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<h3 class="section">24.3 Nonlinear Programming</h3>
<p>Octave can also perform general nonlinear minimization using a
successive quadratic programming solver.
<!-- ./optimization/sqp.m -->
<p><a name="doc_002dsqp"></a>
<div class="defun">
— Function File: [<var>x</var>, <var>obj</var>, <var>info</var>, <var>iter</var>, <var>nf</var>, <var>lambda</var>] = <b>sqp</b> (<var>x, phi, g, h, lb, ub, maxiter, tolerance</var>)<var><a name="index-sqp-1803"></a></var><br>
<blockquote><p>Solve the nonlinear program
<pre class="example"> min phi (x)
x
</pre>
<p>subject to
<pre class="example"> g(x) = 0
h(x) >= 0
lb <= x <= ub
</pre>
<p class="noindent">using a successive quadratic programming method.
<p>The first argument is the initial guess for the vector <var>x</var>.
<p>The second argument is a function handle pointing to the objective
function. The objective function must be of the form
<pre class="example"> y = phi (x)
</pre>
<p class="noindent">in which <var>x</var> is a vector and <var>y</var> is a scalar.
<p>The second argument may also be a 2- or 3-element cell array of
function handles. The first element should point to the objective
function, the second should point to a function that computes the
gradient of the objective function, and the third should point to a
function to compute the hessian of the objective function. If the
gradient function is not supplied, the gradient is computed by finite
differences. If the hessian function is not supplied, a BFGS update
formula is used to approximate the hessian.
<p>If supplied, the gradient function must be of the form
<pre class="example"> g = gradient (x)
</pre>
<p class="noindent">in which <var>x</var> is a vector and <var>g</var> is a vector.
<p>If supplied, the hessian function must be of the form
<pre class="example"> h = hessian (x)
</pre>
<p class="noindent">in which <var>x</var> is a vector and <var>h</var> is a matrix.
<p>The third and fourth arguments are function handles pointing to
functions that compute the equality constraints and the inequality
constraints, respectively.
<p>If your problem does not have equality (or inequality) constraints,
you may pass an empty matrix for <var>cef</var> (or <var>cif</var>).
<p>If supplied, the equality and inequality constraint functions must be
of the form
<pre class="example"> r = f (x)
</pre>
<p class="noindent">in which <var>x</var> is a vector and <var>r</var> is a vector.
<p>The third and fourth arguments may also be 2-element cell arrays of
function handles. The first element should point to the constraint
function and the second should point to a function that computes the
gradient of the constraint function:
<pre class="example"> [ d f(x) d f(x) d f(x) ]
transpose ( [ ------ ----- ... ------ ] )
[ dx_1 dx_2 dx_N ]
</pre>
<p>The fifth and sixth arguments are vectors containing lower and upper bounds
on <var>x</var>. These must be consistent with equality and inequality
constraints <var>g</var> and <var>h</var>. If the bounds are not specified, or are
empty, they are set to -<var>realmax</var> and <var>realmax</var> by default.
<p>The seventh argument is max. number of iterations. If not specified,
the default value is 100.
<p>The eighth argument is tolerance for stopping criteria. If not specified,
the default value is <var>eps</var>.
<p>Here is an example of calling <code>sqp</code>:
<pre class="example"> function r = g (x)
r = [ sumsq(x)-10;
x(2)*x(3)-5*x(4)*x(5);
x(1)^3+x(2)^3+1 ];
endfunction
function obj = phi (x)
obj = exp(prod(x)) - 0.5*(x(1)^3+x(2)^3+1)^2;
endfunction
x0 = [-1.8; 1.7; 1.9; -0.8; -0.8];
[x, obj, info, iter, nf, lambda] = sqp (x0, @phi, @g, [])
x =
-1.71714
1.59571
1.82725
-0.76364
-0.76364
obj = 0.053950
info = 101
iter = 8
nf = 10
lambda =
-0.0401627
0.0379578
-0.0052227
</pre>
<p>The value returned in <var>info</var> may be one of the following:
<dl>
<dt>101<dd>The algorithm terminated because the norm of the last step was less
than <code>tol * norm (x))</code> (the value of tol is currently fixed at
<code>sqrt (eps)</code>—edit <samp><span class="file">sqp.m</span></samp> to modify this value.
<br><dt>102<dd>The BFGS update failed.
<br><dt>103<dd>The maximum number of iterations was reached (the maximum number of
allowed iterations is currently fixed at 100—edit <samp><span class="file">sqp.m</span></samp> to
increase this value).
</dl>
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<p class="noindent"><strong>See also:</strong> <a href="doc_002dqp.html#doc_002dqp">qp</a>.
</p></blockquote></div>
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