intfminbnd Solves a multi-variable optimization problem on a bounded interval xopt = intfminbnd(f,intcon,x1,x2) xopt = intfminbnd(f,intcon,x1,x2,options) [xopt,fopt] = intfminbnd(.....) [xopt,fopt,exitflag]= intfminbnd(.....) [xopt,fopt,exitflag,output]=intfminbnd(.....) [xopt,fopt,exitflag,gradient,hessian]=intfminbnd(.....) f : A function, representing the objective function of the problem. x_{1} : A vector, containing the lower bound of the variables of size (1 X n) or (n X 1) where n is number of variables. If it is empty it means that the lower bound is -\infty. x_{2} : A vector, containing the upper bound of the variables of size (1 X n) or (n X 1) or (0 X 0) where n is the number of variables. If it is empty it means that the upper bound is \infty. intcon : A vector of integers, representing the variables that are constrained to be integers. options : A list, containing the options 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 function on bounded interval specified by : Find the minimum of f(x) such that \begin{eqnarray} &\mbox{min}_{x} & f(x)\\ & \text{Subjected to:}\\ & x_{1} \ < x \ < x_{2} \\ \end{eqnarray} intfminbnd 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 the user to know the status of the optimization which is returned by Bonmin. The values it can take and what they indicate is described below: 0 : Optimal Solution Found 1 : Maximum Number of Iterations Exceeded. Output may not be optimal. 2 : Maximum amount of CPU Time exceeded. Output may not be optimal. 3 : Stop at Tiny Step. 4 : Solved To Acceptable Level. 5 : Converged to a point of local infeasibility. 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 intfminbnd have been provided below. You will find a series of problems and the appropriate code snippets to solve them. We start with a simple objective function. Find x in R^6 such that it minimizes: \begin{eqnarray} \mbox{min}_{x}\ f(x) = sin(x_{1}) + sin(x_{2}) + sin(x_{3}) + sin(x_{4}) + sin(x_{5}) + sin(x_{6}) \end{eqnarray} \\\text{Subjected to:}\\ \begin{eqnarray} \hspace{70pt} &-2 &\leq x{1}, x{2}, x{3}, x{4}, x{5}, x{6} &\leq 2\\ \end{eqnarray}\\ \text{With integer constraints as: }\\ \begin{eqnarray} \begin{array}{ccc} [2 & 3 & 4] \\ \end{array} \end{eqnarray} Here we solve a bounded objective function in R^6. We use this function to illustrate how we can further enhance the functionality of fminbnd 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. Unbounded Problems: Find x in R^2 such that it minimizes: \begin{eqnarray} f(x) = -((x_{1}-1)^{2}+(x_{2}-1)^{2}) \end{eqnarray} \\\text{Subjected to:}\\ \begin{eqnarray} -\infty &\leq x_{1} &\leq \infty\\ -\infty &\leq x_{2} &\leq \infty \end{eqnarray}\\ \text{With integer constraints as: } \\ \begin{eqnarray} \begin{array}{cccccc} [1 & 2] \\ \end{array} \end{eqnarray} Harpreet Singh