# Constrained Minimization¶

O2scl

## Constrained minimization introduction¶

Note

The functionality provided by these legacy classes is probably better found in either https://github.com/stevengj/nlopt or https://petsc.org/main/docs/manual/tao/ .

O₂scl reimplements the Open Optimization Library (OOL) available at http://ool.sourceforge.net. The associated classes allow constrained minimization when the constraint can be expressed as a hyper-cubic constraint on all of the independent variables. The routines have been rewritten and reformatted for C++ in order to facilitate the use of member functions and user-defined vector types as arguments. The base class is mmin_constr and there are two different constrained minimzation algorithms implemented in mmin_constr_pgrad, mmin_constr_spg. (The mmin_constr_gencan minimizer is not yet finished). The O₂scl implementation should be essentially identical to the most recently released version of OOL.

The constrained minimization classes operate in a similar way to the other multi-dimensional minimization classes (which are derived from mmin_base). The constraints are specified with the function:

mmin_constr::set_constraints(size_t nc, vec_t &lower,


and the minimization can be performed by calling either o2scl::mmin_base::mmin() or o2scl::mmin_base::mmin_de() (if the gradient is provided by the user). The method in mmin_constr_gencan requires a Hessian vector product and the user can specify this product for the minimization by using o2scl::mmin_constr::mmin_hess(). The Hessian product function can be specified as an object of type ool_hfunct in a similar way to the other function objects in O₂scl.

There are five error codes defined in mmin_constr which are specific to the classes derived from OOL.

The class anneal_gsl can handle some kinds of constraints by ignoring proposed steps which cause the user-specified function to return a non-zero value.

Also, a simple way of implementing constraints is to add a function to the original which increases the value outside of the allowed region. This can be done with the functions o2scl::constraint() and o2scl::lower_bound(). There are two analogous functions, o2scl::cont_constraint() and o2scl::cont_lower_bound(), which continuous and differentiable versions. Where possible, it is better to use the constrained minimization routines described above.

## Constrained minimization example¶

This example minimizes the function

$f(x,y) = \left[x^2 \log(x)+1\right]\left[\sqrt{y}(y-1)+1\right)]$

which is undefined for $$x<0$$ and $$y<0$$. The function is also minimized by mmin_simp2, which goes outside the allowed region where the function is undefined.

/* Example: ex_conmin.cpp
-------------------------------------------------------------------
This gives an example of the use of a constrained minimizer. This
code finds the global minimum of a two-dimensional function which
is not well-defined outside the region of interest.
*/
#include <boost/numeric/ublas/vector.hpp>
#include <gsl/gsl_math.h>
#include <gsl/gsl_blas.h>
#include <o2scl/test_mgr.h>
#include <o2scl/mmin_constr_spg.h>
#include <o2scl/mmin_simp2.h>

using namespace std;
using namespace o2scl;

typedef boost::numeric::ublas::vector<double> ubvector;

double func(size_t nv, const ubvector &x) {
if (x[0]<0.0 || x[1]<0.0 || x[0]>100.0 || x[1]>100.0) {
cout << "Outside constraint region." << endl;
}
double ret=(log(x[0])*x[0]*x[0]+1.0)*(sqrt(x[1])*(x[1]-1.0)+1.0);
return ret;
}

int dfunc(size_t nv, ubvector &x, ubvector &g) {
g[0]=(x[0]+2.0*x[0]*log(x[0]))*(sqrt(x[1])*(x[1]-1.0)+1.0);
g[1]=(log(x[0])*x[0]*x[0]+1.0)*(sqrt(x[1])+(x[1]-1.0)/2.0/sqrt(x[1]));
return 0;
}

int main(void) {
test_mgr t;
t.set_output_level(1);

cout.setf(ios::scientific);

static const size_t nv=2;

// Specify the function to minimize and its gradient
multi_funct mff11=func;

// The unconstrained minimizer
mmin_simp2<> gm1;
// The constrained minimizer
mmin_constr_spg<> omp;

// The constraints and the location of the minimum
ubvector c1(nv), c2(nv), x(nv);
double fmin;

cout << "Simple minimizer: " << endl;

// Initial guess
for(size_t i=0;i<nv;i++) {
x[i]=10.0;
}

// Minimize
gm1.mmin(nv,x,fmin,mff11);
cout << endl;

cout << "Constrained minimizer: " << endl;

// Initial guess
for(size_t i=0;i<nv;i++) {
x[i]=10.0;
}

// Set constraints
for(size_t i=0;i<nv;i++) {
c1[i]=1.0e-9;
c2[i]=100.0;
}
omp.set_constraints(nv,c1,c2);

// Minimize
omp.mmin_de(nv,x,fmin,mff11,gff);

// Output results
cout << x[0] << " " << x[1] << " " << fmin << endl;

// Test the constrained minimizer results
t.test_rel(x[0],0.60655,1.0e-4,"x0");
t.test_rel(x[1],1.0/3.0,1.0e-4,"x1");

t.report();
return 0;
}