Special cases of nonlinear programs with quadratic constraints that arise in engineering applications are investigated. These applications include refinery blending problems in petrochemical processes, component placement problems in circuit design, minimax location problems in industrial engineering, and modular design problems in production and manufacturing. The primary focus is on the development of a new algorithm for obtaining (approximate) global solutions for these problems. A paramount issue in applying an algorithm to a given problem is the question of existence of a solution when the feasible region is not empty. New conditions will be derived for quadratically constrained applications and techniques for checking these conditions will be developed. Two general approaches will be explored in developing algorithms: outer approximation and branch and bound. Outer approximation includes, but is not limited to cutting (plane) methods. Bounding will be accomplished by the use of convex envelopes and by nonlinear duality theory. The results that are anticipated from this research effort should contribute significantly to the efficient (real time) solution of a larger class of quadratically constrained problems than currently possible. Many of the ideas and constructions that will be reported are not well known and have the potential to be extended to problems beyond the scope of this study and, hence, should contribute to a better understanding of the practicality of solving real instances of nonlinearly constrained optimization problems.
