Approximated approximations for SAO

We propose to replace a number of popular approximations by their diagonal quadratic Taylor series expansions. The resulting separable quadratic approximations are easily convexified, and are well suited for use in dual sequential approximate optimization (SAO) algorithms. Global convergence of the resulting SAO algorithms may be enforced in a natural way using conservatism. The approximated approximation approach is explicitly illustrated for (i) reciprocal and exponential intervening variables, (ii) the intervening variables used in the method of moving asymptotes (MMA), (iii) the intervening variables used in CONLIN, and (iv) the TANA-3 approximations. The use of intermediate responses for use in, for example, truss and frame-like structures, is also discussed. Key advantages of replacing nonlinear approximations by their diagonal quadratic approximations are that these approximated approximations can all be used simultaneously in a single dual statement; the dual does not depend on the form of the original approximations. In addition, in a dual setting, the resulting subproblems yield simple analytical relationships between the primal and dual variables, which is often not the case with the original nonlinear approximations. An important example hereof is the exponential approximation. Although the diagonal quadratic approximations may differ notably from their original counterparts, they typically are quite similar in a sufficiently small search subregion, which relates to the move limits commonly used in SAO anyway. © 2009 Springer-Verlag