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Transcript
Comparative Computer Results of a New Complementary Pivot Algorithm for Solving Equality Constrained Optimization Problems, Mathematical Programming, 18(2), 168-185, 1980. This paper reports the development of a new algorithm for solving the general constrained optimization problem (that of optimizing an objective function subject to both equality and inequality constraints). The approach is based on the complementary pivoting algorithms that have been developed to solve certain classes of fixed-point problems. The specific approach is to use the equality constraints to solve for some of the variables in terms of the remaining ones, thus enabling one to eliminate the quality constraints altogether. The result, under certain circumstances, is an optimization problem that can be transformed into a fixed-point problem in such a way that a complementary pivoting code may be used to search for a fixed point. Seventeen test problems have been solved by this method and the results compared against those obtained from GRG (Generalized Reduced Gradient method). The results of these tests indicate that the fixed-point approach is robust (all seventeen problems were solved by this method whereas GRG solved sixteen). As to the computer times, the fixed-point code proved to be as fast or faster than GRG on the lower-dimensional problems; however, as the dimension increased, the trend reversed and on a forty-dimensional problem, GRG was approximately eleven times faster. The conclusion from these tests is that when the dimension of the original problem can be reduced sufficiently by the equality constraints, the fixed-point approach appears to be more effective than GRG.