Kirjahaku

Real life phenomena in engineering, natural, or medical sciences are often described by a mathematical model with the goal to analyze numerically the behaviour of the system. Advantages of mathematical models are their cheap availability, the possibility of studying extreme situations that cannot be handled by experiments, or of simulating real systems during the design phase before constructing a first prototype. Moreover, they serve to verify decisions, to avoid expensive and time consuming experimental tests, to analyze, understand, and explain the behaviour of systems, or to optimize design and production. As soon as a mathematical model contains differential dependencies from an additional parameter, typically the time, we call it a dynamical model. There are two key questions always arising in a practical environment: 1 Is the mathematical model correct? 2 How can I quantify model parameters that cannot be measured directly? In principle, both questions are easily answered as soon as some experimental data are available. The idea is to compare measured data with predicted model function values and to minimize the differences over the whole parameter space. We have to reject a model if we are unable to find a reasonably accurate fit. To summarize, parameter estimation or data fitting, respectively, is extremely important in all practical situations, where a mathematical model and corresponding experimental data are available to describe the behaviour of a dynamical system.

Real life phenomena in engineering, natural, or medical sciences are often described by a mathematical model with the goal to analyze numerically the behaviour of the system. Advantages of mathematical models are their cheap availability, the possibility of studying extreme situations that cannot be handled by experiments, or of simulating real systems during the design phase before constructing a first prototype. Moreover, they serve to verify decisions, to avoid expensive and time consuming experimental tests, to analyze, understand, and explain the behaviour of systems, or to optimize design and production. As soon as a mathematical model contains differential dependencies from an additional parameter, typically the time, we call it a dynamical model. There are two key questions always arising in a practical environment: 1 Is the mathematical model correct? 2 How can I quantify model parameters that cannot be measured directly? In principle, both questions are easily answered as soon as some experimental data are available. The idea is to compare measured data with predicted model function values and to minimize the differences over the whole parameter space. We have to reject a model if we are unable to find a reasonably accurate fit. To summarize, parameter estimation or data fitting, respectively, is extremely important in all practical situations, where a mathematical model and corresponding experimental data are available to describe the behaviour of a dynamical system.

...The increasing importance of mathematical programming for the solution of complex nonlinear systems arising in practical situations requires the development of qualified optimization software. In recent years, a lot of effort has been made to implement efficient and reliable optimization programs and we can observe a wide distribution of these programs both for research and industrial applications. In spite of their practical importance only a few attempts have been made in the past to come to comparative conclusions and to give a designer the possibility to decide which optimization program could solve his individual problems in the most desirable way. Box [BO 1966J, Huang, Levy [HL 1970J, Himmelblau [HI 1971J, Dumi- tru [DU 1974], and More, Garbow, Hillstrom [MG 1978] for example compared algorithms for unres~ricied u~~illii~Gtiv~ y~~~le~~, B~~n [BD 1970], McKeown [MK 1975], and Ramsin, Wedin [RW 1977l studied codes for nonlinear least squares problems. Codes for the linear case are compared by Bartels [BA 1975.J and Schittkowski, Stoer [SS 1979J. Extensive tests for geometric programming algorithms are found in Dembo [DE 1976bJ, Rijckaert [RI 1977], and Rijckaert, Martens [RM 1978J.

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.