Abstract: This paper reviews mainly the advances in numerical optimization methods developed since the late eighties of the 20th century for complex optimization problems in engineering. It focuses on several critical aspects of the optimization methodology including mathematical formulations, approximation functions and solution strategies for optimization problems. And a concept of fidelity for approximate problems and its significance are presented and discussed. It is pointed out that the formulation DFOP V2 based on the dual theory and envelope function is superior to other formulations in respects of simplifying optimization procedure and raising computational efficiency, especially for large scale problems; as compared with Taylor expansions, multi point approximate functions may approach the original functions with high quality in a wide region of design variable space; and the solution strategy of two level approximation concept is much beneficial to solving complex problems with high non linearity. Suggestions are given for further developing numerical optimum methods which are of convenience and efficiency in applications for engineers.

Key words: op t im izat ion, mathemat ical p rogramm ing, app roximat ion concep ts