Scielo RSS <![CDATA[Computational & Applied Mathematics]]> vol. 30 num. 1 lang. en <![CDATA[SciELO Logo]]> <![CDATA[<b>Special Issue on Nonlinear Programming dedicated to the ALIO-INFORMS Joint International Meeting 2010</b>]]> <![CDATA[<b>Regularity results for semimonotone operators</b>]]> We introduce the concept of ρ-semimonotone point-to-set operators in Hilbert spaces. This notion is symmetrical with respect to the graph of T, as is the case for monotonicity, but not for other related notions, like e.g. hypomonotonicity, of which our new class is a relaxation. We give a necessary condition for ρ-semimonotonicity of T in terms of Lispchitz continuity of [T + ρ-11]-1 and a sufficient condition related to expansivity of T. We also establish surjectivity results for maximal ρ-semimonotone operators. <![CDATA[<b>Derivative-free methods for nonlinear programming with general lower-level constraints</b>]]> 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, Martínez 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. <![CDATA[<b>An SLP algorithm and its application to topology optimization</b>]]> We introduce a globally convergent sequential linear programming method for nonlinear programming. The algorithm is applied to the solution of classic topology optimization problems, as well as to the design of compliantmechanisms. The numerical results suggest that the new algorithm is faster than the globally convergent version of the method of moving asymptotes, a popular method for mechanical engineering applications proposed by Svanberg. <![CDATA[<b>An inexact subgradient algorithm for Equilibrium Problems</b>]]> We present an inexact subgradient projection type method for solving a nonsmooth Equilibrium Problem in a finite-dimensional space. The proposed algorithm has a low computational cost per iteration. Some numerical results are reported. <![CDATA[<b>Solution of a truss topology bilevel programming problem by means of an inexact restoration method</b>]]> We formulate a truss topology optimization problem as a bilevel programming problem and solve it by means of a line search type inexact restoration algorithm. We discuss details of the implementation and show results of numerical experiments. <![CDATA[<b>Stochastic Newton-like methods for computing equilibria in general equilibrium models</b>]]> Calculating an equilibrium point in general equilibrium models in many cases reduces to solving a nonlinear system of equations. Taking model parameter values as random variables with a known distribution increases the level of information provided by the model but makes computation of equilibrium points even more challenging. We propose a computationally efficient procedure based on application of the fixed Newton method for a sequence of equilibrium problems generated by simulation of parameters values. The convergence conditions of the method are derived. The numerical results presented are obtained using the neoclassic exchange model and the spatial price equilibrium model. The results show a clear difference in the quality of information obtained by solving a sequence of problems if compared with the single equilibrium problem. At the same time the proposed numerical procedure is affordable. <![CDATA[<b>Solving the dual subproblem of the Method of Moving Asymptotes using a trust-region scheme</b>]]> An alternative strategy to solve the subproblems of the Method of Moving Asymptotes (MMA) is presented, based on a trust-region scheme applied to the dual of the MMA subproblem. At each iteration, the objective function of the dual problem is approximated by a regularized spectral model. A globally convergent modification to the MMA is also suggested, in which the conservative condition is relaxed by means of a summable controlled forcing sequence. Another modification to the MMA previously proposed by the authors [Optim. Methods Softw., 25 (2010), pp. 883-893] is recalled to be used in the numerical tests. This modification is based on the spectral parameter for updating the MMA models, so as to improve their quality. The performed numerical experiments confirm the efficiency of the indicated modifications, especially when jointly combined. <![CDATA[<b>Active-set strategy in Powell's method for optimization without derivatives</b>]]> In this article we present an algorithm for solving bound constrained optimization problems without derivatives based on Powell's method [38] for derivative-free optimization. First we consider the unconstrained optimization problem. At each iteration a quadratic interpolation model of the objective function is constructed around the current iterate and this model is minimized to obtain a new trial point. The whole process is embedded within a trust-region framework. Our algorithm uses infinity norm instead of the Euclidean norm and we solve a box constrained quadratic subproblem using an active-set strategy to explore faces of the box. Therefore, a bound constrained optimization algorithm is easily extended. We compare our im_ plementation with NEWUOA and BOBYQA, Powell's algorithms for unconstrained and bound constrained derivative free optimization respectively. Numerical experiments show that, in general, our algorithm require less functional evaluations than Powell's algorithms. <![CDATA[<b>Duality results for stationary problems of open pit mine planning in a continuous function framework</b>]]> Open Pit Mine Planning problems are usually considered in a Mixed Integer Programming context. Characterizing each attainable profile by a continuous function yields a continuous framework. It allows for a more detailed modeling of slope constraints and other material properties of slanted layers. Although the resulting nonlinear programming problems are in general non-convex and non-differentiable, they provide certain advantages as one can directly compute sensitivities of optimal solutions w.r.t. small data perturbations. In this work duality results are derived for the stationary problems of the continuous framework employing an additional condition called convex-likeness. <![CDATA[<b>Two derivative-free methods for solving underdetermined nonlinear systems of equations</b>]]> In this paper, two different approaches to solve underdetermined nonlinear system of equations are proposed. In one of them, the derivative-free method defined by La Cruz, Martínez and Raydan for solving square nonlinear systems is modified and extended to cope with the underdetermined case. The other approach is a Quasi-Newton method that uses the Broyden update formula and the globalized line search that combines the strategy of Grippo, Lampariello and Lucidi with the Li and Fukushima one. Global convergence results for both methods are proved and numerical experiments are presented.