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7  Problem Reference

The following sections contain detailed descriptions of all types of TOMNETProblem present in TOMNET.

The final parts describe four interfaces that are central when modeling any function or constraint that is not linear or quadratic:
  1. IFunction: interface to Function (see 7.8).
  2. IDFunction: interface to Differentiable Function (see 7.9).
  3. IConstraints: interface to Constraints (see 7.10).
  4. IDConstraints: interface to Differentiable Constraints (see 7.11).
These Interfaces function as a sort of templates that help the solver evaluate the objective function and constraints.

7.1  class LinearProblem 

Description
One of the basic classes of TOMNET is LinearProblem. It is used for modeling TOMNETProblem's with a linear objective function and only linear constraints. See linear programming in 3.1.6.

Calling Syntax
LinearProblem Prob = new LinearProblem(c, A, b_L, b_U, x_L, x_U,
  x_0, Name, double.NegativeInfinity, null, null);

Constructor Inputs
 
double[] c The vector c in the objective function.
 
object A The linear constraints (a double[] or a Sparse).
 
double[] b_L The lower bounds for the linear constraints.
double[] b_U The upper bounds for the linear constraints.
 
double[] x_L Lower bounds on x.
double[] x_U Upper bounds on x.
double[] x_0 Starting point x.
 
String Name The name of the problem.
 
double fLowBnd A lower bound on the function value at optimum if known. If not known set to NegativeInfinity.
 
double[] f_opt Optimal function value(s), if known (Stationary points). If not known set Null.
double[] x_opt The x-values corresponding to the given f_opt, if known. If not known set Null.

7.2  class QuadraticProblem 

Description
The class QuadraticProblem is used for modeling a TOMNETProblem with a quadratic objective function and only linear constraints. See quadratic programming in 3.1.2.

Calling Syntax
QuadraticProblem Prob = QuadraticProblem.qpAssign(F, c, A,
  b_L, b_U, x_L, x_U, x_0, Name);

Constructor Inputs
 
object F The matrix F in the objective function (a double[] or a Sparse).
 
double[] c The vector c in the objective function.
 
object A The linear constraints (a double[] or a Sparse).
 
double[] b_L The lower bounds for the linear constraints.
double[] b_U The upper bounds for the linear constraints.
 
double[] x_L Lower bounds on x.
double[] x_U Upper bounds on x.
double[] x_0 Starting point x.
 
String Name The name of the problem.
 

7.3  class NonlinearProblem 

Description
A NonlinearProblem is used to model a problem with a general objective function and linear constraints. The user needs to implement a class that inherits from either the interface IDFunction (see 7.9) or IFunction (see 7.8) to model the objective function.

It is always better to use IDFunction and describe the derivatives analytically. If the derivatives are impossible or very hard to model numerical differentiation can be used by implementing an IFunction, this may however result in loss of robustness.

For an example of how to use the IFunction the reader is encouraged to see nlpQG.cs  in tomnet/quickguide  and compare NonlinearProblem with ConProblem in 7.4.

To create an unconstrained optimization problem (see unconstrained optimization 3.1.1) call the constructor with Null-parameters instead of A, b_L and b_U.

Calling Syntax
//
// User-defined objective function
//
public class myNLPFunc : IDFunction    { /* ... */ }

//
// NonlinearProblem needs an instance of myNLPFunc
//
myNLPFunc F = new myNLPFunc();
NonlinearProblem Prob = new NonlinearProblem(F, A, b_L, b_U,
  x_L, x_U, x_0, Name, double.NegativeInfinity, null, null);

Constructor Inputs
 
IFunction F User-defined objective function (see 7.8 and 7.9).
 
object A The linear constraints (a double[] or a Sparse).
 
double[] b_L The lower bounds for the linear constraints.
double[] b_U The upper bounds for the linear constraints.
 
double[] x_L Lower bounds on x.
double[] x_U Upper bounds on x.
double[] x_0 Starting point x.
 
String Name The name of the problem.
 
double fLowBnd A lower bound on the function value at optimum if known. If not known set to NegativeInfinity.
 
double[] f_opt Optimal function value(s), if known (Stationary points). If not known set Null.
double[] x_opt The x-values corresponding to the given f_opt, if known. If not known set Null.

7.4  class ConProblem 

Description
One of the more general classes for modeling a problem in TOMNET is ConProblem. It requires coding of an objective function and nonlinear constraints by the user.

For an extensive example the reader is encouraged to see nlpQG.cs  in tomnet/quickguide . Also see constrained nonlinear programming in 3.1.3.

The user needs to implement a class that inherits from either the interface IFunction (see 7.8) or IDFunction (see 7.9) to model the objective function. Also the nonlinear constraints must be modeled in a class that inherits from IConstraints (see 7.10) or IDConstraints (see 7.11).

Calling Syntax
//
// User-defined objective function class myConFunc
// and nonlinear constraints class myConCons.
//
public class myConFunc : IDFunction    { /* ... */ }
public class myConCons : IDConstraints { /* ... */ }

//
// ConProblem needs an instance of IDFunction and IDConstraints
//
myConFunc myF = new myConFunc();
myConCons myC = new myConCons();
TOMNETProblem Prob = new ConProblem(myF, myC, c_L, c_U, x_L, x_U,
  x_0, Name, double.NegativeInfinity, null, null);

Constructor Inputs
 
IFunction F User-defined objective function (see 7.8 and 7.9).
 
IConstraints C User-defined nonlinear constraints (see 7.10 and 7.11).
 
double[] c_L The lower bounds for the nonlinear constraints.
double[] c_U The upper bounds for the nonlinear constraints.
 
double[] x_L Lower bounds on x.
double[] x_U Upper bounds on x.
double[] x_0 Starting point x.
 
String Name The name of the problem.
 
double fLowBnd A lower bound on the function value at optimum if known. If not known set to NegativeInfinity.
 
double[] f_opt Optimal function value(s), if known (Stationary points). If not known set Null.
double[] x_opt The x-values corresponding to the given f_opt, if known. If not known set Null.

7.5  class ConLinProblem 

Description
One of the more general classes for modeling a problem in TOMNET is ConLinProblem. It requires coding of the objective function and nonlinear constraints by the user. See 3.1.1 for a formal definition.

For a detailed example the reader is encouraged to see A.2.

The user needs to implement a class that inherits from either the interface IFunction (see 7.8) or IDFunction (see 7.9) to model the objective function. Also the nonlinear constraints must be modeled in a class that inherits from IConstraints (see 7.10) or IDConstraints (see 7.11).

Calling Syntax
//
// User-defined objective function class myConFunc
// and nonlinear constraints class myConCons.
//
public class myConFunc : IDFunction    { /* ... */ }
public class myConCons : IDConstraints { /* ... */ }

//
// ConLinProblem needs an myConFunc and a myConCons
//
myConFunc myF = new myConFunc();
myConCons myC = new myConCons();

ConLinProblem Prob = new ConLinProblem(myF, A, b_L, b_U,
  myC, c_L, c_U, x_L, x_U, x_0, Name, double.NegativeInfinity,
  null, null);

Constructor Inputs
 
IFunction F User-defined objective function (see 7.8 and 7.9).
 
IConstraints A Linear Constraints (a Sparse or a double[]).
 
double[] b_L The lower bounds for the linear constraints.
double[] b_U The upper bounds for the linear constraints.
 
IConstraints C User-defined nonlinear constraints (see 7.10 and 7.11).
 
double[] c_L The lower bounds for the non linear constraints.
double[] c_U The upper bounds for the non linear constraints.
 
double[] x_L Lower bounds on x.
double[] x_U Upper bounds on x.
double[] x_0 Starting point x.
 
String Name The name of the problem.
 
double fLowBnd A lower bound on the function value at optimum if known. If not known set to NegativeInfinity.
 
double[] f_opt Optimal function value(s), if known (Stationary points). If not known set Null.
double[] x_opt The x-values corresponding to the given f_opt, if known. If not known set Null.

7.6  class LinearLeastSquaresProblem 

Description
The LinearLeastSquaresProblem efficiently models all linear least squares problem (see 3.1.8 for definitions).

Calling Syntax
LinearLeastSquaresProblem Prob = new LinearLeastSquaresProblem(
  n, m, k, C, d, A, b_L, b_U, x_0, x_L, x_U);

Constructor Inputs
 
int n Number of variables.
int m Number of linear constraints.
int k Number of residuals.
 
double[] C C matrix of the objective function.
double[] y The observations.
 
double[] A The linear constraints.
 
double[] b_L The lower bounds for the linear constraints.
double[] b_U The upper bounds for the linear constraints.
 
double[] x_L Lower bounds on x.
double[] x_U Upper bounds on x.
double[] x_0 Starting point x.

7.7  class NonlinearLeastSquaresProblem 

Description
In TOMNET NonlinearLeastSquaresProblem can be used to model a nonlinear least squares problem with nonlinear constraints. See constrained nonlinear least squares problem in 3.1.9 for definitions. Also see nllsQG.cs  in tomnet/quickguide  for a comprehensive example.

Calling Syntax
NonlinearLeastSquaresProblem Prob = new
  NonlinearLeastSquaresProblem(R, C, A, b_L, b_U, x_L, x_U, x_0,
  c_L, c_U, y, t, "NllsProb", double.NegativeInfinity, null,
  null);

Constructor Inputs
 
IFunction F User-defined objective function (see 7.8 and 7.9).
IConstraints C User-defined nonlinear constraints (see 7.10 and 7.11).
 
object A The linear constraints.
 
double[] b_L The lower bounds for the linear constraints.
double[] b_U The upper bounds for the linear constraints.
 
double[] x_L Lower bounds on x.
double[] x_U Upper bounds on x.
double[] x_0 Starting point x.
 
double[] c_L The lower bounds for the nonlinear constraints.
double[] c_U The upper bounds for the nonlinear constraints.
 
double[] y The observations.
double[] t Array of time points of observations.
 
String Name The name of the problem.
 
double fLowBnd A lower bound on the function value at optimum if known. If not known set to NegativeInfinity.
 
double[] f_opt Optimal function value(s), if known (Stationary points). If not known set Null.
double[] x_opt The x-values corresponding to the given f_opt, if known. If not known set Null.


7.8  Interface IFunction 

Description
An IFunction is used to model a problem with any scalar objective function. It is preferable to use IDFunction (see 7.9) and define the gradient of the objective function to gain robustness and speed.

Calling Syntax
To use an IFunction the method Evaluate must be implemented. 
public class myObjective : IFunction
{
  public void Evaluate(double[] fx, double[] x)
  {
    // ...
  }
}

Methods to Implement
 
void Evaluate(double[] fx, double[] x) User-defined objective function.

Examples
See A.2 for an example of how to use IFunction, IDFunction, IConstraints and/or IDConstraints.

7.9  Interface IDFunction 

Description
An IDFunction is used to model a any scalar objective function where the gradients are known.

IDFunction inherits from IFunction so any IDFunction is also an IFunction.

Calling Syntax
To use an IDFunction two methods needs to be implemented: Evaluate (for the function value) and Grad for the values of the gradient. 
public class myObjective : IDFunction
{
  public void Evaluate(double[] fx, double[] x)
  {
    // ...
  }

  public void Grad(double[] gx, double[] x)
  {
    // ...
  }
}

Methods to Implement
 
void Evaluate(double[] fx, double[] x) User-defined objective function.
void Grad(double[] gx, double[] x) User-defined gradient.

Examples
See A.2 for an example of how to use IFunction, IDFunction, IConstraints and/or IDConstraints.

7.10  Interface IConstraints 

Description
An IConstraints is used to model a problem with any nonlinear constraints. It is preferable to use IDConstraints (see 7.11) and define the derivatives of the constraints to gain robustness and speed.

Calling Syntax
To use an IConstraints two Evaluate methods must be implemented. 
public class myConstraints : IConstraints
{
  public void Evaluate(double[] cx, double[] x)
  {
    // ...
  }

  public double[] Evaluate(double[] x)
  {
    // ...
  }
}

Methods to Implement
 
void Evaluate(double[] cx, double[] x) Evalate the constraints.
double[] Evaluate(double[] x) Evalate the constraints.

Examples
See A.2 for an example of how to use IFunction, IDFunction, IConstraints and/or IDConstraints.

7.11  Interface IDConstraints 

Description
An IDConstraints is used to model a any constraints function where the gradients can be expressed analytically.

IDConstraints inherits from IConstraints so any IDConstraints is also an IConstraints.

Calling Syntax
To use an IDConstraints the methods Evaluate and dc must be implemented. 
public class myConstraints : IDConstraints
{
  public  void Evaluate(double[] cx, double[] x)
  {
    // ...
  }

  public double[] Evaluate(double[] x)
  {
    // ...
  }

  public void dc(Jacobian dcx, double[] x)
  {
    // ...
  }
}

Methods to Implement
 
void Evaluate(double[] cx, double[] x) Evaluate the constraints.
double[] Evaluate(double[] x) Evaluate the constraints.
void dc(Jacobian dcx, double[] x) Evaluate the derivatives of the constraints.

Examples
See A.2 for an example of how to use IFunction, IDFunction, IConstraints and/or IDConstraints.

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