Abstract: Based on interval analysis and immune principles, some properties of solutions on nonlinear interval number programming are investigated, and an immune optimization approach as well as its theoretical foundations are explored. Firstly, the concept of optimal solution for such nonlinear programming is developed based on the version of optimal-valued interval. Some properties of efficient solutions on interval-valued programming are found, while an inherent solution relation is obtained between such nonlinear programming and interval natural extension programming. This derives an efficient pathway to find the optimal solution in terms of sufficient conditions acquired. Secondly, based on simplified metaphors of the immune response, a micro-immune optimization approach is proposed with the characteristics of small populations, few adjustable parameters, simple and non-master-slave structures. It is also proven to be convergent with low computational complexity. Comparatively numerical results show that such an efficient and effective approach is potential to nonlinear interval number programming problems with low or somewhat high dimensions.

Key words: nonlinear interval number programming, interval analysis, optimal-valued interval, immune optimization, non-nested optimization