<html lang="en"> <head> <title>Nonlinear Programming - GNU Octave</title> <meta http-equiv="Content-Type" content="text/html"> <meta name="description" content="GNU Octave"> <meta name="generator" content="makeinfo 4.13"> <link title="Top" rel="start" href="index.html#Top"> <link rel="up" href="Optimization.html#Optimization" title="Optimization"> <link rel="prev" href="Quadratic-Programming.html#Quadratic-Programming" title="Quadratic Programming"> <link rel="next" href="Linear-Least-Squares.html#Linear-Least-Squares" title="Linear Least Squares"> <link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage"> <meta http-equiv="Content-Style-Type" content="text/css"> <style type="text/css"><!-- pre.display { font-family:inherit } pre.format { font-family:inherit } pre.smalldisplay { font-family:inherit; font-size:smaller } pre.smallformat { font-family:inherit; font-size:smaller } pre.smallexample { font-size:smaller } pre.smalllisp { font-size:smaller } span.sc { font-variant:small-caps } span.roman { font-family:serif; font-weight:normal; } span.sansserif { font-family:sans-serif; font-weight:normal; } --></style> </head> <body> <div class="node"> <a name="Nonlinear-Programming"></a> <p> Next: <a rel="next" accesskey="n" href="Linear-Least-Squares.html#Linear-Least-Squares">Linear Least Squares</a>, Previous: <a rel="previous" accesskey="p" href="Quadratic-Programming.html#Quadratic-Programming">Quadratic Programming</a>, Up: <a rel="up" accesskey="u" href="Optimization.html#Optimization">Optimization</a> <hr> </div> <h3 class="section">25.3 Nonlinear Programming</h3> <p>Octave can also perform general nonlinear minimization using a successive quadratic programming solver. <!-- sqp scripts/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>x0, phi</var>)<var><a name="index-sqp-2386"></a></var><br> — Function File: [<small class="dots">...</small>] = <b>sqp</b> (<var>x0, phi, g</var>)<var><a name="index-sqp-2387"></a></var><br> — Function File: [<small class="dots">...</small>] = <b>sqp</b> (<var>x0, phi, g, h</var>)<var><a name="index-sqp-2388"></a></var><br> — Function File: [<small class="dots">...</small>] = <b>sqp</b> (<var>x0, phi, g, h, lb, ub</var>)<var><a name="index-sqp-2389"></a></var><br> — Function File: [<small class="dots">...</small>] = <b>sqp</b> (<var>x0, phi, g, h, lb, ub, maxiter</var>)<var><a name="index-sqp-2390"></a></var><br> — Function File: [<small class="dots">...</small>] = <b>sqp</b> (<var>x0, phi, g, h, lb, ub, maxiter, tol</var>)<var><a name="index-sqp-2391"></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 sequential quadratic programming method. <p>The first argument is the initial guess for the vector <var>x0</var>. <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>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>When supplied, the gradient function <var>phi</var><code>{2}</code> must accept one vector argument and return a vector. When supplied, the Hessian function <var>phi</var><code>{3}</code> must accept one vector argument and return a matrix. <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>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, <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>The seventh argument <var>maxiter</var> specifies the maximum number of iterations. The default value is 100. <p>The eighth argument <var>tol</var> specifies the tolerance for the stopping criteria. The default value is <code>sqrt(eps)</code>. <p>The value returned in <var>info</var> may be one of the following: <dl> <dt>101<dd>The algorithm terminated normally. Either all constraints meet the requested tolerance, or the stepsize, delta <var>x</var>, is less than <var>tol</var><code> * norm (x)</code>. <br><dt>102<dd>The BFGS update failed. <br><dt>103<dd>The maximum number of iterations was reached. </dl> <p>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> <!-- Texinfo @sp should work but in practice produces ugly results for HTML. --> <!-- A simple blank line produces the correct behavior. --> <!-- @sp 1 --> <p class="noindent"><strong>See also:</strong> <a href="doc_002dqp.html#doc_002dqp">qp</a>. </p></blockquote></div> </body></html>