arXiv subjects

Douglas S. Gonçalves

Publications and source records attributed to Douglas S. Gonçalves.

7 recordsLinked to original sources

arXiv2025

A Cubic Regularization Method for Multiobjective Optimization

This work introduces a new cubic regularization method for nonconvex unconstrained multiobjective optimization problems. At each iteration of the method, a model associated with the cubic regularization of each component of the objective function is minimized. This model allows approximations for the first- and second-order derivatives, which must satisfy suitable error conditions. One interesting feature of the proposed algorithm is that the regularization parameter of the model and the accuracy of the derivative approximations are jointly adjusted using a nonmonotone line search criterion. Implementations of the method, where derivative information is computed using finite difference strategies, are discussed. It is shown that, under the assumption that the Hessians of the objectives are globally Lipschitz continuous, the method requires at most $\mathcal{O}(Cε^{-3/2})$ iterations to generate an $ε$-approximate Pareto critical. In particular, if the first- and second-order derivative information is computed using finite differences based solely on function values, the method requires at most $\mathcal{O}(n^{1-β}ε^{-3/2})$ iterations, corresponding to $\mathcal{O}\left(mn^{3-β}\varepsilon^{-\frac{3}{2}}\right)$ function evaluations, where $n$ is the dimension of the domain of the objective function, $m$ is the number of objectives, and $β\in [0,1]$ is a constant associated with the stepsize used in the finite-difference approximation. We further discuss the global convergence and local convergence rate of the method. Specifically, under the local convexity assumption, we show that the method achieves superlinear convergence when the first derivative is computed exactly, and quadratic convergence when both first- and second-order derivatives are exact.

math.OC↗

arXiv2025

On the regularization property of Levenberg-Marquardt method with Singular Scaling for nonlinear inverse problems

Recently, in Applied Mathematics and Computation 474 (2024) 128688, a Levenberg-Marquardt method (LMM) with Singular Scaling was analyzed and successfully applied in parameter estimation problems in heat conduction where the use of a particular singular scaling matrix (semi-norm regularizer) provided approximate solutions of better quality than those of the classic LMM. Here we propose a regularization framework for the Levenberg-Marquardt method with Singular Scaling (LMMSS) applied to nonlinear inverse problems with noisy data. Assuming that the noise-free problem admits exact solutions (zero-residual case), we consider the LMMSS iteration where the regularization effect is induced by the choice of a possibly singular scaling matrix and an implicit control of the regularization parameter. The discrepancy principle is used to define a stopping index that ensures stability of the computed solutions with respect to data perturbations. Under a new Tangent Cone Condition, we prove that the iterates obtained with noisy data converge to a solution of the unperturbed problem as the noise level tends to zero. This work represents a first step toward the analysis of regularizing properties of the LMMSS method and extends previous results in the literature on regularizing LM-type methods.

math.NA↗

arXiv2024

Improved convergence rates for the multiobjective Frank-Wolfe method

This paper analyzes the convergence rates of the {\it Frank-Wolfe } method for solving convex constrained multiobjective optimization. We establish improved convergence rates under different assumptions on the objective function, the feasible set, and the localization of the limit point of the sequence generated by the method. In terms of the objective function values, we firstly show that if the objective function is strongly convex and the limit point of the sequence generated by the method lies in the relative interior of the feasible set, then the algorithm achieves a linear convergence rate. Next, we focus on a special class of problems where the feasible constraint set is $(α,q)$-uniformly convex for some $α>0$ and $q \geq 2$, including, in particular, $\ell_p$-balls for all $p>1$. In this context, we prove that the method attains: (i) a rate of $\mathcal{O}(1/k^\frac{q}{q-1})$ when the objective function is strongly convex; and (ii) a linear rate (if $q=2$) or a rate of $\mathcal{O}(1/k^{\frac{q}{q-2}})$ (if $q>2$) under an additional assumption, which always holds if the feasible set does not contain an unconstrained weak Pareto point. We also discuss enhanced convergence rates for the algorithm in terms of an optimality measure. Finally, we provide some simple examples to illustrate the convergence rates and the set of assumptions.

math.OC↗

arXiv2022

First-order methods for the convex hull membership problem

The convex hull membership problem (CHMP) consists in deciding whether a certain point belongs to the convex hull of a finite set of points, a decision problem with important applications in computational geometry and in foundations of linear programming. In this study, we review, compare and analyze first-order methods for CHMP, namely, Frank-Wolfe type methods, Projected Gradient methods and a recently introduced geometric algorithm, called Triangle Algorithm (TA). We discuss the connections between this algorithm and Frank-Wolfe, showing that TA can be interpreted as an inexact Frank-Wolfe. Despite this similarity, TA is strongly based on a theorem of alternatives known as distance duality. By using this theorem, we propose suitable stopping criteria for CHMP to be integrated into Frank-Wolfe type and Projected Gradient, specializing these methods to the membership decision problem. Interestingly, Frank-Wolfe integrated with such stopping criteria coincides with a greedy version of the Triangle Algorithm which is, in its turn, equivalent to an algorithm due to von Neumann. We report numerical experiments on random instances of CHMP, carefully designed to cover different scenarios, that indicate which algorithm is preferable according to the geometry of the convex hull and the relative position of the query point. Concerning potential applications, we present two illustrative examples, one related to linear programming feasibility problems and another related to image classification problems.

math.OC↗

arXiv2022

A two-phase rank-based algorithm for low-rank matrix completion

Matrix completion aims to recover an unknown low-rank matrix from a small subset of its entries. In many applications, the rank of the unknown target matrix is known in advance. In this paper, first we revisit a recently proposed rank-based heuristic for "known-rank" matrix completion and establish a condition under which the generated sequence is quasi-Fejér convergent to the solution set. Then, by including an acceleration mechanism similar to Nesterov's acceleration, we obtain a new heuristic. Even though the convergence of such heuristic cannot be granted in general, it turns out that it can be very useful as a warm-start phase, providing a suitable estimate for the regularization parameter and a good starting-point, to an accelerated Soft-Impute algorithm. Numerical experiments with both synthetic and real data show that the resulting two-phase rank-based algorithm can recover low-rank matrices, with relatively high precision, faster than other well-established matrix completion algorithms.

math.OC↗

arXiv2019

Levenberg-Marquardt methods with inexact projections for constrained nonlinear systems

In this paper, we first propose a new Levenberg-Marquardt method for solving constrained (and not necessarily square) nonlinear systems. Basically, the method combines the unconstrained Levenberg-Marquardt method with a type of feasible inexact projection. The local convergence of the new method as well as results on its rate are established by using an error bound condition, which is weaker than the standard full-rank assumption. We further present and analyze a global version of the first method by means of a nonmonotone line search technique. Finally, numerical experiments illustrating the practical advantages of the proposed schemes are reported.

math.OC↗

arXiv2012

Local solutions of Maximum Likelihood Estimation in Quantum State Tomography

Maximum likelihood estimation is one of the most used methods in quantum state tomography, where the aim is to reconstruct the density matrix of a physical system from measurement results. One strategy to deal with positivity and unit trace constraints is to parameterize the matrix to be reconstructed in order to ensure that it is physical. In this case, the negative log-likelihood function in terms of the parameters, may have several local minima. In various papers in the field, a source of errors in this process has been associated to the possibility that most of these local minima are not global, so that optimization methods could be trapped in the wrong minimum, leading to a wrong density matrix. Here we show that, for convex negative log-likelihood functions, all local minima of the unconstrained parameterized problem are global, thus any minimizer leads to the maximum likelihood estimation for the density matrix. We also discuss some practical sources of errors.

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