. arXiv admin note: text overlap with arXiv:1403.2462</span> </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 73.4-41.4 132.7-96.3 142.8-104.7 173.4-128.7 5.8-4.5 9.2-11.5 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