A function imbedding technique for a class of global optimization problems: one dimensional global optimization

Global optimal solutions for nonconvex problems are found by first imbedding a nonconvex function into a higher-dimensional convex function and then by transforming the original problem into the problem of finding the mini-max solution of a related Lagrangian function. The imbedding is constructed by using the space of quadratic functions which are lower bounds for the original function. The Lagrangian function is constructed so that the associate dual cost function is concave and so that the global optimal solution can be obtained from the saddle point of the Lagrangian, which can be found using ordinary numerical methods. The duality gap for this Lagrangian is shown to vanish. For the case of one-dimensional global optimization, a working algorithm is presented