Márcia A. Gomes-Ruggiero

Comparing algorithms for solving sparse nonlinear systems of equations

This paper describes implementations of eight algorithms of Newton and quasi-Newton type for solving large sparse systems of nonlinear equations. For linear algebra calculations, a symbolic manipulation is used, as well as a static data structure introduced recently by George and Ng, which allows a partial pivoting strategy for solving linear systems. A numerical comparison of the implemented methods is presented.

Spectral Projected Gradient Method with Inexact Restoration for Minimization with Nonconvex Constraints

This work takes advantage of the spectral projected gradient direction within the inexact restoration framework to address nonlinear optimization problems with nonconvex constraints. The proposed strategy includes a convenient handling of the constraints, together with nonmonotonic features to speed up convergence. The numerical performance is assessed by experiments with hard-spheres problems, pointing out that the inexact restoration framework provides an adequate environment for the extension of the spectral projected gradient method for general nonlinearly constrained optimization.

Solving nonlinear systems of equations by means of quasi-neston methods with a nonmonotone stratgy ∗

A nonmonotone strategy for solving nonlinear systems of equations is introduced. The idea consists of combining efficient local methods with an algorithm that reduces monotonically the squared norm of the system in a proper way. The local methods used are Newtons method and two quasi-Newton algorithms. Global iterations are based on recently introduced box-constrained minimization algorithms. Numerical experiments are presented

Quantum State Tomography with incomplete data: Maximum Entropy and Variational Quantum Tomography

Whenever we do not have an informationally complete set of measurements, the estimate of a quantum state can not be uniquely determined. In this case, among the density matrices compatible with the available data, it is commonly preferred that one which is the most uncommitted with the missing information. This is the purpose of the Maximum Entropy estimation (MaxEnt) and the Variational Quantum Tomography (VQT). Here, we propose a variant of Variational Quantum Tomography and show its relationship with Maximum Entropy methods in quantum tomographies with incomplete set of measurements. We prove their equivalence in case of eigenbasis measurements, and through numerical simulations we stress their similar behavior. Hence, in the modified VQT formulation we have an estimate of a quantum state as unbiased as in MaxEnt and with the benefit that VQT can be more efficiently solved by means of linear semidefinite programs.

Nonmonotone Strategy for Minimization of Quadratics with Simple Constraints

An algorithm for quadratic minimization with simple bounds is introduced, combining, as many well-known methods do, active set strategies and projection steps. The novelty is that here the criterion for acceptance of a projected trial point is weaker than the usual ones, which are based on monotone decrease of the objective function. It is proved that convergence follows as in the monotone case. Numerical experiments with bound-constrained quadratic problems from CUTE collection show that the modified method is in practice slightly more efficient than its monotone counterpart and has a performance superior to the well-known code LANCELOT for this class of problems.

A numerical study on large-scale nonlinear solvers

Abstract We consider the solution of several nonlinear systems that come from the discretization of two-dimensional boundary value problems using well-known algorithms based on the quasi-Newton idea: Newtons method, Broydens method and the Column-Updating method. Numerical results can be useful for researchers to indicate the performances that should be improved by future algorithms or implementations.

Discrete Newton’s method with local variations for solving large-scale nonlinear systems

A globally convergent discrete Newton method is proposed for solving large-scale nonlinear systems of equations. Advantage is taken from discretization steps so that the residual norm can be reduced while the Jacobian is approximated, besides the reduction at Newtonian iterations. The Curtis–Powell–Reid (CPR) scheme for discretization is used for dealing with sparse Jacobians. Global convergence is proved and numerical experiments are presented.

A spectral updating for the method of moving asymptotes

A modified version of the method of moving asymptotes is proposed based on the spectral parameter used in the updating of a key parameter of the model. The second-order information present in the spectral parameter is thus included in the model functions that define the rational approximations. Numerical experiments indicate that the idea is promising in the sense that the cost–benefit of computing the spectral parameter is worth it for reducing the total effort of the algorithm when compared with the original version.

A projected gradient method for optimization over density matrices

An ensemble of quantum states can be described by a Hermitian, positive semidefinite and unit trace matrix called density matrix. Thus, the study of methods for optimizing a certain function (energy, entropy) over the set of density matrices has a direct application to important problems in quantum information and computation. We propose a projected gradient method for solving such problems. By exploiting the geometry of the feasible set, which is the intersection of the cone of Hermitian positive semidefinite matrices with the hyperplane defined by the unit trace constraint, we describe an efficient procedure to compute the projection onto this set using the Frobenius norm. Some important applications, such as quantum state tomography, are described and numerical experiments illustrate the effectiveness of the method when compared to previous methods based on fixed-point iterations or semidefinite programming.

A globally convergent inexact Newton method with a new choice for the forcing term

Abstract In inexact Newton methods for solving nonlinear systems of equations, an approximation to the step sk of the Newton’s system J(xk)s=−F(xk) is found. This means that sk must satisfy a condition like ‖F(xk)+J(xk)sk‖≤ηk‖F(xk)‖ for a forcing term ηk∈[0,1). Possible choices for ηk have already been presented. In this work, a new choice for ηk is proposed. The method is globalized using a robust backtracking strategy proposed by Birgin et al. (Numerical Algorithms 32:249–260, 2003), and its convergence properties are proved. Several numerical experiments with boundary value problems are presented. The numerical performance of the proposed algorithm is analyzed by the performance profile tool proposed by Dolan and Moré (Mathematical Programming Series A 91:201–213, 2002). The results obtained show a competitive inexact Newton method for solving academic and applied problems in several areas.