Jorge Nocedal z An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direct use of second order derivatives. This framework permits primal and primal-dual steps, but the paper focuses on the primal version of the new algorithm. An analysis of the convergence properties of this method is presented. Key words: constrained optimization, interior point method, large-scale optimization, nonlinear programming, primal method, primal-dual method, SQP iteration, barrier method, trust region method.

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.

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...

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

Two interior-point algorithms are proposed and analyzed, for the (local) solution of (possibly) indefinite quadratic programming problems. They are of the Newton-KKT variety in that (much like in the case of primal-dual algorithms for linear programming) search directions for the “primal ” variables and the Karush-Kuhn-Tucker (KKT) multiplier estimates are components of the Newton (or quasi-Newton)

We propose and analyze a primal-dual interior point method of the “feasible ” type, with the additional property that the objective function decreases at each iteration. A distinctive feature of the method is the use of different barrier parameter values for each constraint, with the purpose of better steering the constructed sequence away from non-KKT stationary points. Assets of the proposed scheme include relative simplicity of the algorithm and of the convergence analysis, strong global and local convergence properties, and good performance in preliminary tests. In addition, the initial point is allowed to lie on the boundary of the feasible set.

We study an infeasible interior-point trust-region method for constrained minimization. This method uses a logarithmic-barrier function for the slack variables and updates the slack variables using second-order correction. We show that if a certain set containing the iterates is bounded and the origin is not in the convex hull of the nearly active constraint gradients everywhere on this set, then any cluster point of the iterates is a 1st-order stationary point. If the cluster point satisfies an additional assumption (which holds when the constraints are linear or when the cluster point satisfies strict complementarity and a local error bound holds), then it is a 2nd-order stationary point. Key words. Nonlinear program, logarithmic-barrier function, interior-point method, trustregion strategy, 1st- and 2nd-order stationary points, semidefinite programming. 1 Introduction We consider the nonlinear program with inequality constraints: minimize f(x) subject to g(x) = [g 1 (x) g m (...

In this paper we consider a class of nonlinear programming problems that arise from the discretization of optimal control problems with bounds on both the state and the control variables. For this class of problems, we analyze constraint qualifications and optimality conditions in detail. We derive an affine-scaling and two primal-dual interior-point Newton algorithms by applying, in an interior-point way, Newton's method to equivalent forms of the first-order optimality conditions. Under appropriate assumptions, the interior-point Newton algorithms are shown to be locally well-defined with a q-quadratic rate of local convergence. By using the structure of the problem, the linear algebra of these algorithms can be reduced to the null space of the Jacobian of the equality constraints. The similarities between the three algorithms are pointed out, and their corresponding versions for the general nonlinear programming problem are discussed.

We analyze the properties that optimization algorithms must possess in order to prevent convergence to non-stationary points for the merit function. We show that demanding the exact satisfaction of constraint linearizations results in difficulties in a wide range of optimization algorithms. Feasibility control is a mechanism that prevents convergence to spurious solutions by ensuring that sufficient progress towards feasibility is made, even in the presence of certain rank deficiencies. The concept of feasibility control is studied in this paper in the context of Newton methods for nonlinear systems of equations and equality constrained optimization, as well as in interior methods for nonlinear programming. This work was supported by National Science Foundation grant CDA-9726385 and by Department of Energy grant DE-FG02-87ER25047-A004. y To appear in the proceedings of the Foundations of Computational Mathematics Meeting held in Oxford, England, in July 1999 z Department o...

This paper addresses the local convergence properties of the ane-scaling interior-point algorithm for nonlinear programming. The analysis of local convergence is developed in terms of parameters that control the interiorpoint scheme and the size of the residual of the linear system that provides the step direction. The analysis follows the classical theory for quasi-Newton methods and addresses q-linear, qsuperlinear, and q-quadratic rates of convergence.