The paper describes an interior-point algorithm for nonconvex nonlinear programming which is a direct extension of interior--point methods for linear and quadratic programming. Major modifications include a merit function and an altered search direction to ensure that a descent direction for the merit function is obtained. Preliminary numerical testing indicates that the method is robust. Further, numerical comparisons with MINOS and LANCELOT show that the method is efficient, and has the promise of greatly reducing solution times on at least some classes of models.

The design and implementation of a new algorithm for solving large nonlinear programming problems is described. It follows a barrier approach that employs sequential quadratic programming and trust regions to solve the subproblems occurring in the iteration. Both primal and primal-dual versions of the algorithm are developed, and their performance is illustrated in a set of numerical tests. Key words: constrained optimization, interior point method, large-scale optimization, nonlinear programming, primal method, primal-dual method, successive quadratic programming, trust region method.

Abstract. Interior methods are an omnipresent, conspicuous feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interior-point techniques were popular during the 1960s for solving nonlinearly constrained problems. However, their use for linear programming was not even contemplated because of the total dominance of the simplex method. Vague but continuing anxiety about barrier methods eventually led to their abandonment in favor of newly emerging, apparently more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost without exception regarded as a closed chapter in the history of optimization. This picture changed dramatically with Karmarkar’s widely publicized announcement in 1984 of a fast polynomial-time interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, interior methods have advanced so far, so fast, that their influence has transformed both the theory and practice of constrained optimization. This article provides a condensed, selective look at classical material and recent research about interior methods for nonlinearly constrained optimization.

Using a simple analytical example, we demonstrate that a class of interior point methods for general nonlinear programming, including some current methods, is not globally convergent. It is shown that those algorithms do produce limit points that are neither feasible nor stationary points of some measure of the constraint violation, when applied to a well-posed problem. 1 Introduction Over the past decade a variety of interior point methods for nonconvex nonlinear programming (NLP) have been proposed and found to be efficient in practice (see e.g. [1]--[4], [6]--[8], [10]--[12]). Based on earlier work [5], these methods come in different varieties, such as primal or primal-dual methods, line search or trust region methods, with different merit functions, different strategies to update the barrier parameter, etc. For some algorithms, theoretical global convergence properties have been proved. It has been shown that under certain assumptions the considered method converges to a loca...

In this paper, the filter technique of Fletcher and Leyffer (1997) is used to globalize the primaldual interior-point algorithm for nonlinear programming, avoiding the use of merit functions and the updating of penalty parameters. The new algorithm decomposes the primal-dual step obtained from the perturbed first-order necessary conditions into a normal and a tangential step, whose sizes are controlled by a trust-region type parameter. Each entry in the filter is a pair of coordinates: one resulting from feasibility and centrality, and associated with the normal step; the other resulting from optimality (complementarity and duality), and related with the tangential step. Global convergence to first-order critical points is proved for the new primal-dual interior-point filter algorithm.

An exact-penalty-function-based scheme---inspired from an old idea due to Mayne and Polak (Math. Prog., vol. 11, 1976, pp. 67--80)---is proposed for extending to general smooth constrained optimization problems any given feasible interior-point method for inequality constrained problems. It is shown that the primal-dual interior-point framework allows for a simpler penalty parameter update rule than that discussed and analyzed by the originators of the scheme in the context of first order methods of feasible direction. Strong global and local convergence results are proved under mild assumptions. In particular, (i) the proposed algorithm does not su#er a common pitfall # Department of Electrical and Computer Engineering and Institute for Systems Research, University of Maryland, College Park, MD 20742, USA + IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA # Applied Physics Laboratory, Laurel, MD 20723, USA Alphatech, Arlington, VA 22203, USA recently pointed out by Wachter and Biegler; and (ii) the positive definiteness assumption on the Hessian estimate, made in the original version of the algorithm, is relaxed, allowing for the use of exact Hessian information, resulting in local quadratic convergence. Promising numerical results are reported.

Jorge Nocedal z We study the local convergence of a primal-dual interior point method for nonlinear programming. A linearly convergent version of this algorithm has been shown in [2] to be capable of solving large and di cult non-convex problems. But for the algorithm to reach its full potential, it must converge rapidly to the solution. In this paper we describe how to design the algorithm so that it converges superlinearly on regular problems. Key words: constrained optimization, interior point method, large-scale optimization, nonlinear programming, primal method, primal-dual method, successive quadratic programming.

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 stationarity 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

Interior methods are a central, striking feature of the constrained optimization landscape today, but it was not always so. Primarily in the form of barrier methods, interiorpoint techniques were widely used during the 1960s to solve nonlinearly constrained problems. However, their use for linear programming was not even contemplated because of the total dominance of the simplex method. During the 1970s, barrier methods were superseded by newly emerging, apparently more efficient alternatives such as augmented Lagrangian and sequential quadratic programming methods. By the early 1980s, barrier methods were almost universally regarded as a closed chapter in the history of optimization. This picture changed dramatically in the mid-1980s. In 1984, Karmarkar announced a fast polynomial-time interior method for linear programming; in 1985, a formal connection was established between his method and classical barrier methods. Since then, the new incarnations of interior methods ha...

A slack-based feasible interior point method is described which can be derived as a modification of infeasible methods. The modification is minor for most line search methods, but trust region methods require special attention. It is shown how the Cauchy point, which is often computed in trust region methods, must be modified so that the feasible method is effective for problems containing both equality and inequality constraints. The relationship between slack-based methods and traditional feasible methods is discussed. Numerical results showing the relative performance of feasible versus infeasible interior point methods are presented.