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<a name="Minimizers"></a>
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<p>
Previous: <a href="Solvers.html#Solvers" accesskey="p" rel="prev">Solvers</a>, Up: <a href="Nonlinear-Equations.html#Nonlinear-Equations" accesskey="u" rel="up">Nonlinear Equations</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Minimizers-1"></a>
<h3 class="section">20.2 Minimizers</h3>
<a name="index-local-minimum"></a>
<a name="index-finding-minimums"></a>
<p>Often it is useful to find the minimum value of a function rather than just
the zeroes where it crosses the x-axis. <code>fminbnd</code> 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 <code>fminunc</code> or <code>fminsearch</code>. The two
functions use different internal algorithms and some knowledge of the objective
function is required. For functions which can be differentiated,
<code>fminunc</code> is appropriate. For functions with discontinuities, or for
which a gradient search would fail, use <code>fminsearch</code>.
See <a href="Optimization.html#Optimization">Optimization</a>, for minimization with the presence of constraint
functions. Note that searches can be made for maxima by simply inverting the
objective function
(<code>Fto_max = -Fto_min</code>).
</p>
<a name="XREFfminbnd"></a><dl>
<dt><a name="index-fminbnd"></a><em><var>x</var> =</em> <strong>fminbnd</strong> <em>(<var>fun</var>, <var>a</var>, <var>b</var>)</em></dt>
<dt><a name="index-fminbnd-1"></a><em><var>x</var> =</em> <strong>fminbnd</strong> <em>(<var>fun</var>, <var>a</var>, <var>b</var>, <var>options</var>)</em></dt>
<dt><a name="index-fminbnd-2"></a><em>[<var>x</var>, <var>fval</var>, <var>info</var>, <var>output</var>] =</em> <strong>fminbnd</strong> <em>(…)</em></dt>
<dd><p>Find a minimum point of a univariate function.
</p>
<p><var>fun</var> must be a function handle or name.
</p>
<p>The starting interval is specified by <var>a</var> (left boundary) and <var>b</var>
(right boundary). The endpoints must be finite.
</p>
<p><var>options</var> is a structure specifying additional parameters which
control the algorithm. Currently, <code>fminbnd</code> recognizes these options:
<code>"Display"</code>, <code>"FunValCheck"</code>, <code>"MaxFunEvals"</code>,
<code>"MaxIter"</code>, <code>"OutputFcn"</code>, <code>"TolX"</code>.
</p>
<p><code>"MaxFunEvals"</code> proscribes the maximum number of function evaluations
before optimization is halted. The default value is 500.
The value must be a positive integer.
</p>
<p><code>"MaxIter"</code> proscribes the maximum number of algorithm iterations
before optimization is halted. The default value is 500.
The value must be a positive integer.
</p>
<p><code>"TolX"</code> specifies the termination tolerance for the solution <var>x</var>.
The default is <code>1e-4</code>.
</p>
<p>For a description of the other options, see <a href="Linear-Least-Squares.html#XREFoptimset">optimset</a>.
To initialize an options structure with default values for <code>fminbnd</code>
use <code>options = optimset ("fminbnd")</code>.
</p>
<p>On exit, the function returns <var>x</var>, the approximate minimum point, and
<var>fval</var>, the function evaluated <var>x</var>.
</p>
<p>The third output <var>info</var> reports whether the algorithm succeeded and may
take one of the following values:
</p>
<ul>
<li> 1
The algorithm converged to a solution.
</li><li> 0
Iteration limit (either <code>MaxIter</code> or <code>MaxFunEvals</code>) exceeded.
</li><li> -1
The algorithm was terminated by a user <code>OutputFcn</code>.
</li></ul>
<p>Programming Notes: The search for a minimum is restricted to be in the
finite interval bound by <var>a</var> and <var>b</var>. If you have only one initial
point to begin searching from then you will need to use an unconstrained
minimization algorithm such as <code>fminunc</code> or <code>fminsearch</code>.
<code>fminbnd</code> internally uses a Golden Section search strategy.
</p>
<p><strong>See also:</strong> <a href="Solvers.html#XREFfzero">fzero</a>, <a href="#XREFfminunc">fminunc</a>, <a href="#XREFfminsearch">fminsearch</a>, <a href="Linear-Least-Squares.html#XREFoptimset">optimset</a>.
</p></dd></dl>
<a name="XREFfminunc"></a><dl>
<dt><a name="index-fminunc"></a><em></em> <strong>fminunc</strong> <em>(<var>fcn</var>, <var>x0</var>)</em></dt>
<dt><a name="index-fminunc-1"></a><em></em> <strong>fminunc</strong> <em>(<var>fcn</var>, <var>x0</var>, <var>options</var>)</em></dt>
<dt><a name="index-fminunc-2"></a><em>[<var>x</var>, <var>fval</var>, <var>info</var>, <var>output</var>, <var>grad</var>, <var>hess</var>] =</em> <strong>fminunc</strong> <em>(<var>fcn</var>, …)</em></dt>
<dd><p>Solve an unconstrained optimization problem defined by the function
<var>fcn</var>.
</p>
<p><var>fcn</var> should accept a vector (array) defining the unknown variables, and
return the objective function value, optionally with gradient.
<code>fminunc</code> attempts to determine a vector <var>x</var> such that
<code><var>fcn</var> (<var>x</var>)</code> is a local minimum.
</p>
<p><var>x0</var> determines a starting guess. The shape of <var>x0</var> is preserved in
all calls to <var>fcn</var>, but otherwise is treated as a column vector.
</p>
<p><var>options</var> is a structure specifying additional parameters which
control the algorithm. Currently, <code>fminunc</code> recognizes these options:
<code>"AutoScaling"</code>, <code>"FinDiffType"</code>, <code>"FunValCheck"</code>,
<code>"GradObj"</code>, <code>"MaxFunEvals"</code>, <code>"MaxIter"</code>,
<code>"OutputFcn"</code>, <code>"TolFun"</code>, <code>"TolX"</code>, <code>"TypicalX"</code>.
</p>
<p>If <code>"AutoScaling"</code> is <code>"on"</code>, the variables will be
automatically scaled according to the column norms of the (estimated)
Jacobian. As a result, <code>"TolFun"</code> becomes scaling-independent. By
default, this option is <code>"off"</code> because it may sometimes deliver
unexpected (though mathematically correct) results.
</p>
<p>If <code>"GradObj"</code> is <code>"on"</code>, it specifies that <var>fcn</var>–when
called with two output arguments—also returns the Jacobian matrix of
partial first derivatives at the requested point.
</p>
<p><code>"MaxFunEvals"</code> proscribes the maximum number of function evaluations
before optimization is halted. The default value is
<code>100 * number_of_variables</code>, i.e., <code>100 * length (<var>x0</var>)</code>.
The value must be a positive integer.
</p>
<p><code>"MaxIter"</code> proscribes the maximum number of algorithm iterations
before optimization is halted. The default value is 400.
The value must be a positive integer.
</p>
<p><code>"TolX"</code> specifies the termination tolerance for the unknown variables
<var>x</var>, while <code>"TolFun"</code> is a tolerance for the objective function
value <var>fval</var>. The default is <code>1e-6</code> for both options.
</p>
<p>For a description of the other options, see <code>optimset</code>.
</p>
<p>On return, <var>x</var> is the location of the minimum and <var>fval</var> contains
the value of the objective function at <var>x</var>.
</p>
<p><var>info</var> may be one of the following values:
</p>
<dl compact="compact">
<dt>1</dt>
<dd><p>Converged to a solution point. Relative gradient error is less than
specified by <code>TolFun</code>.
</p>
</dd>
<dt>2</dt>
<dd><p>Last relative step size was less than <code>TolX</code>.
</p>
</dd>
<dt>3</dt>
<dd><p>Last relative change in function value was less than <code>TolFun</code>.
</p>
</dd>
<dt>0</dt>
<dd><p>Iteration limit exceeded—either maximum number of algorithm iterations
<code>MaxIter</code> or maximum number of function evaluations <code>MaxFunEvals</code>.
</p>
</dd>
<dt>-1</dt>
<dd><p>Algorithm terminated by <code>OutputFcn</code>.
</p>
</dd>
<dt>-3</dt>
<dd><p>The trust region radius became excessively small.
</p></dd>
</dl>
<p>Optionally, <code>fminunc</code> can return a structure with convergence
statistics (<var>output</var>), the output gradient (<var>grad</var>) at the
solution <var>x</var>, and approximate Hessian (<var>hess</var>) at the solution
<var>x</var>.
</p>
<p>Application Notes: If the objective function is a single nonlinear equation
of one variable then using <code>fminbnd</code> is usually a better choice.
</p>
<p>The algorithm used by <code>fminunc</code> 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
<code>fminsearch</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFfminbnd">fminbnd</a>, <a href="#XREFfminsearch">fminsearch</a>, <a href="Linear-Least-Squares.html#XREFoptimset">optimset</a>.
</p></dd></dl>
<a name="XREFfminsearch"></a><dl>
<dt><a name="index-fminsearch"></a><em><var>x</var> =</em> <strong>fminsearch</strong> <em>(<var>fun</var>, <var>x0</var>)</em></dt>
<dt><a name="index-fminsearch-1"></a><em><var>x</var> =</em> <strong>fminsearch</strong> <em>(<var>fun</var>, <var>x0</var>, <var>options</var>)</em></dt>
<dt><a name="index-fminsearch-2"></a><em><var>x</var> =</em> <strong>fminsearch</strong> <em>(<var>problem</var>)</em></dt>
<dt><a name="index-fminsearch-3"></a><em>[<var>x</var>, <var>fval</var>, <var>exitflag</var>, <var>output</var>] =</em> <strong>fminsearch</strong> <em>(…)</em></dt>
<dd>
<p>Find a value of <var>x</var> which minimizes the multi-variable function
<var>fun</var>.
</p>
<p>The search begins at the point <var>x0</var> 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 <code>fminunc</code> fails.
</p>
<p>Options for the search are provided in the parameter <var>options</var> using the
function <code>optimset</code>. Currently, <code>fminsearch</code> accepts the options:
<code>"Display"</code>, <code>"FunValCheck"</code>,<code>"MaxFunEvals"</code>,
<code>"MaxIter"</code>, <code>"OutputFcn"</code>, <code>"TolFun"</code>, <code>"TolX"</code>.
</p>
<p><code>"MaxFunEvals"</code> proscribes the maximum number of function evaluations
before optimization is halted. The default value is
<code>200 * number_of_variables</code>, i.e., <code>200 * length (<var>x0</var>)</code>.
The value must be a positive integer.
</p>
<p><code>"MaxIter"</code> proscribes the maximum number of algorithm iterations
before optimization is halted. The default value is
<code>200 * number_of_variables</code>, i.e., <code>200 * length (<var>x0</var>)</code>.
The value must be a positive integer.
</p>
<p>For a description of the other options, see <code>optimset</code>. To initialize
an options structure with default values for <code>fminsearch</code> use
<code>options = optimset ("fminsearch")</code>.
</p>
<p><code>fminsearch</code> may also be called with a single structure argument
with the following fields:
</p>
<dl compact="compact">
<dt><code>objective</code></dt>
<dd><p>The objective function.
</p>
</dd>
<dt><code>x0</code></dt>
<dd><p>The initial point.
</p>
</dd>
<dt><code>solver</code></dt>
<dd><p>Must be set to <code>"fminsearch"</code>.
</p>
</dd>
<dt><code>options</code></dt>
<dd><p>A structure returned from <code>optimset</code> or an empty matrix to
indicate that defaults should be used.
</p></dd>
</dl>
<p>The field <code>options</code> is optional. All others are required.
</p>
<p>On exit, the function returns <var>x</var>, the minimum point, and <var>fval</var>,
the function value at the minimum.
</p>
<p>The third output <var>exitflag</var> reports whether the algorithm succeeded and
may take one of the following values:
</p>
<dl compact="compact">
<dt>1</dt>
<dd><p>if the algorithm converged
(size of the simplex is smaller than <code>TolX</code> <strong>AND</strong> the step in
function value between iterations is smaller than <code>TolFun</code>).
</p>
</dd>
<dt>0</dt>
<dd><p>if the maximum number of iterations or the maximum number of function
evaluations are exceeded.
</p>
</dd>
<dt>-1</dt>
<dd><p>if the iteration is stopped by the <code>"OutputFcn"</code>.
</p></dd>
</dl>
<p>The fourth output is a structure <var>output</var> containing runtime
about the algorithm. Fields in the structure are <code>funcCount</code>
containing the number of function calls to <var>fun</var>, <code>iterations</code>
containing the number of iteration steps, <code>algorithm</code> with the name of
the search algorithm (always:
<code>"Nelder-Mead simplex direct search"</code>), and <code>message</code>
with the exit message.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">fminsearch (@(x) (x(1)-5).^2+(x(2)-8).^4, [0;0])
</pre></div>
<p>Note: If you need to find the minimum of a single variable function it is
probably better to use <code>fminbnd</code>.
</p>
<p><strong>See also:</strong> <a href="#XREFfminbnd">fminbnd</a>, <a href="#XREFfminunc">fminunc</a>, <a href="Linear-Least-Squares.html#XREFoptimset">optimset</a>.
</p></dd></dl>
<p>The function <code>humps</code> is a useful function for testing zero and
extrema finding functions.
</p>
<a name="XREFhumps"></a><dl>
<dt><a name="index-humps"></a><em><var>y</var> =</em> <strong>humps</strong> <em>(<var>x</var>)</em></dt>
<dt><a name="index-humps-1"></a><em>[<var>x</var>, <var>y</var>] =</em> <strong>humps</strong> <em>(<var>x</var>)</em></dt>
<dd><p>Evaluate a function with multiple minima, maxima, and zero crossings.
</p>
<p>The output <var>y</var> is the evaluation of the rational function:
</p>
<div class="example">
<pre class="example"> 1200*<var>x</var>^4 - 2880*<var>x</var>^3 + 2036*<var>x</var>^2 - 348*<var>x</var> - 88
<var>y</var> = - ---------------------------------------------
200*<var>x</var>^4 - 480*<var>x</var>^3 + 406*<var>x</var>^2 - 138*<var>x</var> + 17
</pre></div>
<p><var>x</var> may be a scalar, vector or array. If <var>x</var> is omitted, the
default range [0:0.05:1] is used.
</p>
<p>When called with two output arguments, [<var>x</var>, <var>y</var>], <var>x</var> will
contain the input values, and <var>y</var> will contain the output from
<code>humps</code>.
</p>
<p>Programming Notes: <code>humps</code> has two local maxima located near <var>x</var> =
0.300 and 0.893, a local minimum near <var>x</var> = 0.637, and zeros near
<var>x</var> = -0.132 and 1.300. <code>humps</code> is a useful function for testing
algorithms which find zeros or local minima and maxima.
</p>
<p>Try <code>demo humps</code> to see a plot of the <code>humps</code> function.
</p>
<p><strong>See also:</strong> <a href="Solvers.html#XREFfzero">fzero</a>, <a href="#XREFfminbnd">fminbnd</a>, <a href="#XREFfminunc">fminunc</a>, <a href="#XREFfminsearch">fminsearch</a>.
</p></dd></dl>
<hr>
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Previous: <a href="Solvers.html#Solvers" accesskey="p" rel="prev">Solvers</a>, Up: <a href="Nonlinear-Equations.html#Nonlinear-Equations" accesskey="u" rel="up">Nonlinear Equations</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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