The Unconstrained Nonlinear Programming Solver: Conditions of Optimality :: SAS/OR(R) 9.22 User's Guide: Mathematical Programming

The Unconstrained Nonlinear Programming Solver

Conditions of Optimality

Before beginning the discussion, the following notation is presented for easy reference:

$\text{[math]}$: is the dimension of $\text{[math]}$ (that is, the number of decision variables).
$\text{[math]}$: is the iteration (that is, the vector of $\text{[math]}$ decision variables).
$\text{[math]}$: is the objective function.
$\text{[math]}$: is the gradient of the objective function.
$\text{[math]}$: is the Hessian matrix of the objective function.

Denote the feasible region as $\text{[math]}$ . In unconstrained problems, any point $\text{[math]}$ is a feasible point. Therefore, the set $\text{[math]}$ is the entire $\text{[math]}$ space.

A point $\text{[math]}$ is a local solution of the problem if there exists a neighborhood $\text{[math]}$ of $\text{[math]}$ such that

$\text{[math]}$

Further, a point $\text{[math]}$ is a strict local solution if strict inequality holds in the preceding case; that is,

$\text{[math]}$

A point $\text{[math]}$ is a global solution of the problem if no point in $\text{[math]}$ has a smaller function value than $\text{[math]}$ ); that is,

$\text{[math]}$

All the algorithms in the NLPU solver find a local minimum of an optimization problem.

Unconstrained Optimization

The following conditions hold for unconstrained optimization problems:

First-order necessary conditions: If $\text{[math]}$ is a local solution and $\text{[math]}$ is continuously differentiable in some neighborhood of $\text{[math]}$ , then

$\text{[math]}$
Second-order necessary conditions: If $\text{[math]}$ is a local solution and $\text{[math]}$ is twice continuously differentiable in some neighborhood of $\text{[math]}$ , then $\text{[math]}$ is positive semidefinite.
Second-order sufficient conditions: If $\text{[math]}$ is twice continuously differentiable in some neighborhood of $\text{[math]}$ and $\text{[math]}$ and $\text{[math]}$ is positive definite, then $\text{[math]}$ is a strict local solution.

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