A NONLINEAR CONE SEPARATION THEOREM AND SCALARIZATION IN NONCONVEX VECTOR OPTIMIZATION

作者：Kasimbeyli Refail^*

来源：SIAM Journal on Optimization, 2010, 20(3): 1591-1619.

DOI：10.1137/070694089

• cone separation theorem
• cone approximation
• augmented dual cones
• Bishop-Phelps cones
• sublinear functions
• nonconvex vector optimization
• conic scalarization
• multiobjective optimization

In this paper, a special separation property for two closed cones in Banach spaces is proposed, and a nonlinear separation theorem for the cones possessing this property is proved. By extending a usual definition of dual cones, an augmented dual of a cone is introduced. A special class of monotonically increasing sublinear functions is defined by using the elements of the augmented dual cone. Any closed cone possessing the separation property with its epsilon-conic neighborhood is shown to be approximated arbitrarily closely by a zero sublevel set of some function from this class. As an application, a simple and efficient scalarization technique for nonconvex vector optimization problems is suggested, and it is shown that any properly minimal point of a set in a Banach space can be calculated by minimizing a certain sublinear functional.

• 出版日期2010