More Test Examples for Nonlinear Programming Codes

verfasst von: Prof. Dr. Klaus Schittkowski

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Economics and Mathematical Systems

Enthalten in: Professional Book Archive

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This collection of 188 nonlinear programming test examples is a supplement of the test problem collection published by Hock and Schittkowski [2]. As in the former case, the intention is to present an extensive set of nonlinear programming problems that were used by other authors in the past to develop, test or compare optimization algorithms. There is no distinction between an "easy" or "difficult" test problem, since any related classification must depend on the underlying algorithm and test design. For instance, a nonlinear least squares problem may be solved easily by a special purpose code within a few iterations, but the same problem can be unsolvable for a general nonlinear programming code due to ill-conditioning. Thus one should consider both collections as a possible offer to choose some suitable problems for a specific test frame. One difference between the new collection and the former one pub lished by Hock and Schittkowski [2], is the attempt to present some more realistic or "real world" problems. Moreover a couple of non linear least squares test problems were collected which can be used e. g. to test data fitting algorithms. The presentation of the test problems is somewhat simplified and numerical solutions are computed only by one nonlinear programming code, the sequential quadratic programming algorithm NLPQL of Schittkowski [3]. But both test problem collections are implemeted in the same way in form of special FORTRAN subroutines, so that the same test programs can be used.

Inhaltsverzeichnis

Frontmatter

Chapter I. The Documentation Scheme

Abstract

The development of algorithms for solving the nonlinear programming problem

$$ \begin{gathered} \min f(x) \hfill \\ g_j (x) \geqslant 0,\quad j = 1, \ldots ,m_1 \hfill \\ x \in \mathbb{R}^n :g_j (x) = 0,\quad j = m_1 + 1, \ldots ,m \hfill \\ x_1 \leqslant x \leqslant x_u \hfill \\ \end{gathered} $$

(NLP)

with continuously differentiable functions f and g₁,…,g_m requires the availability of test examples. The intention is to present a collection of optimization problems which can be used by a test designer to choose from either to develop a new algorithm or to become familar with an existing code written by somebody else. All problems were found in the literature and have been used in the past to test or compare optimization software.

Klaus Schittkowski

Chapter II. Usage of the Fortran Subroutines

Abstract

This chapter describes the organization of the FORTRAN subroutines and informs the user on the way how to execute the test problems. Since it is assumed that at least a subset of the problems is used within a series of test runs for different optimization programs, the problems are coded in a very flexible manner. For example, it is possible to execute an arbitrary subset of the restrictions. To distinguish between linear and nonlinear constraints, we have to define a fixed succession of the restrictions in the following way:

$$ g_j (x) \geqslant 0,\quad j = 1, \ldots ,m_{11} ,\quad linear\,functions, $$

$$ g_j (x) \geqslant 0,\quad j = m_{11} + 1, \ldots ,m_1 ,\quad nonlinear\,functions, $$

$$ g_j (x) = 0,\quad j = m_1 + 1, \ldots ,m_{21} ,\quad linear\,functions, $$

$$ g_j (x) = 0,\quad j = m_{21} + 1, \ldots ,m,\quad nonlinear\,functions. $$

Klaus Schittkowski

Chapter III. Condensed Information on the Test Problems

Abstract

To give a first survey of the test examples and their solution properties, a comprehensive list of all problems is presented in Table 1. Besides the current problem number and the classification number OCD-Kr-s as described in Chapter I, we report the dimension n, the number of all inequality constraints m₁, the number of all equality constraints m-m₁, and the number of all bounds b. If linear restrictions exist, their number is given in the brackets behind m₁ or m-m₁, respectively. The column headed by x_o gives the information, whether the starting point x_o is feasible (T) or not (F). Moreover, the objective function value f(x*) is displayed, where x* denotes either the exact solution, if known in advance, or a solution computed numerically by the nonlinear programming code NLPQL. Any numerical data not given in full length, are assumed to be exact, i.e. the remaining digits are zero.

Klaus Schittkowski

Chapter IV. The Test Problems

Abstract

Using the documentation scheme proposed in Chapter I, we present a detailed description of 188 problems for testing nonlinear programming algorithms. The corresponding FORTRAN-subroutines are available on request. Extensive lists of constant data, solution results or function evaluatives are shown in Appendix A.

Klaus Schittkowski

Backmatter

Titel: More Test Examples for Nonlinear Programming Codes
verfasst von: Prof. Dr. Klaus Schittkowski
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-61582-5
Print ISBN: 978-3-540-17182-9
DOI: https://doi.org/10.1007/978-3-642-61582-5

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter I. The Documentation Scheme

Chapter II. Usage of the Fortran Subroutines

Chapter III. Condensed Information on the Test Problems

Chapter IV. The Test Problems

Backmatter