An Interior Algorithm for Nonlinear Optimization That Combines Line Search and Trust Region Steps R.A. Waltz * J.L. Morales † J. Nocedal * D. Orban * September 8, 2004 Abstract An interior-point method for nonlinear programming is presented. It enjoys the flexibility of switching between a line search method that computes steps by factoring the primal-dual equations and a trust region method that uses a conjugate gradient iteration. Steps computed by direct factorization are always tried first, but if they are deemed ineffective, a trust region iteration that guarantees progress toward stationar- ity is invoked. To demonstrate its effectiveness, the algorithm is implemented in the Knitro [6, 28] software package and is extensively tested on a wide selection of test problems. 1 Introduction In this paper we describe an interior method for nonlinear programming and discuss its software implementation and numerical performance. A typical iteration computes a pri- mary step by solving the primal-dual equations (using direct linear algebra) and performs a line search to ensure decrease in a merit function. In order to obtain global convergence in the presence of nonconvexity and Hessian or Jacobian rank deficiencies, the primary step is replaced, under certain circumstances, by a safeguarding trust region step. The algorithm can use exact second derivatives of the objective function and constraints or quasi-Newton approximations. The motivation for this paper is to develop a new interior point algorithm, implemented in the Knitro software package [28], which is more robust and efficient than either a pure trust region or a pure line search interior approach. The algorithm implemented in the first release of Knitro [6] is a trust region method that uses a null-space decomposition and a projected conjugate gradient iteration to compute the step. This iterative approach has the advantage that the Hessian of the Lagrangian need not be formed or factored, which is * Department of Electrical and Computer Engineering, Northwestern University. These authors were supported by National Science Foundation grants CCR-9987818, ATM-0086579, and CCR-0219438 and Department of Energy grant DE-FG02-87ER25047-A004. † Departamento de Matem´aticas, ITAM, M´ exico. This author was supported by Asociaci´on Mexicana de Cultura, A.C. and CONACyT grant 39372-A. 1