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 propose performance profiles --- distribution functions for a performance metric --- as a tool for benchmarking and comparing optimization software. We show that performance profiles combine the best features of other tools for performance evaluation. 1 Introduction The benchmarking of optimization software has recently gained considerable visibility. Hans Mittlemann's [13] work on a variety of optimization software has frequently uncovered deficiencies in the software and has generally led to software improvements. Although Mittelmann's efforts have gained the most notice, other researchers have been concerned with the evaluation and performance of optimization codes. As recent examples, we cite [1, 2, 3, 4, 6, 12, 17]. The interpretation and analysis of the data generated by the benchmarking process are the main technical issues addressed in this paper. Most benchmarking efforts involve tables displaying the performance of each solver on each problem for a set of metrics such...

. This paper describes a software package, called LOQO, which implements a primaldual interior-point method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex and general nonlinear programming, since a detailed paper describing these extensions were published recently elsewhere. In particular, we emphasize the importance of establishing and maintaining symmetric quasidefiniteness of the reduced KKT system. We show that the industry standard MPS format can be nicely formulated in such a way to provide quasidefiniteness. Computational results are included for a variety of linear and quadratic programming problems. 1. INTRODUCTION LOQO is a software package for solving general (smooth) nonlinear optimization problems. It implements an infeasible-primal-dual path-following method. For linear programming, such methods were first proposed independently by Lust...

Abstract. In this paper, we present global and local convergence results for an interior-point method for nonlinear programming and analyze the computational performance of its implementation. The algorithm uses an ℓ1 penalty approach to relax all constraints, to provide regularization, and to bound the Lagrange multipliers. The penalty problems are solved using a simplified version of Chen and Goldfarb’s strictly feasible interior-point method [12]. The global convergence of the algorithm is proved under mild assumptions, and local analysis shows that it converges Q-quadratically for a large class of problems. The proposed approach is the first to simultaneously have all of the following properties while solving a general nonconvex nonlinear programming problem: (1) the convergence analysis does not assume boundedness of dual iterates, (2) local convergence does not require the Linear Independence Constraint Qualification, (3) the solution of the penalty problem is shown to locally converge to optima that may not satisfy the Karush-Kuhn-Tucker conditions, and (4) the algorithm is applicable to mathematical programs with equilibrium constraints. Numerical testing on a set of general nonlinear programming problems, including degenerate problems and infeasible problems, confirm the theoretical results. We also provide comparisons to a highly-efficient nonlinear solver and thoroughly analyze the effects of enforcing theoretical convergence guarantees on the computational performance of the algorithm. 1.

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 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

As the World Wide Web experiences increasing commercial and mission-critical use, server systems are expected to deliver high and predictable performance. The phenomenal improvement in microprocessor speeds, coupled with the deployment of clusters of commodity workstations has enabled server systems to meet the continually increasing performance demands in a cost-effective and scalable manner. However, as the volume, variety and sophistication of services oered by server systems increase, eective support for providing dierentiated and predictable quality of service has also become important. For example, it is often desirable to dierentiate between the resources allocated to virtual web sites hosted on a server system so as to provide predictable performance to individual sites, regardless of the load imposed upon others. Server systems lack adequate support for providing predictable performance to hosted services in terms of metrics that are meaningful to server applications, such...