A starting-point strategy for nonlinear interior methods

AbstractThis paper presents a strategy for choosing the initial point, slacks, and multipliers in interior methods for nonlinear programming. It consists of first computing a Newton-like step to estimate the magnitude of these three variables and then shifting the slacks and multipliers so that they are sufficiently positive. The new strategy has the option of respecting the initial estimate of the solution given by the user, and attempts to avoid the introduction of artificial nonconvexities. Numerical experiments on a large test set illustrate the performance of the strategy

A one-phase interior point method for nonconvex optimization

The work of Wachter and Biegler suggests that infeasible-start interior point methods (IPMs) developed for linear programming cannot be adapted to nonlinear optimization without significant modification, i.e., using a two-phase or penalty method. We propose an IPM that, by careful initialization and updates of the slack variables, is guaranteed to find a first-order certificate of local infeasibility, local optimality or unboundedness of the (shifted) feasible region. Our proposed algorithm differs from other IPM methods for nonconvex programming because we reduce primal feasibility at the same rate as the barrier parameter. This gives an algorithm with more robust convergence properties and closely resembles successful algorithms from linear programming. We implement the algorithm and compare with IPOPT on a subset of CUTEst problems. Our algorithm requires a similar median number of iterations, but fails on only 9% of the problems compared with 16% for IPOPT. Experiments on infeasible variants of the CUTEst problems indicate superior performance for detecting infeasibility. The code for our implementation can be found at https://github.com/ohinder/OnePhase .Comment: fixed typo in sign of dual multiplier in KKT syste

Moment conditions for the quadratic regression model with measurement error

We consider a new estimator for the quadratic errors-in-variables model that exploits higher-order moment conditions under the assumption that the distribution of the measurement error is symmetric and free of excess kurtosis. Our approach contributes to the literature by not requiring any side information and by straightforwardly allowing for one or more error-free control variables. We propose a Wald-type statistical test, based on an auxiliary method-of-moments estimator, to verify a necessary condition for our estimator's consistency. We derive the asymptotic properties of the estimator and the statistical test and illustrate their finite-sample properties by means of a simulation study and an empirical application to existing data from the literature. Our simulations show that the method-of-moments estimator performs well in terms of bias and variance and even exhibits a certain degree of robustness to the distributional assumptions about the measurement error. In the simulation experiments where such robustness is not present, our statistical test already has high power for relatively small samples

Algorithme intelligent d'optimisation d'un design structurel de grande envergure

RÉSUMÉ L’implémentation d’un système automatisé d’aide à la décision en design et d’optimisation structurelle peut donner un avantage significatif à toute industrie oeuvrant dans le domaine du design de pièces mécaniques. En effet, en fournissant des idées de solutions au designer ou en améliorant les solutions existantes pendant qu’il n’est pas au travail, ce système lui permet de réduire le temps de conception, ou encore d’explorer davantage de possibilités de design dans un même délai de réalisation. Cette thèse présente une approche originale permettant l’automatisation d’un processus de design basée sur le raisonnement par cas (RPC), mieux connu sous l’appellation « Case-Based Reasoning » ou CBR. Cette approche a été développée avec l’optique de nécessiter le moins de ressources possible pour l’entretien et le fonctionnement du système. Elle nécessite toutefois une quantité appréciable de ressources lors de l’implémentation, et convient par conséquent davantage aux problèmes de design de grande envergure pour lesquels on envisage à moyen terme de répondre à plusieurs ensembles de spécifications différentes. Dans un premier temps, le processus de RPC utilise une banque de données contenant toutes les solutions antérieures connues aux problèmes de design similaires. Ensuite, une sélection de solutions de la banque de données est choisie en comparant les spécifications actuelles du problème avec celles des solutions antérieures. Un réseau de neurones substitut est alors utilisé pour produire une solution en adaptant les solutions antérieures aux spécifications actuelles. Les solutions émergeant du RPC servent ensuite à générer chacune un îlot de solutions initiales pour un algorithme génétique oeuvrant lors de la phase de raffinement du processus.----------ABSTRACT The implementation of an automated decision support system in the field of design and structural optimisation can give a significant advantage to any industry working on mechanical designs. Indeed, by providing solution ideas to a designer or by upgrading existing design solutions while the designer is not at work, the system may reduce the project cycle time, or allow more time to produce a better design. This thesis presents a new approach to automate a design process based on Case-Based Reasoning (CBR), in combination with a new genetic algorithm named Genetic Algorithm with Territorial core Evolution (GATE). This approach was developed in order to reduce the operating cost of the process. However, as the system implementation cost is quite expensive, the approach is better suited for large scale design problem, and particularly for design problems that the designer plans to solve for many different specification sets. First, the CBR process uses a databank filled with every known solution to similar design problems. Then, the closest solutions to the current problem in term of specifications are selected. After this, during the adaptation phase, an artificial neural network (ANN) interpolates amongst known solutions to produce an additional solution to the current problem using the current specifications as inputs. Each solution produced and selected by the CBR is then used to initialize the population of an island of the genetic algorithm. The algorithm will optimise the solution further during the refinement phase