Efficiency of unconstrained minimization techniques in nonlinear analysis

Unconstrained minimization algorithms have been critically evaluated for their effectiveness in solving structural problems involving geometric and material nonlinearities. The algorithms have been categorized as being zeroth, first, or second order depending upon the highest derivative of the function required by the algorithm. The sensitivity of these algorithms to the accuracy of derivatives clearly suggests using analytically derived gradients instead of finite difference approximations. The use of analytic gradients results in better control of the number of minimizations required for convergence to the exact solution

The Davidon-Fletcher-Powell penalty function method: A generalized iterative technique for solving parameter optimization problems

The Fletcher-Powell version of the Davidon variable metric unconstrained minimization technique is described. Equations that have been used successfully with the Davidon-Fletcher-Powell penalty function technique for solving constrained minimization problems and the advantages and disadvantages of using them are discussed. The experience gained in the behavior of the method while iterating is also related

Function minimization without derivatives by a sequence of quadratic programming problems

Function minimization without derivatives by sequence of quadratic programming problem

An acceleration technique for a conjugate direction algorithm for nonlinear regression

A linear acceleration technique, LAT, is developed which is applied to three conjugate direction algorithms: (1) Fletcher-Reeves algorithm, (2) Davidon-Fletcher-Powell algorithm and (3) Grey\u27s Orthonormal Optimization Procedure (GOOP). Eight problems are solved by the three algorithms mentioned above and the Levenberg-Marquardt algorithm. The addition of the LAT algorithm improves the rate of convergence for the GOOP algorithm in all problems attempted and for some problems using the Fletcher-Reeves algorithm and the Davidon-Fletcher-Powell algorithm. Using the number of operations to perform function and derivative evaluations, the algorithms mentioned above are compared. Although the GOOP algorithm is relatively unknown outside of the optics literature, it was found to be competitive with the other successful algorithms. A proof of convergence of the accelerated GOOP algorithm for nonquadratic problems is also developed --Abstract, page ii

The application op non-linear optimisation techniques in geophysics

Non-linear optimisation techniques form an important subject in non-linear programming. They work by searching for an optimum of a function in the hyperspace of its variable parameters. The purpose of the present work is to test the applicability of the techniques to solving non-linear geophysical problems. A problem from each of the major branches of geophysics is considered. The problem of fitting continental edges is also considered. Direct search methods are slow but are robust and, therefore useful in the early stages of the search. Gradient methods are fast and are efficient in the proximity of the optimum. A gravity or magnetic anomaly due to a two-dimensional polygonal model has a unique solution in theory. In practice, ambiguity arises from the presence of several factors and takes the form of a scatter of local minima and elongated ‘valleys’, in the hyperspace. The solution becomes less ambiguous as the influence of these factors gets less and as more parameters in the model are specified. The techniques are used successfully to interpret two-dimensional gravity and magnetic anomalies. Their efficiency, and flexibility make it possible to tackle a wide range of gravity and magnetic problems. The required computer time can be reduced by careful programming. The techniques are useful in interpreting surface wave dispersion data; the large degree of ambiguity associated with the problem may be overcome by specifying several parameters. A fast curve matching process is deviced for interpreting apparent resistivity curves. The method of outputting the results reduces the effect of equivalence. A method of fitting continental edges, by minimising the gaps and overlaps between them, is given. The ambiguity in the precise position of the pole of rotation is illustrated using the same concept adopted in the gravity, magnetic and seismic problems

On methods for minimizing a function without calculating its derivatives

Imperial Users onl

Mixed nonderivative algorithms for unconstrained optimization

A general technique is developed to restart nonderivative algorithms in unconstrained optimization. Application of the technique is shown to result in mixed algorithms which are considerably more robust than their component procedures. A general mixed algorithm is developed and its convergence is demonstrated. A uniform computational comparison is given for the new mixed algorithms and for a collection of procedures from the literature --Abstract, page ii