This paper deals with a class of nonsmooth interval-valued optimization problems (NIVOPs) whose objective and constraints are interval-valued functions (IVFs). For this, we propose a new concept of symmetric invexity (S-invexity) for IVFs based on gH-symmetrically derivative, and deduce a sufficient optimality condition for non-dominated solutions. Furthermore, we establish appropriate duality theorems for Wolfe and Mond-Weir type duals. Examples are also presented to illustrate corresponding results.

In this work, two generalized quantum integral identities are proved by using some parameters. By utilizing these equalities we present several parameterized quantum inequalities for convex mappings. These quantum inequalities generalize many of the important inequalities that exist in the literature, such as quantum trapezoid inequalities, quantum Simpson's inequalities and quantum Newton's inequalities. We also give some new midpoint type inequalities as special cases. The results in this work naturally generalize the results for the Riemann integral.