A starting point strategy for nonlinear interior methods

This paper presents a strategy for choosing the initial point slacks and multipliers in interior methods for nonlinear programming It consists of rst computing a Newton like step to estimate the magnitude of these three variables and then shifting the slacks and multipliers so that they are su ciently 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 arti cial non convexities Numerical experiments on a large test set illustrate the performance of the strategy Introduction It is well known that interior methods for linear and quadratic programming perform poorly and can even fail if the starting point is unfavorable To overcome this problem it is common to employ heuristics for choosing an initial value for the variables slacks and multipliers see e g These heuristics have proved to be generally successful in practice and have been incorporated into com mercial linear programming packages In this paper we study initial point strategies for nonlinear programming This topic has not received much attention in spite of the fact that nonlinear interior methods can be as sensitive as their linear counterparts to a poor initial guess The heuristics developed for linear and quadratic programming cannot be ex tended directly to nonlinear problems First of all in linear programming an initial estimate of the solution is typically not provided by the user Moreover since the objective function and constraints are de ned everywhere there is great freedom in selecting initial values and some of the most popular strategies often choose very large values for the variables slacks and possibly multipliers see and the references therein In contrast nonlinear programming algorithms compute only local minimizers and accept user supplied initial estimates that often lie in the vicinity of a minimizer of interest Therefore initial point strategies should either respect the user supplied estimate or compute one that is not too distant from it Even large initial values of the multipliers should be avoided since they may introduce unnecessary non convexities in the problem as we discuss later on The initial point heuristics presented in this paper aim to preserve user supplied information are readily computable and allow interior methods to perform e ciently on a wide range of problems Interior Point Framework We will consider the solution of nonlinear pro gramming problems of the form minimize f x subject to h x g x Computer Science Department University of Wisconsin at Madison Madison Wisconsin Department of Electrical and Computer Engineering Northwestern University Evanston IL USA This author was supported by National Science Foundation grants CCR ATM and CCR and Department of Energy grant DE FG ER A zDepartment of Mathematics Facult es Universitaires Notre Dame de la Paix rue de Bruxelles B Namur Belgium This author was supported by the Belgian National Fund for Scienti c Research where f IR IR h IR IR and g IR IR are twice continuously di erentiable Introducing a vector of slack variables s we can restate as minimize f x subject to h x g x s s The rst order optimality conditions of can be written as