Abstract
It is well known that optimization problem model has many applications arising from matrix completion, image processing, statistical learning, economics, engineering sciences, and so on. And convex programming problem is closely related to variational inequality problem. The so-called alternative direction of multiplier method (ADMM) is an importance class of numerical methods for solving convex programming problem. When analyzing the rate of convergence of various ADMMs, an error bound condition is usually required. The error bound can be obtained when the isolated calmness of the inverse of the KKT mapping of the related problem holds at the given KKT point. This paper is to study the isolated calmness of the inverse KKT mapping onto the mixed variational inequality problem with nonlinear term defined by norm function and indicator function of a convex polyhedral set, respectively. We also consider the isolated calmness of the inverse KKT mapping onto classical variational inequality problem with equality and inequality constrains under strict Mangasarian-Fromovitz constraint qualification condition. The results obtained here are new and very interesting.
Highlights
In this paper, let us consider the following mixed variational inequality problem: find x ∈ Rn such that⟨F (x), y − x⟩ + φ (y) − φ (x) ≥ 0, ∀y ∈ Rn, (1)where F : Rn → Rn is a twice continuously differentiable mapping and φ : Rn → R ∪ {+∞} is a proper convex lower semicontinuous functional
It is well known that optimization problem model has many applications arising from matrix completion, image processing, statistical learning, economics, engineering sciences, and so on
The error bound can be obtained when the isolated calmness of the inverse of the KKT mapping of the related problem holds at the given KKT point
Summary
Let us consider the following mixed variational inequality problem: find x ∈ Rn such that. Zhang [10] proved the isolated calmness of the KKT mapping under the strict Robinson constraint qualification (SRCQ) and the second-order sufficient optimality condition for the nonlinear SDP problem. The problem considered is a variational inequality, not a composite optimization problem; and we use second-order sufficient optimality conditions and the strict Robinson constraint qualification to ensure the isolated calmness of the solution mapping. We mainly discuss the isolated calmness of the inverse KKT mapping on problem (1) with nonlinear term φ defined by norm function and the indicator function of a convex polyhedral set, respectively. We consider the isolated calmness of the inverse KKT mapping on classical variational inequality problem with equality and inequality constrains under strict Mangasarian-Fromovitz constraint qualification condition
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