This paper introduces a new class of weak subdifferentiability for multifunctions through a gauge function $\rho $ , namely, weak $(\rho ,k)$ -subdifferential $(k>0)$ , and investigates some of its properties. The introduced weak $(\rho ,k)$ -subdifferential is a natural extension of the concept of subdifferentiability of scalar paraconvex functions introduced by Jourani (Control Cybern. 25:721–737, 1996) to multifunctions. It is also weaker than the concept of weak subdifferentiability of multifunctions in the sense of Tanino and Sawaragi (J. Math. Anal. Appl. 167:84–97, 1992), they become equivalent when the multifunction is cone-convex. Our new subdifferential enabled us to contribute to generalizing the sum rule formulas given in Theorem 3.1 (Lin in J. Math. Anal. Appl. 186:30–51, 1994), Theorem 3.2 (Taa in J. Math. Anal. Appl. 283:398–415, 2003), and Theorem 3.1 (Taa in Nonlinear Anal. 74:7312–7324, 2011) to the class of the so-called cone- $\rho $ -paraconvex multifunctions instead of cone-convex multifunctions under a general qualification condition. This latter generalizes the Azé’s well-known qualification condition (Azé in Arch. Math. 62:554–561, 1994) formulated for real-valued functions to multifunctions. As a consequence, we extend the well-known sum rule formulas by Rockafellar (Convex Analysis, Princeton University J., 1970), Attouch-Brezis (Aspects of Mathematics and Its Applications, Elsevier Science Publishers B.V., North Holland, pp. 125–133, 1986), and Azé (Arch. Math. 62:554–561, 1994) to extended real-valued paraconvex functions. An application is given to establish the existence of the Lagrange-Kuhn-Tucker multipliers for a constrained nonconvex set-valued optimization problem.
