The Interior Point Nonlinear Programming Solver: Basic Definitions and Notation :: SAS/OR(R) 9.2 User's Guide: Mathematical Programming

The Interior Point Nonlinear Programming Solver -- Experimental

Basic Definitions and Notation

The gradient of a function $f: \mathbb{r}^n \mapsto \mathbb{r}$ is the vector of all the first partial derivatives of , and is denoted by

$\nabla f(x) = (\frac{\partial f}{\partial x_{1}}, \frac{\partial f}{\partial x_{2}}, ..., \frac{\partial f}{\partial x_{n}})^{\rm t}$

where the superscript T denotes the transpose of a vector.

The Hessian matrix of , denoted by $\nabla^2 f(x)$ , or simply by , is an symmetric matrix whose element is the second partial derivative of with respect to $x_{i}$ and $x_{j}$ . That is, $h_{i,j}(x) = \frac{\partial^2 f(x)}{\partial x_{i} \partial x_{j}}$ .

Consider the vector function, $c: r^n \mapsto r^{p+q}$ , whose first elements are the equality constraint functions $h_{i}(x), i=1,2,..,p$ , and whose last elements are the inequality constraint functions $g_{i}(x), i=1,2,..,q$ . That is,

$c(x) = (h(x) : g(x))^{\rm t} = (h_{1}(x), ..., h_{p}(x) : g_{1}(x), ..., g_{q}(x))^{\rm t}$

The matrix whose th column is the gradient of the th element of is called the Jacobian matrix of (or simply the Jacobian of the NLP problem) and is denoted by . We can also use $j_{h}(x)$ to denote the Jacobian matrix of the equality constraints and use $j_{g}(x)$ to denote the Jacobian matrix of the inequality constraints. It is easy to see that $j(x) = (j_{h}(x) : j_{g}(x))$ .

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