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OPT++

/** \page ConstrainedProblems  Constrained minimization 

Consider the general nonlinear programming problem 

<em> minimize </em> \f[ f(x) \f]
<em> subject to </em> \f[ h(x) = 0, \f]
<em>  </em> \f[ g(x) \ge 0. \f]

<p>
In this section, we provide the framework for the general nonlinear 
programming problem.  We also examine the construction of linear, nonlinear 
and compound constraints.  
</p>
<UL>
<li> \ref  LinearConstraints
<li> \ref  NonLinearConstraints
<li> \ref  CompoundConstraintsDoc
</UL>
If a constraint has finite lower and upper bounds, OPT++ treats it as two
separate constraints.  In the computation of the residuals and gradients,
constraints with finite lower bounds appear first followed by those with
finte upper bounds. Consequently, in the optimization summary, you will
see the constraint count is double the original number of constraints.

<p> 
Now, we have all the necessary tools to build a nonlinear programming problem.
</p>

<UL>
<li> \ref  Problem 
<li> \ref  Fragments
</UL>

\section Problem Creating a general nonlinear programming problem 

Let's consider Problem 65 from the Hock and Schittkowski suite of test
problems:

<em> minimize </em> 
\f[ (x_1 - x_2)^2 + (1/9)(x_1 + x_2 - 10)^2 + (x_3 - 5)^2 \f]
<em> subject to </em> \f[ x_1^2 + x_2^2 + x_3^2 \le 48, \f]
<em> </em>  \f[-4.5 \le x_1 \le 4.5, \f]
<em> </em>  \f[-4.5 \le x_2 \le 4.5, \f]
<em> </em>  \f[ -5.0 \le x_3 \le 5.0 \f]

Step 1: Build your constraints. The constraint set consists of bounds on
the variables plus one nonlinear inequality.
\code

   // Number of bounds and number of nonlinear constraints
   int numBds = 3, ncnln = 1;

   ColumnVector lower(numBds), upper(numBds); 

   // Construct the nonlinear equation
   NLP* chs65      = new NLP( new NLF2(numBds,ncnln,ineq_hs65,init_hs65) );
   Constraint ineq = new NonLinearInequality(chs65);

   // Construct the bound constraints 
   lower << -4.5 << -4.5 << -5.0;
   upper <<  4.5 <<  4.5 <<  5.0 ;
   Constraint bc   = new BoundConstraint(n,lower,upper); 

   // Construct the compound constraint 
   CompoundConstraint* constraints = new CompoundConstraint(ineq,bc);
\endcode

Step 2: Create a constrained problem where the objective function has
analytic Hessians.
\code
   NLF2 hs65_problem(n,hs65,init_hs65,constraints);
\endcode

\section Fragments  Specifying the optimization method 
In OPT++, the only freely available method to solve a general nonlinear problem 
is OptNIPS, a nonlinear interior-point method. However, we do provide an
wrapper to NPSOL, a sequential quadratic programming algorithm. For
more information, go to the 
<a href ="http://www.sbsi-sol-optimize.com/NPSOL.htm>NPSOL website</a>.
<ol>
	<li> \ref tstnips
	<li> \ref tstnpsol
</ol>

<p> <a href="ParallelOptimization.html">
	Next Section: Parallel optimization
</a> |  <a href="Classification.html">Back to Solvers Page</a> </p> 

Last revised <em> July 13, 2006</em>
*/