Maria A. Diniz-Ehrhardt

Derivative-free methods for nonlinear programming with general lower-level constraints

Augmented Lagrangian methods for derivative-free continuous optimization with constraints are introduced in this paper. The algorithms inherit the convergence results obtained by Andreani, Birgin, Martinez and Schuverdt for the case in which analytic derivatives exist and are available. In particular, feasible limit points satisfy KKT conditions under the Constant Positive Linear Dependence (CPLD) constraint qualification. The form of our main algorithm allows us to employ well established derivative-free subalgorithms for solving lower-level constrained subproblems. Numerical experiments are presented.

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 parallel projection method for overdetermined nonlinear systems of equations

We consider overdetermined nonlinear systems of equationsF(x)=0, whereF: ℝn → ℝm,m≥n. For this type of systems we define “weighted least square distance” (WLSD) solutions, which represent an alternative to classical least squares solutions and to other solutions based on residual normas. We introduce a generalization of the classical method of Cimmino for linear systems and we prove local convergence results. We introduce a practical strategy for improving the global convergence properties of the method. Finally, numerical experiments are presented.

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.

Numerical analysis of leaving-face parameters in bound-constrained quadratic minimization ∗

In this work we focus our attention on the quadratic subproblem of trust-region algorithms for large-scale bound-constrained minimization. An approach that combines a mild active set strategy with gradient projection techniques is employed in the solution of large-scale bound-constrained quadratic problems. To fill in some gaps that have appeared in previous work, we propose and analyze heuristics which dynamically choose the parameters in charge of the decision of leaving or not the current face of the feasible set. The numerical analysis is based on problem from CUTE collection and randomly generated convex problems with controlled conditioning and degeneracy. The practical consequences of an appropriate decision of such parameters have shown to be crucial, particularly when dual degenerate problems are solved.

Parallel Projection Methods and the Resolution of Ill-Posed Problems

Abstract In this paper, we consider a modification of the parallel projection method for solving overdetermined nonlinear systems of equations introduced recently by Diniz-Ehrhardt and Martinez [1]. This method is based on the classical Cimminos algorithm for solving linear systems. The components of the function are divided into small blocks, as an attempt to correct the intrinsic ill-conditioning of the system, and the new iteration is a convex combination of the projections onto the linear manifolds defined by different blocks. The modification suggested here was motivated by the application of the method to the resolution of a nonlinear Fredholm first kind integral equation. We prove convergence results and we report numerical experiments.

Successive Projection Methods for the Solution of Overdetermined Nonlinear Systems

We analyze a generalization of the classical Kaczmarz method for overdetermined nonlinear systems of equations with a convex constraint, where the feasible region is, in general, empty. We prove a local convergence theorem to fixed points of the algorithmic mapping. We defined a stopping rule for ill-conditioned problems, based on the behavior of the increment norm ∥x k+1 − x k ∥. We show numerical experiments.

A pattern search and implicit filtering algorithm for solving linearly constrained minimization problems with noisy objective functions

ABSTRACT PSIFA – Pattern Search and Implicit Filtering Algorithm – is a derivative-free algorithm that has been designed for linearly constrained problems with noise in the objective function. It combines some elements of the pattern search approach of Lewis and Torczon with ideas from the method of implicit filtering of Kelley enhanced with a further analysis of the current face and a simple extrapolation strategy for updating the step length. The feasible set is explored by PSIFA without any particular assumption about its description, being the equality constraints handled in their original formulation. Besides, compact bounds for the variables are not mandatory. The global convergence analysis is presented, encompassing the degenerate case, under mild assumptions. Numerical experiments with linearly constrained problems from the literature were performed. Additionally, problems with the feasible set defined by polyhedral 3D cones with several degrees of degeneration at the solution were addressed, including noisy functions that are not covered by the theoretical hypotheses. To put PSIFA in perspective, comparative tests have been prepared, with encouraging results.

A computer model for particle-like simulation in broiler houses

The behavior of chickens in broiler houses is affected by many conditions that involve, from the dimensions and design of the house, to the real-time control of temperature and humidity. A lot of experience is available which provides useful recommendations leading to reasonable efficiency of broiler houses. Trying different alternatives in practice is, or course, very expensive. For this reason, computer simulation becomes an extremely useful tool. Presently, standard computers make it possible to build and run simulation models in which each broiler is considered as a single “particle” whose behavior is subject to the interaction with other broilers and the environment. The objective of this paper is to introduce, discuss, and analyze a new computational model, that, to the best of our knowledge, is the first of this type. The model considers the displacements of the chickens as being analogous to the motion of physico-chemical particles (atoms or molecules), relying on Langevin dynamics and taking temperature, air speed, humidity, house dimensions, chicken population, availability of eaters, and water drinking systems into consideration. The parameters of our model were tunned both using extreme and standard assumptions on the behavior of chickens in a broiler house.