Abstract
In this paper, we study a concept on the calm B-differentiability, a new kind of generalized differentiabilities for a given vector function introduced by Ye and Zhou in 2017, of the projector onto the circular cone. Then, we discuss its applications in mathematical programming problems with circular cone complementarity constraints. Here, this problem can be considered to be a generalization of mathematical programming problems with second-order cone complementarity constraints, and thus it includes a large class of mathematical models in optimization theory. Consequently, the obtained results for this problem are generalized, and then corresponding results for some special mathematical problems can be implied from them directly. For more detailed information, we will first prove the calmly B-differentiable property of the projector onto the circular cone. This result is not easy to be shown by simply resorting to those of the projection operator onto the second-order cone. By virtue of exploiting variational techniques, we next establish the exact formula for the regular (Fréchet) normal cone (this concept was proposed by Kruger and Mordukhovich in 1980) to the circular cone complementarity set. Note that this set can be considered to be a generalization of the second-order cone complementarity set. In finally, the exact formula for the regular (Fréchet) normal cone to the circular cone complementarity set would be useful for us to study first-order necessary optimality conditions for mathematical programming problems with circular cone complementarity constraints. Our obtained results in the paper are new, and they are generalized to some existing ones in the literature.
Highlights
The second-order cone programming (SOCP) problem plays an important role in the optimization theory and has attracted much attention from mathematicians, see, e.g., 1–7
Inspired by the second-order cone, many researchers have investigated optimization and complementarity problems where their constraints are involved in second-order cones. It is called the second-order cone complementarity problem (SOCCP), which includes a large class of optimization problems such as quadratically constrained problems, the second-order cone programming, and nonlinear complementarity problem
Ye and Zhou gave exact formulas for the proximal/regular (Fréchet)/limiting normal cone to the second-order cone complementarity set by using the projection operator onto second-order cones and the generalized differentiability called the calm B-differentiability
Summary
The second-order cone programming (SOCP) problem plays an important role in the optimization theory and has attracted much attention from mathematicians, see, e.g., 1–7. Let us define the circular cone complementarity set as X = λ1(x) u1x + λ2(x)u2x , where the spectral values λ1(x), λ2(x) and the spectral vectors u1x , u2x are defined respectively by λ1(x) := x0 − ||xr|| cot θ , λ2(x) := x0 + ||xr|| tan θ , Cot θ ] [−x]r tan θ xr which means that x = ΠKθ (x − y).
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