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<a name="Nonlinear-Programming"></a>
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Next: <a href="Linear-Least-Squares.html#Linear-Least-Squares" accesskey="n" rel="next">Linear Least Squares</a>, Previous: <a href="Quadratic-Programming.html#Quadratic-Programming" accesskey="p" rel="prev">Quadratic Programming</a>, Up: <a href="Optimization.html#Optimization" accesskey="u" rel="up">Optimization</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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<hr>
<a name="Nonlinear-Programming-1"></a>
<h3 class="section">25.3 Nonlinear Programming</h3>
<p>Octave can also perform general nonlinear minimization using a
successive quadratic programming solver.
</p>
<a name="XREFsqp"></a><dl>
<dt><a name="index-sqp"></a><em>[<var>x</var>, <var>obj</var>, <var>info</var>, <var>iter</var>, <var>nf</var>, <var>lambda</var>] =</em> <strong>sqp</strong> <em>(<var>x0</var>, <var>phi</var>)</em></dt>
<dt><a name="index-sqp-1"></a><em>[…] =</em> <strong>sqp</strong> <em>(<var>x0</var>, <var>phi</var>, <var>g</var>)</em></dt>
<dt><a name="index-sqp-2"></a><em>[…] =</em> <strong>sqp</strong> <em>(<var>x0</var>, <var>phi</var>, <var>g</var>, <var>h</var>)</em></dt>
<dt><a name="index-sqp-3"></a><em>[…] =</em> <strong>sqp</strong> <em>(<var>x0</var>, <var>phi</var>, <var>g</var>, <var>h</var>, <var>lb</var>, <var>ub</var>)</em></dt>
<dt><a name="index-sqp-4"></a><em>[…] =</em> <strong>sqp</strong> <em>(<var>x0</var>, <var>phi</var>, <var>g</var>, <var>h</var>, <var>lb</var>, <var>ub</var>, <var>maxiter</var>)</em></dt>
<dt><a name="index-sqp-5"></a><em>[…] =</em> <strong>sqp</strong> <em>(<var>x0</var>, <var>phi</var>, <var>g</var>, <var>h</var>, <var>lb</var>, <var>ub</var>, <var>maxiter</var>, <var>tol</var>)</em></dt>
<dd><p>Minimize an objective function using sequential quadratic programming (SQP).
</p>
<p>Solve the nonlinear program
</p>
<div class="example">
<pre class="example">min phi (x)
x
</pre></div>
<p>subject to
</p>
<div class="example">
<pre class="example">g(x) = 0
h(x) >= 0
lb <= x <= ub
</pre></div>
<p>using a sequential quadratic programming method.
</p>
<p>The first argument is the initial guess for the vector <var>x0</var>.
</p>
<p>The second argument is a function handle pointing to the objective function
<var>phi</var>. The objective function must accept one vector argument and
return a scalar.
</p>
<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 that computes
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>
<p>When supplied, the gradient function <code><var>phi</var>{2}</code> must accept one
vector argument and return a vector. When supplied, the Hessian function
<code><var>phi</var>{3}</code> must accept one vector argument and return a matrix.
</p>
<p>The third and fourth arguments <var>g</var> and <var>h</var> are function handles
pointing to functions that compute the equality constraints and the
inequality constraints, respectively. If the problem does not have
equality (or inequality) constraints, then use an empty matrix ([]) for
<var>g</var> (or <var>h</var>). When supplied, these equality and inequality
constraint functions must accept one vector argument and return a vector.
</p>
<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:
</p>
<div class="example">
<pre class="example"> [ d f(x) d f(x) d f(x) ]
transpose ( [ ------ ----- ... ------ ] )
[ dx_1 dx_2 dx_N ]
</pre></div>
<p>The fifth and sixth arguments, <var>lb</var> and <var>ub</var>, contain lower and
upper bounds on <var>x</var>. These must be consistent with the equality and
inequality constraints <var>g</var> and <var>h</var>. If the arguments are vectors
then <var>x</var>(i) is bound by <var>lb</var>(i) and <var>ub</var>(i). A bound can also
be a scalar in which case all elements of <var>x</var> will share the same
bound. If only one bound (lb, ub) is specified then the other will
default to (-<var>realmax</var>, +<var>realmax</var>).
</p>
<p>The seventh argument <var>maxiter</var> specifies the maximum number of
iterations. The default value is 100.
</p>
<p>The eighth argument <var>tol</var> specifies the tolerance for the stopping
criteria. The default value is <code>sqrt (eps)</code>.
</p>
<p>The value returned in <var>info</var> may be one of the following:
</p>
<dl compact="compact">
<dt>101</dt>
<dd><p>The algorithm terminated normally.
All constraints meet the specified tolerance.
</p>
</dd>
<dt>102</dt>
<dd><p>The BFGS update failed.
</p>
</dd>
<dt>103</dt>
<dd><p>The maximum number of iterations was reached.
</p>
</dd>
<dt>104</dt>
<dd><p>The stepsize has become too small, i.e.,
delta <var>x</var>,
is less than <code><var>tol</var> * norm (x)</code>.
</p></dd>
</dl>
<p>An example of calling <code>sqp</code>:
</p>
<div class="example">
<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></div>
<p><strong>See also:</strong> <a href="Quadratic-Programming.html#XREFqp">qp</a>.
</p></dd></dl>
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