This paper introduces semismooth and semiconvex functions and discusses their properties in the context of nonsmooth nonconvex constrained optimization. These functions are locally Lipschitz and have generalized gradients. The author presents an optimization algorithm that uses generalized gradients and converges to stationary points if the functions are semismooth. If the functions are semiconvex and a constraint qualification is satisfied, then a stationary point is optimal. The paper shows that the pointwise maximum or minimum over a compact family of continuously differentiable functions is a semismooth function, and the pointwise maximum over a compact family of semiconvex functions is a semiconvex function. It also demonstrates that a semismooth composition of semismooth functions is semismooth and provides a chain rule for generalized gradients. The paper further shows that semiconvex functions behave similarly to convex functions with respect to maximization, while pseudoconvex functions do not due to the loss of differentiability. The paper concludes with theorems on stationarity and optimality, showing that under certain conditions, stationarity implies optimality for semiconvex functions.This paper introduces semismooth and semiconvex functions and discusses their properties in the context of nonsmooth nonconvex constrained optimization. These functions are locally Lipschitz and have generalized gradients. The author presents an optimization algorithm that uses generalized gradients and converges to stationary points if the functions are semismooth. If the functions are semiconvex and a constraint qualification is satisfied, then a stationary point is optimal. The paper shows that the pointwise maximum or minimum over a compact family of continuously differentiable functions is a semismooth function, and the pointwise maximum over a compact family of semiconvex functions is a semiconvex function. It also demonstrates that a semismooth composition of semismooth functions is semismooth and provides a chain rule for generalized gradients. The paper further shows that semiconvex functions behave similarly to convex functions with respect to maximization, while pseudoconvex functions do not due to the loss of differentiability. The paper concludes with theorems on stationarity and optimality, showing that under certain conditions, stationarity implies optimality for semiconvex functions.