The NLP Procedure |
PROC NLP solves
where is the objective function and the 's are the constraint functions.
A point is feasible if it satisfies all the constraints. The feasible region is the set of all the feasible points. A feasible point is a global solution of the preceding problem if no point in has a smaller function value than ). A feasible point is a local solution of the problem if there exists some open neighborhood surrounding in that no point has a smaller function value than ). Nonlinear programming algorithms cannot consistently find global minima. All the algorithms in PROC NLP find a local minimum for this problem. If you need to check whether the obtained solution is a global minimum, you may have to run PROC NLP with different starting points obtained either at random or by selecting a point on a grid that contains .
Every local minimizer of this problem satisfies the following local optimality conditions:
Most of the optimization algorithms in PROC NLP use iterative techniques that result in a sequence of points , that converges to a local solution . At the solution, PROC NLP performs tests to confirm that the (projected) gradient is close to zero and that the (projected) Hessian matrix is positive definite.
An important tool in the analysis and design of algorithms in constrained optimization is the Lagrangian function, a linear combination of the objective function and the constraints:
The coefficients are called Lagrange multipliers. This tool makes it possible to state necessary and sufficient conditions for a local minimum. The various algorithms in PROC NLP create sequences of points, each of which is closer than the previous one to satisfying these conditions.
Assuming that the functions and are twice continuously differentiable, the point is a local minimum of the nonlinear programming problem, if there exists a vector that meets the following conditions.
1. First-order Karush-Kuhn-Tucker conditions:
2. Second-order conditions: Each nonzero vector that satisfies
Most of the algorithms to solve this problem attempt to find a combination of vectors and for which the gradient of the Lagrangian function with respect to is zero.
The first- and second-order conditions of optimality are based on first and second derivatives of the objective function and the constraints .
The gradient vector contains the first derivatives of the objective function with respect to the parameters as follows:
The symmetric Hessian matrix contains the second derivatives of the objective function with respect to the parameters as follows:
For least-squares problems, the Jacobian matrix contains the first-order derivatives of the objective functions with respect to the parameters as follows:
The Jacobian matrix contains the first-order derivatives of the nonlinear constraint functions , with respect to the parameters , as follows:
PROC NLP provides three ways to compute derivatives:
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