The gradient of a function is the vector of all the first partial derivatives of f and is denoted by

where the superscript T denotes the transpose of a vector.

The Hessian matrix of f, denoted by , or simply by , is an symmetric matrix whose element is the second partial derivative of with respect to and . That is, .

Consider the vector function, , whose first p elements are the equality constraint functions , and whose last q elements are the inequality constraint functions . That is,

The matrix whose ith row is the gradient of the ith element of is called the Jacobian matrix of (or simply the Jacobian of the NLP problem) and is denoted by . You can also use to denote the Jacobian matrix of the equality constraints and use to denote the Jacobian matrix of the inequality constraints. It is easy to see that