238 research outputs found
On the Triality Theory for a Quartic Polynomial Optimization Problem
This paper presents a detailed proof of the triality theorem for a class of
fourth-order polynomial optimization problems. The method is based on linear
algebra but it solves an open problem on the double-min duality left in 2003.
Results show that the triality theory holds strongly in a tri-duality form if
the primal problem and its canonical dual have the same dimension; otherwise,
both the canonical min-max duality and the double-max duality still hold
strongly, but the double-min duality holds weakly in a symmetrical form. Four
numerical examples are presented to illustrate that this theory can be used to
identify not only the global minimum, but also the largest local minimum and
local maximum.Comment: 16 pages, 1 figure; J. Industrial and Management Optimization, 2011.
arXiv admin note: substantial text overlap with arXiv:1104.297
Global Solutions to Nonconvex Optimization of 4th-Order Polynomial and Log-Sum-Exp Functions
This paper presents a canonical dual approach for solving a nonconvex global
optimization problem governed by a sum of fourth-order polynomial and a
log-sum-exp function. Such a problem arises extensively in engineering and
sciences. Based on the canonical duality-triality theory, this nonconvex
problem is transformed to an equivalent dual problem, which can be solved
easily under certain conditions. We proved that both global minimizer and the
biggest local extrema of the primal problem can be obtained analytically from
the canonical dual solutions. As two special cases, a quartic polynomial
minimization and a minimax problem are discussed. Existence conditions are
derived, which can be used to classify easy and relative hard instances.
Applications are illustrated by several nonconvex and nonsmooth examples
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