intfminunc Solves an unconstrainted multi-variable mixed integer non linear programming optimization problem 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(.....) 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. 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. Search the minimum of a multi-variable mixed integer non linear programming unconstrained optimization problem specified by : Find the minimum of f(x) such that \begin{eqnarray} &\mbox{min}_{x} & f(x) & x_{i} \in \!\, \mathbb{Z}, i \in \!\, I \end{eqnarray} intfminunc calls Bonmin, which is an optimization library written in C++, to solve the bound optimization problem. 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. The default values for the various items are given as: 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. We begin with the minimization of a simple non-linear function. Find x in R^2 such that it minimizes: \begin{eqnarray} \mbox{min}_{x}\ f(x) = x_{1}^{2} + x_{2}^{2} \end{eqnarray}\\ \text{With integer constraints as: } \\ \begin{eqnarray} \begin{array}{c} [1] \\ \end{array} \end{eqnarray} 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. \begin{eqnarray} \mbox{min}_{x}\ f(x) = 100\boldsymbol{\cdot} (x_{2} - x_{1}^{2})^{2} + (1-x_{1})^{2} \end{eqnarray}\\ \text{With integer constraints as: } \\ \begin{eqnarray} \begin{array}{c} [2] \\ \end{array} \end{eqnarray} Unbounded Problems: Find x in R^2 such that it minimizes: \begin{eqnarray} f(x) = -x_{1}^{2} - x_{2}^{2} \end{eqnarray}\\ \text{With integer constraints as: } \\ \begin{eqnarray} \begin{array}{c} [1] \\ \end{array} \end{eqnarray}