In this paper we will establish some necessary condition and sufficient condition respectively for a set-valued mapping to have the Lipschitz-like property relative to a closed set by employing regular normal cone and limiting normal cone of a restricted graph of the setvalued mapping. We will obtain a complete characterization for a set-valued mapping to have the Lipschitz-property relative to a closed and convex set by virtue of the projection of the coderivative onto a tangent cone. Furthermore, by introducing a projectional coderivative of set-valued mappings, we establish a verifiable generalized Mordukhovich criterion for the Lipschitz-like property relative to a closed and convex set. We will study the representation of the graphical modulus of a set-valued mapping relative to a closed and convex set by using the outer norm of the corresponding projectional coderivative value. For an extended real-valued function, we will apply the obtained results to investigate its Lipschitz continuity relative to a closed and convex set and the Lipschitz-like property of a level-set mapping relative to a half line.

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The linearized alternating direction methods of multipliers (L-ADMM) for solving convex minimization problems with two separable blocks in the objective functions is efficient. And its extended version (\(m \geqslant 3\)) is convergent under some mild conditions. Recently, The L-ADMM inspires much attention in analyzing its theoretical convergence rate. However, the research on its convergence rate is still in its infancy. In this paper, we consider the convergence rate of L-ADMM when solving the convex optimization problems that the subdifferentials of the underlying functions are piecewise linear multifunctions. Based on the error bound, we establish the linear convergence rate.