A Direct Search Optimization Method That Models the Objective and Constraint Functions by Linear Interpolation
An iterative algorithm is proposed for nonlinearly constrained optimization calculations when there are no derivatives. Each iteration forms linear approximations to the objective and constraint functions by interpolation at the vertices of a simplex and a trust region bound restricts each change to the variables. Thus a new vector of variables is calculated, which may replace one of the current vertices, either to improve the shape of the simplex or because it is the best vector that has been found so far, according to a merit function that gives attention to the greatest constraint violation. The trust region radius ρ is never increased, and it is reduced when the approximations of a well-conditioned simplex fail to yield an improvement to the variables, until ρ reaches a prescribed value that controls the final accuracy. Some convergence properties and several numerical results are given, but there are no more than 9 variables in these calculations because linear approximations can be highly inefficient. Nevertheless, the algorithm is easy to use for small numbers of variables.
Key wordsDirect search Linear interpolation Nonlinear constraints Optimization without derivatives
Unable to display preview. Download preview PDF.
- R. Fletcher (1987), Practical Methods of Optimization,John Wiley and Sons (Chichester).Google Scholar
- D.M. Himmelblau (1972), Applied Nonlinear Programming, McGraw-Hill (NewYork).Google Scholar
- W. Hock and K. Schittkowski (1980), Test Examples for Nonlinear Programming Codes, Lecture Notes in Economics and Mathematical Systems 187,Springer-Verlag (Berlin).Google Scholar
- M.J.D. Powell (1978), “A fast algorithm for nonlinearly constrained optimization calculations”, in Numerical Analysis, Dundee 1977, Lecture Notes in Mathematics 630, ed. G.A. Watson, Springer-Verlag (Berlin), pp. 144–157.Google Scholar