Klaus Schittkowski

NLPQL: A fortran subroutine solving constrained nonlinear programming problems

NLPQL is a FORTRAN implementation of a sequential quadratic programming method for solving nonlinearly constrained optimization problems with differentiable objective and constraint functions. At each iteration, the search direction is the solution of a quadratic programming subproblem. This paper discusses the organization of NLPQL, including the formulation of the subproblem and the information that must be provided by a user. A summary is given of the performance of different algorithmic options of NLPQL on a collection of test problems (115 hand-selected or application problems, 320 randomly generated problems). The performance of NLPQL is compared with that of some other available codes.

Test Examples for Nonlinear Programming Codes

The increasing importance of nonlinear programming software requires an enlarged set of test examples. The purpose of this note is to point out how an interested mathematical programmer could obtain computer programs of more than 120 constrained nonlinear programming problems which have been used in the past to test and compare optimization codes.

On the convergence of a sequential quadratic programming method with an augmented lagrangian line search function 2

Sequential quadratic programming methods as developed by Wilson, Han, and Powell have gained considerable attention in the last few years mainly because of their outstanding numerical performance. Although the theoretical convergence aspects of this method and its various modifications have been investigated in the literature, there still remain some open questions which will be treated in this paper. The convergence theory to be presented, takes into account the additional variable introduced in the quadratic programming subproblem to avoid inconsistency, the one-dimensional minimization procedure, and, in particular, an “ active set” strategy to avoid the recalculation of unnecessary gradients. This paper also contains a detailed mathematical description of a nonlinear programming algorithm which has been implemented by the author. the usage of the code and detailed numerical test results are presented in [5].

The nonlinear programming method of Wilson, Han, and Powell with an augmented Lagrangian type line search function

SummaryThe paper represents an outcome of an extensive comparative study of nonlinear optimization algorithms. This study indicates that quadratic approximation methods which are characterized by solving a sequence of quadratic subproblems recursively, belong to the most efficient and reliable nonlinear programming algorithms available at present. The purpose of this paper is to analyse the theoretical convergence properties and to investigate the numerical performance in more detail. In Part 1, the exactL1-penalty function of Han and Powell is replaced by a differentiable augmented Lagrange function for the line search computation to the able to prove the global convergence and to show that the steplength one is chosen in the neighbourhood of a solution. In Part 2, the quadratic subproblem is exchanged by a linear least squares problem to improve the efficiency, and to test the dependence of the performance from different solution methods for the quadratic or least squares subproblems.

A Comparative Performance Evaluation of 27 Nonlinear Programming Codes

The numerical performance of 27 computer programs, which are all designed to solve the general constrained nonlinear optimization problem, is to be evaluated. In contrast to Schittkowski [34], where besides of one exception, the same codes are compared on randomly generated test problems, the test examples are now given by the 115 hand-selected and real life problems published in Hock and Schittkowski [19]. The different type of the test examples requires the development of a special evaluation system based on priority theory. Detailed numerical results are presented allowing a quantitative comparison of the performance criteria efficiency and reliability.ZusammenfassungEs soll das Leistungsvermögen von 27 Computerprogrammen numerisch ermittelt werden, die alle zur Lösung des allgemeinen restringierten, nichtlinearen Optimierungsproblems entwickelt wurden. Im Gegensatz zu Schittkowski [34], wo bis auf eine Ausnahme dieselben Programme an Hand von zufallsmäßig erzeugten Testbeispielen verglichen wurden, sind die Testbeispiele jetzt die von Hock und Schittkowski [19] veröffentlichten 115 Optimierungsprobleme, die entweder per Hand konstruiert wurden oder einen praktischen Hintergrund besitzen. Die unterschiedliche Art dieser Testbeispiele erfordert die Entwicklung eines speziellen Auswertungssystems, das auf Prioritätstheorie basiert. Detaillierte numerische Resultate werden präsentiert, die einen quantitativen Vergleich der Leistungskriterien Effizienz und Zuverlässigkeit erlauben.

Numerical Data Fitting in Dynamical Systems

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Numerical comparison of nonlinear programming algorithms for structural optimization

For FE-based structural optimization systems, a large variety of different numerical algorithms is available, e.g. sequential linear programming, sequential quadratic programming, convex approximation, generalized reduced gradient, multiplier, penalty or optimality criteria methods, and combinations of these approaches. The purpose of the paper is to present the numerical results of a comparative study of eleven mathematical programming codes which represent typical realizations of the mathematical methods mentioned. They are implemented in the structural optimization system MBB-LAGRANGE, which proceeds from a typical finite element analysis. The comparative results are obtained from a collection of 79 test problems. The majority of them are academic test cases, the others possess some practicalreal life background. Optimization is performed with respect to sizing of trusses and beams, wall thicknesses, etc., subject to stress, displacement, and many other constraints. Numerical comparison is based on reliability and efficiency measured by calculation time and number of analyses needed to reach a certain accuracy level.

Computational mathematical programming

Integer Programming.- Model Building in Linear and Integer Programming.- LP-Based Combinatorial Problem Solving.- Network Optimization.- Reflections on Geometric Programming.- Principles of Sequential Quadratic Programming Methods for Solving Nonlinear Programs.- Model Building and Practical Aspects of Nonlinear Programming.- Comparative Performance Evaluation, Experimental Design, and Generation of Test Problems in Nonlinear Optimization.- On Converting Optimal Control Problems into Nonlinear Programming Problems.- A Stochastic Approach to Global Optimization.- Algorithmic Procedures for Stochastic Optimization.- Nondifferentiable Optimization.- Parallel Computing in Optimization.- Software for Mathematical Programming.

A trust region SQP algorithm for mixed-integer nonlinear programming

We propose a modified sequential quadratic programming method for solving mixed-integer nonlinear programming problems. Under the assumption that integer variables have a smooth influence on the model functions, i.e., that function values do not change drastically when in- or decrementing an integer value, successive quadratic approximations are applied. The algorithm is stabilized by a trust region method with Yuan’s second order corrections. It is not assumed that the mixed-integer program is relaxable or, in other words, function values are evaluated only at integer points. The Hessian of the Lagrangian function is approximated by a quasi-Newton update formula subject to the continuous and integer variables. Numerical results are presented for a set of 80 mixed-integer test problems taken from the literature. The surprising result is that the number of function evaluations, the most important performance criterion in practice, is less than the number of function calls needed for solving the corresponding relaxed problem without integer variables.

Very large scale optimization by sequential convex programming

We introduce a method for constrained nonlinear programming that is widely used in mechanical engineering and that is known under the name SCP for sequential convex programming. The algorithm consists of solving a sequence of convex and separable subproblems, where an augmented Lagrangian merit function is used for guaranteeing convergence. Originally, SCP methods were developed in structural mechanical optimization, and are particularly applied to solve topology optimization problems. These problems are extremely large and possess dense Hessians of the objective function. The purpose of the article is to show that constrained dense nonlinear programs with 105–106 variables can be solved successfully and that SCP methods can be applied also to optimal control problems based on semilinear elliptic partial differential equations after a full discretization.