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Orizon P. Ferreira

Publications and source records attributed to Orizon P. Ferreira.

17 recordsLinked to original sources
arXiv2025

On projection mappings and the gradient projection method on hyperbolic space forms

This paper presents several new properties of the intrinsic $κ$-projection into $κ$-hyperbolically convex sets of $κ$-hyperbolic space forms, along with closed-form formulas for the intrinsic $κ$-projection into specific $κ$-hyperbolically convex sets. It also discusses the relationship between the intrinsic $κ$-projection, the Euclidean orthogonal projection, and the Lorentz projection. These properties lay the groundwork for analyzing the gradient projection method and hold importance in their own right. Additionally, new properties of the gradient projection method to solve constrained optimization problems in $κ$-hyperbolic space forms are established, considering both constant and backtracking step sizes in the analysis. It is shown that every accumulation point of the sequence generated by the method for both step sizes is a stationary point for the given problem. Additionally, an iteration complexity bound is provided that upper bounds the number of iterations needed to achieve a suitable measure of stationarity for both step sizes. Finally, the properties of the constrained Fermat-Weber problem are explored, demonstrating that the sequence generated by the gradient projection method converges to its unique solution. Numerical experiments on solving the Fermat-Weber problem are presented, illustrating the theoretical findings and demonstrating the effectiveness of the proposed methods.

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arXiv2024

An Inexact Boosted Difference of Convex Algorithm for Nondifferentiable Functions

In this paper, we introduce an inexact approach to the Boosted Difference of Convex Functions Algorithm (BDCA) for solving nonconvex and nondifferentiable problems involving the difference of two convex functions (DC functions). Specifically, when the first DC component is differentiable and the second may be nondifferentiable, BDCA utilizes the solution from the subproblem of the DC Algorithm (DCA) to define a descent direction for the objective function. A monotone linesearch is then performed to find a new point that improves the objective function relative to the subproblem solution. This approach enhances the performance of DCA. However, if the first DC component is nondifferentiable, the BDCA direction may become an ascent direction, rendering the monotone linesearch ineffective. To address this, we propose an Inexact nonmonotone Boosted Difference of Convex Algorithm (InmBDCA). This algorithm incorporates two main features of inexactness: First, the subproblem therein is solved approximately allowing us for a controlled relative error tolerance in defining the linesearch direction. Second, an inexact nonmonotone linesearch scheme is used to determine the step size for the next iteration. Under suitable assumptions, we demonstrate that InmBDCA is well-defined, with any accumulation point of the sequence generated by InmBDCA being a critical point of the problem. We also provide iteration-complexity bounds for the algorithm. Numerical experiments show that InmBDCA outperforms both the nonsmooth BDCA (nmBDCA) and the monotone version of DCA in practical scenarios.

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arXiv2024

Introducao a otimizacao de Portfolio

In this work, we introduce Modern Portfolio Theory using basic concepts from linear algebra, differential calculus, statistics, and optimization. This theory allows us to measure the return and risk of an investment portfolio, serving as a basis for decision-making in the financial market. As an application, we will present four very simple investment strategies that aim to minimize the risk of investing in only two assets, outperforming the CDI (Interbank Deposit Certificate) yield.

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arXiv2023

The difference of convex algorithm on Hadamard manifolds

In this paper, we propose a Riemannian version of the difference of convex algorithm (DCA) to solve a minimization problem involving the difference of convex (DC) function. We establish the equivalence between the classical and simplified Riemannian versions of the DCA. We also prove that, under mild assumptions, the Riemannian version of the DCA is well-defined, and every cluster point of the sequence generated by the proposed method, if any, is a critical point of the objective DC function. Additionally, we establish some duality relations between the DC problem and its dual. To illustrate the effectiveness of the algorithm, we present some numerical experiments.

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arXiv2022

Reducing the projection onto the monotone extended second-order cone to the pool-adjacent-violators algorithm of isotonic regression

This paper introduces the monotone extended second-order cone (MESOC), which is related to the monotone cone and the second-order cone. Some properties of the MESOC are presented and its dual cone is computed. Projecting onto the MESOC is reduced to the pool-adjacent-violators algorithm (PAVA) of isotonic regression. An application of MESOC to portfolio optimisation is provided. Some broad descriptions of possible MESOC-regression models are also outlined.

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arXiv2022

A boosted DC algorithm for non-differentiable DC components with non-monotone line search

We introduce a new approach to apply the boosted difference of convex functions algorithm (BDCA) for solving non-convex and non-differentiable problems involving difference of two convex functions (DC functions). Supposing the first DC component differentiable and the second one possibly non-differentiable, the main idea of BDCA is to use the point computed by the DC algorithm (DCA) to define a descent direction and perform a monotone line search to improve the decreasing the objetive function accelerating the convergence of the DCA. However, if the first DC component is non-differentiable, then the direction computed by BDCA can be an ascent direction and a monotone line search cannot be performed. Our approach uses a non-monotone line search in the BDCA (nmBDCA) to enable a possible growth in the objective function values controlled by a parameter. Under suitable assumptions, we show that any cluster point of the sequence generated by the nmBDCA is a critical point of the problem under consideration and provide some iteration-complexity bounds. Furthermore, if the first DC component is differentiable, we present different iteration-complexity bounds and prove the full convergence of the sequence under the Kurdyka-Łojasiewicz property of the objective function. Some numerical experiments show that the nmBDCA outperforms the DCA such as its monotone version.

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arXiv2021

Newton's Method with GeoGebra

In this work, we present a program in the computational environment, GeoGebra, that enables a graphical study of Newton's Method. Using this computational device, we will analyze Newton's Method convergence applied to various examples of real functions. Then, it will be given a guide to the construction of the program in GeoGebra.

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arXiv2021

On the inexact scaled gradient projection method

The purpose of this paper is to present an inexact version of the scaled gradient projection method on a convex set, which is inexact in two sense. First, an inexact projection on the feasible set is computed, allowing for an appropriate relative error tolerance. Second, an inexact non-monotone line search scheme is employed to compute a step size which defines the next iteration. It is shown that the proposed method has similar asymptotic convergence properties and iteration-complexity bounds as the usual scaled gradient projection method employing monotone line searches.

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arXiv2021

Boosted scaled subgradient method for DC programming

The purpose of this paper is to present a boosted scaled subgradient-type method (BSSM) to minimize the difference of two convex functions (DC functions), where the first function is differentiable and the second one is possibly non-smooth. Although the objective function is in general non-smooth, under mild assumptions, the structure of the problem allows to prove that the negative scaled generalized subgradient at the current iterate is a descent direction from an auxiliary point. Therefore, instead of applying the Armijo linear search and computing the next iterate from the current iterate, both the linear search and the new iterate are computed from that auxiliary point along the direction of the negative scaled generalized subgradient. As a consequence, it is shown that the proposed method has similar asymptotic convergence properties and iteration-complexity bounds as the usual descent methods to minimize differentiable convex functions employing Armijo linear search. Finally, for a suitable scale matrix the quadratic subproblems of BSSM have a closed formula, and hence, the method has a better computational performance than classical DC algorithms which must solve a convex (not necessarily quadratic) subproblem.

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arXiv2021

Inexact gradient projection method with relative error tolerance

A gradient projection method with feasible inexact projections is proposed in the present paper. The inexact projection is performed using a relative error tolerance. Asymptotic convergence analysis and iteration-complexity bounds of the method employing constant and Armijo step sizes are presented. Numerical results are reported illustrating the potential advantages of considering inexact projections instead of exact ones in some medium scale instances of a least squares problem over the spectrohedron.

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arXiv2019

Iteration-complexity and asymptotic analysis of steepest descent method for multiobjective optimization on Riemannian manifolds

The steepest descent method for multiobjective optimization on Riemannian manifolds with lower bounded sectional curvature is analyzed in this paper. The aim of the paper is twofold. Firstly, an asymptotic analysis of the method is presented with three different finite procedures for determining the stepsize, namely, Lipschitz stepsize, adaptive stepsize and Armijo-type stepsize. The second aim is to present, by assuming that the Jacobian of the objective function is componentwise Lipschitz continuous, iteration-complexity bounds for the method with these three stepsizes strategies. In addition, some examples are presented to emphasize the importance of working in this new context. Numerical experiments are provided to illustrate the effectiveness of the method in this new setting and certify the obtained theoretical results.

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arXiv2019

Newton method for finding a singularity of a special class of locally Lipschitz continuous vector fields on Riemannian manifolds

In this paper, we extend some results of nonsmooth analysis from Euclidean context to the Riemannian setting. In particular, we discuss the concept and some properties of locally Lipschitz continuous vector fields on Riemannian settings, such as Clarke generalized covariant derivative, upper semicontinuity and Rademacher theorem. We also present a version of Newton method for finding a singularity of a special class of locally Lipschitz continuous vector fields. Under mild conditions, we establish the well-definedness and local convergence of the sequence generated by the method in a neighborhood of a singularity. In particular, a local convergence result for semismooth vector fields is presented. Furthermore, under Kantorovich-type assumptions the convergence of the sequence generated by the Newton method to a solution is established, and its uniqueness in a suitable neighborhood of the starting point is verified.

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arXiv2019

Inexact Newton method with feasible inexact projections for solving constrained smooth and nonsmooth equations

In this paper, we propose a new method that combines the inexact Newton method with a procedure to obtain a feasible inexact projection for solving constrained smooth and nonsmooth equations. The local convergence theorems are established under the assumption of smoothness or semismoothness of the function that defines the equation and its regularity at the solution. In particular, we show that a sequence generated by the method converges to a solution with linear, superlinear, or quadratic rate, under suitable conditions. Moreover, some numerical experiments are reported to illustrate the practical behavior of the proposed method.

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arXiv2016

On the superlinear convergence of Newton's method on Riemannian manifolds

In this paper we study the Newton's method for finding a singularity of a differentiable vector field defined on a Riemannian manifold. Under the assumption of invertibility of covariant derivative of the vector field at its singularity, we establish the well definition of the method in a suitable neighborhood of this singularity. Moreover, we also show that the generated sequence by Newton method converges for the solution with superlinear rate.

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arXiv2015

Robust Kantorovich's theorem on Newton's method under majorant condition in Riemannian Manifolds

A robust affine invariant version of Kantorovich's theorem on Newton's method, for finding a zero of a differentiable vector field defined on a complete Riemannian manifold, is presented in this paper. In the analysis presented, the classical Lipschitz condition is relaxed by using a general majorant function, which allow to establish existence and local uniqueness of the solution as well as unifying previously results pertaining Newton's method. The most important in our analysis is the robustness, namely, is given a prescribed ball, around the point satisfying Kantorovich's assumptions, ensuring convergence of the method for any starting point in this ball. Moreover, bounds for $Q$-quadratic convergence of the method which depend on the majorant function is obtained.

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arXiv2015

Projection onto simplicial cones by Picard's method

By using Moreau's decomposition theorem for projecting onto cones, the problem of projecting onto a simplicial cone is reduced to finding the unique solution of a nonsmooth system of equations. It is shown that Picard's method applied to the system of equations associated to the problem of projecting onto a simplicial cone generates a sequence that converges linearly to the solution of the system. Numerical experiments are presented making the comparison between Picard's and semi-smooth Newton's methods to solve the nonsmooth system associated with the problem of projecting a point onto a simplicial cone.

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