The Unconstrained Nonlinear Programming Solver |
Overview
The unconstrained nonlinear programming solver (NLPU) is a component of the OPTMODEL procedure, and it can be used for solving general unconstrained nonlinear programming (NLP) problems.
Mathematically, an unconstrained nonlinear programming problem can be stated as follows:
![\displaystyle\mathop{\rm minimize}_{x\in{\mathbb r}^n} & f(x) \](images/nlpu_nlpueq1.gif)
where
![x \in \mathbb{r}^n](images/nlpu_nlpueq2.gif)
is the vector of decision variables and
![f](images/nlpu_nlpueq3.gif)
:
![\mathbb{r}^n\mapsto\mathbb{r}](images/nlpu_nlpueq4.gif)
is the nonlinear objective function.
![f](images/nlpu_nlpueq3.gif)
is assumed to be twice
continuously differentiable, meaning that its second derivatives exist and are continuous in
![\mathbb{r}^n](images/nlpu_nlpueq5.gif)
.
For purely unconstrained optimization, PROC OPTMODEL implements the following algorithms:
- Fletcher-Reeves nonlinear conjugate gradient algorithm for
convex unconstrained optimization
- Polak-Ribière nonlinear conjugate gradient algorithm for
convex unconstrained optimization
- Limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm
for general nonconvex unconstrained optimization (default)
If the objective function is
convex , then the optimal solution is a global optimum; otherwise it is a local optimum. In other words, the optimal solution obtained by the NLPU solver depends on the starting point. An example of a nonlinear function with multiple local optima is displayed in
Figure 11.1.
Figure 11.1: An Example of Multiple Local Optimal Points
The function displayed in Figure 11.1 is
![f(x, y) = {sinc}\,(x^2 + y^2) + {sinc}\,((x - 2)^2 + y^2)](images/nlpu_nlpueq6.gif)
where sinc(.), also called the "sampling function," is a function that arises frequently in signal processing. The function is defined as
![{sinc}(x) & \equiv & \{1 & {for}\, x = 0,\ \displaystyle\frac{{sin}\, x}x & {otherwise}\ . \](images/nlpu_nlpueq7.gif)
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