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### 20.2 Minimizers

Often it is useful to find the minimum value of a function rather than just the zeroes where it crosses the x-axis. `fminbnd` is designed for the simpler, but very common, case of a univariate function where the interval to search is bounded. For unbounded minimization of a function with potentially many variables use `fminunc` or `fminsearch`. The two functions use different internal algorithms and some knowledge of the objective function is required. For functions which can be differentiated, `fminunc` is appropriate. For functions with discontinuities, or for which a gradient search would fail, use `fminsearch`. See Optimization, for minimization with the presence of constraint functions. Note that searches can be made for maxima by simply inverting the objective function (`Fto_max = -Fto_min`).

[x, fval, info, output] = fminbnd (fun, a, b, options)

  Find a minimum point of a univariate function.

  fun should be a function handle or name. a, b specify a starting interval. options is a structure specifying additional options. Currently, `fminbnd` recognizes these options: `"FunValCheck"`, `"OutputFcn"`, `"TolX"`, `"MaxIter"`, `"MaxFunEvals"`. For a description of these options, see optimset.

  On exit, the function returns x, the approximate minimum point and fval, the function value thereof.

  info is an exit flag that can have these values:

  - 1 The algorithm converged to a solution.
  - 0 Maximum number of iterations or function evaluations has been exhausted.
  - -1 The algorithm has been terminated from user output function.

  Notes: The search for a minimum is restricted to be in the interval bound by a and b. If you only have an initial point to begin searching from you will need to use an unconstrained minimization algorithm such as `fminunc` or `fminsearch`. `fminbnd` internally uses a Golden Section search strategy.

  See also: fzero, fminunc, fminsearch, optimset.

fminunc (fcn, x0)
fminunc (fcn, x0, options)
[x, fval, info, output, grad, hess] = fminunc (fcn, …)

  Solve an unconstrained optimization problem defined by the function fcn.

  fcn should accept a vector (array) defining the unknown variables, and return the objective function value, optionally with gradient. `fminunc` attempts to determine a vector x such that `fcn (x)` is a local minimum.

  x0 determines a starting guess. The shape of x0 is preserved in all calls to fcn, but otherwise is treated as a column vector.

  options is a structure specifying additional options. Currently, `fminunc` recognizes these options: `"FunValCheck"`, `"OutputFcn"`, `"TolX"`, `"TolFun"`, `"MaxIter"`, `"MaxFunEvals"`, `"GradObj"`, `"FinDiffType"`, `"TypicalX"`, `"AutoScaling"`.

  If `"GradObj"` is `"on"`, it specifies that fcn, when called with two output arguments, also returns the Jacobian matrix of partial first derivatives at the requested point. `TolX` specifies the termination tolerance for the unknown variables x, while `TolFun` is a tolerance for the objective function value fval. The default is `1e-7` for both options.

  For a description of the other options, see `optimset`.

  On return, x is the location of the minimum and fval contains the value of the objective function at x.

  info may be one of the following values:

  1

    Converged to a solution point. Relative gradient error is less than specified by `TolFun`.

  2

    Last relative step size was less than `TolX`.

  3

    Last relative change in function value was less than `TolFun`.

  0

    Iteration limit exceeded—either maximum number of algorithm iterations `MaxIter` or maximum number of function evaluations `MaxFunEvals`.

  -1

    Algorithm terminated by `OutputFcn`.

  -3

    The trust region radius became excessively small.

  Optionally, `fminunc` can return a structure with convergence statistics (output), the output gradient (grad) at the solution x, and approximate Hessian (hess) at the solution x.

  Application Notes: If the objective function is a single nonlinear equation of one variable then using `fminbnd` is usually a better choice.

  The algorithm used by `fminunc` is a gradient search which depends on the objective function being differentiable. If the function has discontinuities it may be better to use a derivative-free algorithm such as `fminsearch`.

  See also: fminbnd, fminsearch, optimset.

x = fminsearch (fun, x0)
x = fminsearch (fun, x0, options)
x = fminsearch (fun, x0, options, fun_arg1, fun_arg2, …)
[x, fval, exitflag, output] = fminsearch (…)

  Find a value of x which minimizes the function fun.

  The search begins at the point x0 and iterates using the Nelder & Mead Simplex algorithm (a derivative-free method). This algorithm is better-suited to functions which have discontinuities or for which a gradient-based search such as `fminunc` fails.

  Options for the search are provided in the parameter options using the function `optimset`. Currently, `fminsearch` accepts the options: `"TolX"`, `"TolFun"`, `"MaxFunEvals"`, `"MaxIter"`, `"Display"`, `"FunValCheck"`, and `"OutputFcn"`. For a description of these options, see `optimset`.

  Additional inputs for the function fun can be passed as the fourth and higher arguments. To pass function arguments while using the default options values, use `[]` for options.

  On exit, the function returns x, the minimum point, and fval, the function value at the minimum.

  The third return value exitflag is

  1

    if the algorithm converged (size of the simplex is smaller than `options.TolX` AND the step in the function value between iterations is smaller than `options.TolFun`).

  0

    if the maximum number of iterations or the maximum number of function evaluations are exceeded.

  -1

    if the iteration is stopped by the `"OutputFcn"`.

  The fourth return value is a structure output with the fields, `funcCount` containing the number of function calls to fun, `iterations` containing the number of iteration steps, `algorithm` with the name of the search algorithm (always: `"Nelder-Mead simplex direct search"`), and `message` with the exit message.

  Example:

      fminsearch (@(x) (x(1)-5).^2+(x(2)-8).^4, [0;0])

  See also: fminbnd, fminunc, optimset.

The function `humps` is a useful function for testing zero and extrema finding functions.

y = humps (x)
[x, y] = humps (x)

  Evaluate a function with multiple minima, maxima, and zero crossings.

  The output y is the evaluation of the rational function:

              1200*x^4 - 2880*x^3 + 2036*x^2 - 348*x - 88
       y = - ---------------------------------------------
               200*x^4 - 480*x^3 + 406*x^2 - 138*x + 17

  x may be a scalar, vector or array. If x is omitted, the default range [0:0.05:1] is used.

  When called with two output arguments, [x, y], x will contain the input values, and y will contain the output from `humps`.

  Programming Notes: `humps` has two local maxima located near x = 0.300 and 0.893, a local minimum near x = 0.637, and zeros near x = -0.132 and 1.300. `humps` is a useful function for testing algorithms which find zeros or local minima and maxima.

  Try `demo humps` to see a plot of the `humps` function.

  See also: fzero, fminbnd, fminunc, fminsearch.

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