# intfminunc

Solves an unconstrained mixed-integer nonlinear optimization problem.

### Calling Sequence

xopt = intfminunc(f,x0) xopt = intfminunc(f,x0,intcon) xopt = intfminunc(f,x0,intcon,options) [xopt,fopt] = intfminunc(.....) [xopt,fopt,exitflag]= intfminunc(.....) [xopt,fopt,exitflag,gradient,hessian]= intfminunc(.....)

### Input Parameters

- f :
A function, representing the objective function of the problem.

- x0 :
A vector of doubles, containing the starting values of variables of size (1 X n) or (n X 1) where 'n' is the number of Variables.

- intcon :
A vector of integers, representing the variables that are constrained to be integers.

- options :
A list, containing the option for user to specify. See below for details.

### Outputs

- xopt :
A vector of doubles, containing the computed solution of the optimization problem.

- fopt :
A double, containing the the function value at x.

- exitflag :
An integer, containing the flag which denotes the reason for termination of algorithm. See below for details.

- gradient :
A vector of doubles, containing the objective's gradient of the solution.

- hessian :
A matrix of doubles, containing the Lagrangian's hessian of the solution.

### Description

Search the minimum of a multivariable mixed integer nonlinear programming unconstrained optimization problem specified by : Find the minimum of f(x) such that

intfminunc calls Bonmin, which is an optimization library written in C++, to solve the bound optimization problem.

### Options

The options allow the user to set various parameters of the Optimization problem. The syntax for the options is given by:options= list("IntegerTolerance", [---], "MaxNodes",[---], "MaxIter", [---], "AllowableGap",[---] "CpuTime", [---],"gradobj", "off", "hessian", "off" );

- IntegerTolerance : A Scalar, a number with that value of an integer is considered integer.
- MaxNodes : A Scalar, containing the maximum number of nodes that the solver should search.
- CpuTime : A scalar, specifying the maximum amount of CPU Time in seconds that the solver should take.
- AllowableGap : A scalar, that specifies the gap between the computed solution and the the objective value of the best known solution stop, at which the tree search can be stopped.
- MaxIter : A scalar, specifying the maximum number of iterations that the solver should take.
- gradobj : A string, to turn on or off the user supplied objective gradient.
- hessian : A scalar, to turn on or off the user supplied objective hessian.

options = list('integertolerance',1d-06,'maxnodes',2147483647,'cputime',1d10,'allowablegap',0,'maxiter',2147483647,'gradobj',"off",'hessian',"off")

The exitflag allows to know the status of the optimization which is given back by Ipopt.

- 0 : Optimal Solution Found
- 1 : InFeasible Solution.
- 2 : Objective Function is Continuous Unbounded.
- 3 : Limit Exceeded.
- 4 : User Interrupt.
- 5 : MINLP Error.

For more details on exitflag, see the Bonmin documentation which can be found on http://www.coin-or.org/Bonmin

A few examples displaying the various functionalities of intfminunc have been provided below. You will find a series of problems and the appropriate code snippets to solve them.

### Example

We begin with the minimization of a simple nonlinear function.

Find x in R^2 such that it minimizes:

//Example 1: //Objective function to be minimised function y=f(x) y= x(1)^2 + x(2)^2; endfunction //Starting point x0=[2,1]; intcon = [1]; [xopt,fopt]=intfminunc(f,x0,intcon) // Press ENTER to continue |

### Example

We now look at the Rosenbrock function, a non-convex performance test problem for optimization routines. We use this example to illustrate how we can enhance the functionality of intfminunc by setting input options. We can pre-define the gradient of the objective function and/or the hessian of the lagrange function and thereby improve the speed of computation. This is elaborated on in example 2. We also set solver parameters using the options.

///Example 2: //Objective function to be minimised function y=f(x) y= 100*(x(2) - x(1)^2)^2 + (1-x(1))^2; endfunction //Starting point x0=[-1,2]; intcon = [2] //Options options=list("MaxIter", [1500], "CpuTime", [500]); //Calling [xopt,fopt,exitflag,gradient,hessian]=intfminunc(f,x0,intcon,options) // Press ENTER to continue |

### Example

Unbounded Problems: Find x in R^2 such that it minimizes:

//The below problem is an unbounded problem: //Find x in R^2 such that the below function is minimum //f = - x1^2 - x2^2 //Objective function to be minimised function [y, g, h]=f(x) y = -x(1)^2 - x(2)^2; g = [-2*x(1),-2*x(2)]; h = [-2,0;0,-2]; endfunction //Starting point x0=[2,1]; intcon = [1] options = list("gradobj","ON","hessian","on"); [xopt,fopt,exitflag,gradient,hessian]=intfminunc(f,x0,intcon,options) |