Abstract. We present an extension, for nonlinear optimization under linear constraints, of an algorithm for quadratic programming using a trust region idea introduced by Ye and Tse [Math. Programming, 44 (1989), pp. 157-179] and extended by Bonnans and Bouhtou [RAIRO Rech. Opér., 29 (1995), pp. 195-217]. Due to the nonlinearity of the cost, we use a linesearch in order to reduce the step if necessary. We prove that, under suitable hypotheses, the algorithm converges to a point satisfying the first-order optimality system, and we analyze under which conditions the unit stepsize will be asymptotically accepted.Key words. trust region, quadratic model, linesearch, interior points AMS subject classifications. 90C30, 65K05, 49M40 PII. S10526234932506391. Introduction. In this paper, we study an algorithm for minimizing a nonlinear cost under linear constraints. We consider problems with linear equality constraints and nonnegative variables. At each step, a direction is computed by minimizing a convex quadratic model over an ellipsoidal trust region, and then a linesearch of Armijo type is performed in this direction. At each iteration, the ellipsoid of the quadratic problem is so small that it forces the nonnegativity constraints to be satisfied. However, the ellipsoid is not necessarily contained in the set of feasible points.In the case of linear programming (LP) or convex quadratic programming (QP), we may assume the quadratic model to be equal to the cost function. Then the unit step will be accepted by the linesearch. In the case of LP, the algorithm is then reduced to the celebrated Dikin's algorithm [10] (see also Tsuchiya [26]). Ye and Tse [27] have extended this algorithm to convex quadratic programming using the trust region idea. This problem was also considered by Sun [25]. Bonnans and Bouhtou [2] studied such methods for nonconvex quadratic problems by taking a variable size for the trust region. An early extension of trust region algorithms to nonlinear costs is done in Dikin and Zorkalcev [11]. Among the related work, we quote Gonzaga and Carlos [13]. Interior point algorithms for the solution of constrained convex optimization problems have been studied by many other researchers; see, for instance, Den Hertog, Roos, and Terlaky [8] Dennis, Heinkenschloss, and Vicente [9], and Coleman and Li [7]. Gonzaga [14] explores the shape of the trust regions to generate ellipsoidal regions adapted to the shape of the feasible set. The resulting algorithm generates sequences of points in the interior of the feasible set.In this paper, we obtain some results of global convergence, comparable to those obtained in [2] for QP; by global convergence we only mean that the limit points of the sequence generated by the algorithm satisfy the first-order optimality system. The main novelty of the paper, however, is in the local analysis in the vicinity of
