Criteria for Unconstrained Global Optimization |
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Authors: | E Demidenko |
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Institution: | (1) Departments of Biostatistics and Mathematics, Dartmouth College, Hanover, NH, USA |
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Abstract: | We develop criteria for the existence and uniqueness of the 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. A new multimodal function is introduced, the decomposition function, which can
be represented as the composition of a convex function and a nonlinear function from the argument space to a space of larger
dimension. Several general global criteria based on majorization and minorization functions are formulated. |
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Keywords: | Existence Nonconvex optimization Minimization Least squares Maximum likelihood |
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