Criteria for Unconstrained Global Optimization in Nonconvex Problems
Eugene Demidenko, Theodore E. Simos, George Psihoyios, Ch. Tsitouras · AIP conference proceedings · 2007
We develop criteria for the existence and uniqueness of global minima of a continuous bounded function on a noncompact set. Special attention is given to the problem of parameter estimation via minimization of the sum of squares in nonlinear regression and maximum likelihood. Definitions of local convexity and unimodality are given using the level set. A fundamental theorem of nonconvex optimization is formulated: If a function approaches the minimal limiting value at the boundary of the optimization domain from below and its Hessian matrix is positive definite at the point where the gradient vanishes then the function has a unique minimum. It is shown that the local convexity level of the sum of squares is equal to the minimal squared radius of the regression curvature.