fminunc Solves a multi-variable unconstrainted optimization problem xopt = fminunc(f,x0) xopt = fminunc(f,x0,options) [xopt,fopt] = fminunc(.....) [xopt,fopt,exitflag]= fminunc(.....) [xopt,fopt,exitflag,output]= fminunc(.....) [xopt,fopt,exitflag,output,gradient]=fminunc(.....) [xopt,fopt,exitflag,output,gradient,hessian]=fminunc(.....) 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. 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. output : A structure, containing the information about the optimization. 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 an unconstrained optimization problem specified by : Find the minimum of f(x) such that \begin{eqnarray} &\mbox{min}_{x} & f(x)\\ \end{eqnarray} Fminunc calls Ipopt which is an optimization library written in C++, to solve the unconstrained 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("MaxIter", [---], "CpuTime", [---], "GradObj", ---, "Hessian", ---, "GradCon", ---); MaxIter : A Scalar, specifying the Maximum Number of Iterations that the solver should take. CpuTime : A Scalar, specifying the Maximum amount of CPU Time in seconds that the solver should take. Gradient: A function, representing the gradient function of the objective in Vector Form. Hessian : A function, representing the hessian function of the lagrange in the form of a Symmetric Matrix with input parameters as x, objective factor and lambda. Refer to Example 5 for definition of lagrangian hessian function. The default values for the various items are given as: options = list("MaxIter", [3000], "CpuTime", [600]); The exitflag allows the user to know the status of the optimization which is returned by Ipopt. 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 Ipopt documentation which can be found on http://www.coin-or.org/Ipopt/documentation/ The output data structure contains detailed information about the optimization process. It is of type "struct" and contains the following fields. output.Iterations: The number of iterations performed. output.Cpu_Time : The total cpu-time taken. output.Objective_Evaluation: The number of objective evaluations performed. output.Dual_Infeasibility : The Dual Infeasiblity of the final soution. output.Message: The output message for the problem. A few examples displaying the various functionalities of fminunc 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} 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 fminunc 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} Unbounded Problems: Find x in R^2 such that it minimizes: \begin{eqnarray} f(x) = -x_{1}^{2} - x_{2}^{2} \end{eqnarray} R.Vidyadhar , Vignesh Kannan