Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first derivatives are available, and that the constraint gradients are sparse.

We present a primal-dual interior-point algorithm with a filter line-search method for nonlinear programming. Local and global convergence properties of this method were analyzed in previous work. Here we provide a comprehensive description of the algorithm, including the feasibility restoration phase for the filter method, second-order corrections, and inertia correction of the KKT matrix. Heuristics are also considered that allow faster performance. This method has been implemented in the IPOPT code, which we demonstrate in a detailed numerical study based on 954 problems from the CUTEr test set. An evaluation is made of several line-search options, and a comparison is provided with two state-of-the-art interior-point codes for nonlinear programming.

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.

Abstract. Generalized stationary points of the mathematical program with equilibrium constraints (MPEC) are studied to better describe the limit points produced by interior point methods for MPEC. A primal-dual interior-point method is then proposed, which solves a sequence of relaxed barrier problems derived from MPEC. Global convergence results are deduced without assuming strict complementarity or the linear independence constraint qualification for MPEC (MPEC-LICQ). Under certain general assumptions, the algorithm can always find some point with strong or weak stationarity. In particular, it is shown that every limit point of the generated sequence is a strong stationary point of MPEC if the penalty parameter of the merit function is bounded. Otherwise, a certain point with weak stationarity can be obtained. Preliminary numerical results are reported, which include a case analyzed by Leyffer for which the penalty interior-point algorithm failed to find a stationary point. Key words: Global convergence, interior-point methods, mathematical programming with equilibrium constraints, stationary point

This paper describes numerical experience with solving MPECs as NLPs on a large collection of test problems. The key idea is to use off-the-shelf NLP solvers to tackle large instances of MPECs. It is shown that SQP methods are very well suited to solving MPECs and at present outperform Interior Point solvers both in terms of speed and reliability. All NLP solvers also compare very favourably to special MPEC solvers on tests published in the literature.

A new method for derivative-free optimization is presented. It is designed for solving problems in which the objective function is smooth and the number of variables is moderate, but the gradient is not available. The method generates a model that interpolates the objective function at a set of sample points, and uses trust regions to promote convergence. The step-generation subproblem ensures that all the iterates satisfy a geometric condition and are therefore adequate for updating the model. The sample points are updated using a scheme that improves the accuracy of the interpolation model when needed. Two versions of the method are presented: one using linear models and the other using quadratic models. Numerical tests comparing the new approach with established methods for derivate-free optimization are reported. This work was supported by National Science Foundation grant CCR-9987818, and by Department of Energy grant DE-FG02-87ER25047-A004. y Department of Industrial ...

In recent work, the local convergence behavior of path-following interior-point methods and sequential quadratic programming methods for nonlinear programming has been investigated for the case in which the active constraint gradients may fail to be linearly independent at the solution, but the Mangasarian-Fromovitz constraint qualication is satisfied. In this paper, we describe a stabilization of the primal-dual interior-point approach that ensures rapid local convergence under these conditions without enforcing the usual centrality condition associated with path-following methods. The stabilization takes the form of perturbations to the coefficient matrix in the step equations that vanish as the iterates converge to the solution.