OPTIMIZATION OF MECHANICAL SYSTEMS - ON STRATEGY OF NONLINEAR 1ST-ORDER APPROXIMATION

The paper presents an extended approach to non-linear first-order approximation of non-linear programming problems and it explains how to transform an approximate problem into a strictly convex one. The essence of the proposed approximation technique is to rewrite each given function h(j) as a composite g(j)-degrees-PSI(j). The function PSI(j) has to be chosen-the paper explains how to do this-while g(j) is linear approximated with g(j). The approximation of h(j) is then obtained as g(j)-degrees-PSI(j). This approach enables one to obtain approximate functions with variable conservativeness, which implies an adjustable approximate problem. A solution procedure, which replaces the original problem with a sequence of approximate problems, can therefore adjust each succeeding approximate problem to improve the convergence properties. The theory is illustrated with a three parameters controlled approximation. This technique represents, together with an optimality criteria based solution procedure, a powerful and economic tool for solving non-linear programming problems. The three parameters, which influence to a great extent the conservativeness of the approximate functions, are under full control of the optimizer. They are varied automatically during the process of optimization to speed-up the convergence or to prevent oscillations. The benefits gained from the proposed approach are demonstrated on several numerical examples involving structures and a dynamic multibody system.