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The relief indicator method for constrained global optimization
Author(s) -
Thach Phan Thiên,
Tuy Hoàng
Publication year - 1990
Publication title -
naval research logistics (nrl)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.665
H-Index - 68
eISSN - 1520-6750
pISSN - 0894-069X
DOI - 10.1002/1520-6750(199008)37:4<473::aid-nav3220370404>3.0.co;2-o
Subject(s) - separable space , mathematics , mathematical optimization , function (biology) , sequence (biology) , concave function , quadratic equation , convex function , regular polygon , set (abstract data type) , global optimization , decomposition , extreme point , feasible region , point (geometry) , computer science , combinatorics , mathematical analysis , ecology , genetics , geometry , evolutionary biology , biology , programming language
We consider the problem of globally minimizing a continuous, not‐necessarily smooth function f ( x ) over a compact set S in R n . To each real number α we associate a function φ α ( x ), called the relief indicator, such that the function φ α α( x ) + ∥ x ∥ 2 is closed, convex, and a feasible point x is a global optimal solution if and only if 0 = min{φα( x ): x ∈ R n }, where α = f ( x ). Based on this global optimality criterion, an algorithm is then developed which reduces the problem to a sequence of linearly constrained concave quadratic separable programs. We discuss the practical use of these results when n (the number of variables) is small and also show how the method can be applied in the decomposition of certain nonconvex problems.