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.

A comparison of algorithms for maximum entropy parameter estimation Conditional maximum entropy (ME) models provide a general purpose machine learning technique which has been successfully applied to fields as diverse as computer vision and econometrics, and which is used for a wide variety of classification problems in natural language processing. However, the flexibility of ME models is not without cost. While parameter estimation for ME models is conceptually straightforward, in practice ME models for typical natural language tasks are very large, and may well contain many thousands of free parameters. In this paper, we consider a number of algorithms for estimating the parameters of ME models, including iterative scaling, gradient ascent, conjugate gradient, and variable metric methods. Surprisingly, the standardly used iterative scaling algorithms perform quite poorly in comparison to the others, and for all of the test problems, a limited-memory variable metric algorithm outperformed the other choices.

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.

This survey covers the state of the art of techniques for solving general purpose constrained global optimization problems and continuous constraint satisfaction problems, with emphasis on complete techniques that provably nd all solutions (if there are nitely many). The core of the material is presented in sucient detail that the survey may serve as a text for teaching constrained global optimization.

Abstract. Augmented Lagrangian methods with general lower-level constraints are considered in the present research. These methods are useful when efficient algorithms exist for solving subproblems in which the constraints are only of the lower-level type. Inexact resolution of the lower-level constrained subproblems is considered. Global convergence is proved using the Constant Positive Linear Dependence constraint qualification. Conditions for boundedness of the penalty parameters are discussed. The reliability of the approach is tested by means of a comparison against Ipopt and Lancelot B. The resolution of location problems in which many constraints of the lower-level set are nonlinear is addressed, employing the Spectral Projected Gradient method for solving the subproblems. Problems of this type with more than 3 × 10 6 variables and 14 × 10 6 constraints are solved in this way, using moderate computer time. The codes are free for download in www.ime.usp.br/∼egbirgin/tango/

The calculation of the matrix exponential e A maybeoneofthebestknownmatrix problems in numerical computation. It achieved folk status in our community from the paper by Moler and Van Loan, “Nineteen Dubious Ways to Compute the Exponential of a Matrix, ” published in this journal in 1978 (and revisited in this journal in 2003). The matrix exponential is utilized in a wide variety of numerical methods for solving differential equations and many other areas. It is somewhat amazing given the long history and extensive study of the matrix exponential problem that one can improve upon the best existing methods in terms of both accuracy and efficiency, but that is what the SIGEST selection in this issue does. “The Scaling and Squaring Method for the Matrix Exponential Revisited ” by N. Higham, originally published in the SIAM Journal on Matrix Analysis and Applications in 2005, applies a new backward error analysis to the commonly used scaling and squaring method, as well as a new rounding error analysis of the Padé approximant of the scaled matrix. The analysis shows, and the accompanying experimental results verify, that a Padé approximant of a higher order than currently used actually results in a more accurate

This paper describes an active-set algorithm for large-scale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza [10]. The step computation is performed in two stages. In the first stage a linear program is solved to estimate the active set at the solution. The linear program is obtained by making a linear approximation to the ` 1 penalty function inside a trust region. In the second stage, an equality constrained quadratic program (EQP) is solved involving only those constraints that are active at the solution of the linear program.

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

Results are reported of testing a number of existing state of the art solvers for global constrained optimization and constraint satisfaction on a set of over 1000 test problems in up to 1000 variables.

We describe an algorithm for solving nonlinear optimization problems with lower and upper bounds that constrain the variables. The algorithm uses projected gradients to construct a limited memory BFGS matrix and determine a step direction. The algorithm has been implemented and distributed as part of the Toolkit for Advanced Optimization (TAO). We include numerical results demonstrate is eectiveness on a set of large test problems and its scalability to multiple processors.