NAG CPP Interface nagcpp::opt::handle_solve_ipopt (e04st)
Note:this function usesoptional parametersto define choices in the problem specification and in the details of the algorithm. If you wish to use default settings for all of the optional parameters, you need only read Sections 1 to 10 of this document. If, however, you wish to reset some or all of the settings please refer to Section 11 for a detailed description of the algorithm and to Section 12 for a detailed description of the specification of the optional parameters.
handle_solve_ipopt, an interior point method optimization solver, based on the IPOPT software package, is a solver for the NAG optimization modelling suite and is suitable for large scale nonlinear programming (NLP) problems.
${m}_{g}$ is the number of the nonlinear constraints and $g\left(x\right)$, ${l}_{g}$ and ${u}_{g}$ are ${m}_{g}$-dimensional vectors,
${m}_{B}$ is the number of the linear constraints and $B$ is a ${m}_{B}$ by $n$ matrix, ${l}_{B}$ and ${u}_{B}$ are ${m}_{B}$-dimensional vectors,
there are $n$ box constraints and ${l}_{x}$ and ${u}_{x}$ are $n$-dimensional vectors.
The objective $f\left(x\right)$ can be specified in a number of ways: handle_set_linobj for a dense linear function, handle_set_quadobj for a sparse linear or quadratic function and handle_set_nlnobj for a general nonlinear function. In the last case, objfun and objgrd will be used to compute values and gradients of the objective function. Variable box bounds ${l}_{x},{u}_{x}$ can be specified with handle_set_simplebounds. The special case of linear constraints ${l}_{B},B,{u}_{B}$ is handled by handle_set_linconstr while general nonlinear constraints ${l}_{g},g\left(x\right),{u}_{g}$ are specified by handle_set_nlnconstr (both can be specified). Again, in the last case, confun and congrd will be used to compute values and gradients of the nonlinear constraint functions.
Finally, if the you are willing to calculate second derivatives, the sparsity structure of the second partial derivatives of a nonlinear objective and/or of any nonlinear constraints is specified by handle_set_nlnhess while the values of these derivatives themselves will be computed by user-supplied hess. While there is an option (see Hessian Mode) that forces internal approximation of second derivatives, no such option exists for first derivatives which must be computed accurately. If handle_set_nlnhess has been called and hess is used to calculate values for second derivatives, both the objective and all the constraints must be included; it is not possible to provide a subset of these. If handle_set_nlnhess is not called, then internal approximation of second derivatives will take place.
See Section 3.1 in the E04 Chapter Introduction for more details about the NAG optimization modelling suite.
3.1Structure of the Lagrange Multipliers
For a problem consisting of $n$ variable bounds, ${m}_{B}$ linear constraints and
${m}_{g}$ nonlinear constraints (as specified in nvar, nclin and ncnln of handle_set_simplebounds, handle_set_linconstr and handle_set_nlnconstr, respectively), the number of Lagrange multipliers, and consequently the correct value for nnzu, will be $q=2*n+2*{m}_{B}+2*{m}_{g}$. The order these will be found in the u array is
where the $L$ and $U$ subscripts refer to lower and upper bounds, respectively, and the variable bound constraint multipliers come first (if present, i.e., if handle_set_simplebounds was called), followed by the linear constraint multipliers (if present, i.e., if handle_set_linconstr was called) and the nonlinear constraint multipliers (if present, i.e., if handle_set_nlnconstr was called).
Significantly nonzero values for any of these, after the solver has terminated, indicates that the corresponding constraint is active. Significance is judged in the first instance by the relative scale of any value compared to the smallest among them.
4References
Byrd R H, Gilbert J Ch and Nocedal J (2000) A trust region method based on interior point techniques for nonlinear programming Mathematical Programming89 149–185
Byrd R H, Liu G and Nocedal J (1997) On the local behavior of an interior point method for nonlinear programming Numerical Analysis (eds D F Griffiths and D J Higham) Addison–Wesley
Conn A R, Gould N I M, Orban D and Toint Ph L (2000) A primal-dual trust-region algorithm for non-convex nonlinear programming Mathematical Programming87 (2) 215–249
Conn A R, Gould N I M and Toint Ph L (2000) Trust Region Methods SIAM, Philadephia
Fiacco A V and McCormick G P (1990) Nonlinear Programming: Sequential Unconstrained Minimization Techniques SIAM, Philadelphia
Gould N I M, Orban D, Sartenaer A and Toint Ph L (2001) Superlinear convergence of primal-dual interior point algorithms for nonlinear programming SIAM Journal on Optimization11 (4) 974–1002
Hock W and Schittkowski K (1981) Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems187 Springer–Verlag
Hogg J D and Scott J A (2011) HSL MA97: a bit-compatible multifrontal code for sparse symmetric systems RAL Technical Report. RAL-TR-2011-024
Wächter A and Biegler L T (2006) On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming Mathematical Programming106(1) 25–57
Williams P and Lang B (2013) A framework for the $M{R}^{3}$ Algorithm: theory and implementation SIAM J. Sci. Comput.35 740–766
Yamashita H (1998) A globally convergent primal-dual interior-point method for constrained optimization Optimization Methods and Software10 443–469
Communication structure.
An object of either the derived class CommE04RA or its base class NoneCopyableComm can be supplied.
It is recommended that the derived class is used. If the base class is supplied it must
first be initialized via a call to opt::handle_init (e04ra).
2: $\mathbf{objfun}$ – void functionFunction
objfun must calculate the value of the nonlinear objective function $f\left(x\right)$ at a specified value of the $n$-element vector of $x$ variables. If there is no nonlinear objective (e.g., handle_set_linobj and handle_set_quadobj was called to define a linear or quadratic objective function), objfun will never be called by handle_solve_ipopt and objfun may be the dummy function ipopt_dummy_objfun. (ipopt_dummy_objfun is included in the NAG Library.)
void function objfun(const utility::array1D<double,data_handling::ArgIntent::IN> &x, double &fx)
On entry: $n$, the number of variables in the problem. It must be unchanged from the value set during the initialization of the handle by handle_init.
Note:objfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by handle_solve_ipopt. If your code inadvertently does return any NaNs or infinities, handle_solve_ipopt is likely to produce unexpected results.
3: $\mathbf{objgrd}$ – void functionFunction
objgrd must calculate the values of the nonlinear objective function gradients $\frac{\partial f}{\partial x}$ at a specified value of the $n$-element vector of $x$ variables. If there is no nonlinear objective (e.g., handle_set_linobj and handle_set_quadobj was called to define a linear or quadratic objective function), objgrd will never be called by handle_solve_ipopt and objgrd may be the dummy function ipopt_dummy_objgrd. (ipopt_dummy_objgrd is included in the NAG Library.)
void function objgrd(const utility::array1D<double,data_handling::ArgIntent::IN> &x, const types::f77_integernnzfd, utility::array1D<double,data_handling::ArgIntent::OUT> &fdx)
On entry: the number of nonzero elements in the sparse gradient vector of the objective function, as was set in a previous call to handle_set_nlnobj.
On exit: the values of the nonzero elements in the sparse gradient vector of the objective function, in the order specified by idxfd in a previous call to handle_set_nlnobj. ${\mathbf{fdx}}\left(\mathit{i}-1\right)$ will be the gradient $\frac{\partial f}{\partial {x}_{{\mathbf{idxfd}}\left(\mathit{i}-1\right)}}$.
On entry: $n$, the number of variables in the problem. It must be unchanged from the value set during the initialization of the handle by handle_init.
Note:objgrd should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by handle_solve_ipopt. If your code inadvertently does return any NaNs or infinities, handle_solve_ipopt is likely to produce unexpected results.
4: $\mathbf{confun}$ – void functionFunction
confun must calculate the values of the ${m}_{g}$-element vector ${g}_{i}\left(x\right)$ of nonlinear constraint functions at a specified value of the $n$-element vector of $x$ variables. If no nonlinear constraints were registered in this handle, confun will never be called by handle_solve_ipopt and confun may be the dummy function ipopt_dummy_confun. (ipopt_dummy_confun is included in the NAG Library.)
void function confun(const utility::array1D<double,data_handling::ArgIntent::IN> &x, const types::f77_integerncnln, utility::array1D<double,data_handling::ArgIntent::OUT> &gx)
On entry: $n$, the number of variables in the problem. It must be unchanged from the value set during the initialization of the handle by handle_init.
Note:confun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by handle_solve_ipopt. If your code inadvertently does return any NaNs or infinities, handle_solve_ipopt is likely to produce unexpected results.
5: $\mathbf{congrd}$ – void functionFunction
congrd must calculate the nonzero values of the sparse Jacobian of the nonlinear constraint functions $\frac{\partial {g}_{i}}{\partial x}$ at a specified value of the $n$-element vector of $x$ variables. If there are no nonlinear constraints (e.g., handle_set_nlnconstr was never called with the same handle or it was called with ncnln$=0$), congrd will never be called by handle_solve_ipopt and congrd may be the dummy function ipopt_dummy_congrd. (ipopt_dummy_congrd is included in the NAG Library.)
void function congrd(const utility::array1D<double,data_handling::ArgIntent::IN> &x, const types::f77_integernnzgd, utility::array1D<double,data_handling::ArgIntent::OUT> &gdx)
On entry: is the number of nonzero elements in the sparse Jacobian of the constraint functions, as was set in a previous call to handle_set_nlnconstr.
On exit: the nonzero values of the Jacobian of the nonlinear constraints, in the order specified by irowgd and icolgd in an earlier call to handle_set_nlnconstr. ${\mathbf{gdx}}\left(\mathit{i}-1\right)$ will be the gradient $\frac{\partial {g}_{{\mathbf{irowgd}}\left(\mathit{i}-1\right)}}{\partial {x}_{{\mathbf{icolgd}}\left(\mathit{i}-1\right)}}$.
On entry: $n$, the number of variables in the problem. It must be unchanged from the value set during the initialization of the handle by handle_init.
Note:congrd should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by handle_solve_ipopt. If your code inadvertently does return any NaNs or infinities, handle_solve_ipopt is likely to produce unexpected results.
6: $\mathbf{hess}$ – void functionFunction
hess must calculate the nonzero values of one of a set of second derivative quantities:
the Hessian of the Lagrangian function $\sigma {\nabla}^{2}f+{\displaystyle \sum _{i=1}^{{m}_{g}}}{\lambda}_{i}{\nabla}^{2}{g}_{i}$
the Hessian of the objective function ${\nabla}^{2}f$
the Hessian of the constraint functions ${\nabla}^{2}{g}_{i}$
The value of argument idf determines which one of these is to be computed and this, in turn, is determined by earlier calls to handle_set_nlnhess, when the nonzero sparsity structure of these Hessians was registered. Please note that it is not possible to only supply a subset of the Hessians (see $\mathbf{errorid}={\mathbf{6}}$). If there were no calls to handle_set_nlnhess, hess will never be called by handle_solve_ipopt and hess may be the dummy function ipopt_dummy_hess (ipopt_dummy_hess is included in the NAG Library). In this case, the Hessian of the Lagrangian will be approximated by a limited-memory quasi-Newton method (L-BFGS).
On entry: specifies the quantities to be computed in hx.
${\mathbf{idf}}=-1$
The values of the Hessian of the Lagrangian will be computed in hx. This will be the case if handle_set_nlnhess has been called with idf of the same value.
${\mathbf{idf}}=0$
The values of the Hessian of the objective function will be computed in hx. This will be the case if handle_set_nlnhess has been called with idf of the same value.
${\mathbf{idf}}>0$
The values of the Hessian of the idfth constraint function will be computed in hx. This will be the case if handle_set_nlnhess has been called with idf of the same value.
3: $\mathbf{sigma}$ – doubleInput
On entry: if ${\mathbf{idf}}=-1$, the value of the $\sigma $ quantity in the definition of the Hessian of the Lagrangian. Otherwise, sigma should not be referenced.
On entry: if ${\mathbf{idf}}=-1$, the values of the ${\lambda}_{i}$ quantities in the definition of the Hessian of the Lagrangian. Otherwise, lambda should not be referenced.
On exit: the nonzero values of the requested Hessian evaluated at $x$. For each value of idf, the ordering of nonzeros must follow the sparsity structure registered in the handle by earlier calls to handle_set_nlnhess through the arguments irowh and icolh.
On entry: $n$, the number of variables in the problem. It must be unchanged from the value set during the initialization of the handle by handle_init.
On entry: ${m}_{g}$, the number of nonlinear constraints, as specified in an earlier call to handle_set_nlnconstr.
Note:hess should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by handle_solve_ipopt. If your code inadvertently does return any NaNs or infinities, handle_solve_ipopt is likely to produce unexpected results.
7: $\mathbf{monit}$ – void functionFunction
monit is provided to enable you to monitor the progress of the optimization.
monit may be the dummy function ipopt_dummy_monit (ipopt_dummy_monit is included in the NAG Library).
On entry: if ${\mathbf{nnzu}}>0$, u holds the values at the current iteration of Lagrange multipliers (dual variables) for the constraints. See Section 3.1 for layout information.
Note:monit should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by handle_solve_ipopt. If your code inadvertently does return any NaNs or infinities, handle_solve_ipopt is likely to produce unexpected results.
Note: if ${\mathbf{nnzu}}>0$, u holds Lagrange multipliers (dual variables) for the constraints. See Section 3.1 for layout information. If ${\mathbf{nnzu}}=0$, u will not be referenced.
On exit: the final value of Lagrange multipliers $z,\lambda $.
On exit: solver statistics at the end of the final iteration as given in the table below:
$0$
Number of the iterations.
$2$
Number of backtracking trial steps.
$3$
Number of Hessian evaluations.
$4$
Number of objective gradient evaluations.
$7$
Total wall clock time elapsed.
$18$
Number of objective function evaluations.
$19$
Number of constraint function evaluations.
$20$
Number of constraint Jacobian evaluations.
13: $\mathbf{opt}$ – OptionalE04STInput/Output
Optional parameter container, derived from Optional.
5.1Additional Quantities
1: $\mathbf{nvar}$
$n$, the number of variables in the problem.
6Exceptions and Warnings
Errors or warnings detected by the function:
Note: in some cases handle_solve_ipopt may return useful information.
All errors and warnings have an associated numeric error code field, errorid, stored either as a member of the thrown exception object (see errorid), or as a member of
opt.ifail, depending on how errors
and warnings are being handled (see Error Handling for more details).
comm::handle does not belong to the NAG optimization modelling suite, has not been initialized properly or is corrupted.
$\mathbf{errorid}=1$
comm::handle has not been initialized properly or is corrupted.
$\mathbf{errorid}=2$
The problem is already being solved.
$\mathbf{errorid}=2$
This solver does not support the model defined in the handle.
$\mathbf{errorid}=3$
A different solver from the suite has already been used. Initialize a new comm::handle using function.
$\mathbf{errorid}=4$
The information supplied does not match with that previously stored. On entry, ${\mathbf{nvar}}=\u27e8\mathit{value}\u27e9$ must match that given during initialization of the comm::handle, i.e., $\u27e8\mathit{\text{value}}\u27e9$.
$\mathbf{errorid}=5$
On entry, ${\mathbf{nnzu}}=\u27e8\mathit{value}\u27e9$. Constraint: ${\mathbf{nnzu}}=\u27e8\mathit{value}\u27e9$ or $0$.
$\mathbf{errorid}=5$
On entry, ${\mathbf{nnzu}}=\u27e8\mathit{value}\u27e9$. Constraint: no constraints present, so ${\mathbf{nnzu}}$ must be $0$.
$\mathbf{errorid}=6$
On entry, a nonlinear objective function has been defined but no objective Hessian sparsity structure has been defined through function.
$\mathbf{errorid}=6$
On entry, a nonlinear constraint function has been defined but no constraint Hessian sparsity structure has been defined through function, for constraint number $\u27e8\mathit{\text{value}}\u27e9$.
$\mathbf{errorid}=7$
Either change the option ‘Hessian Mode’ or provide a proper hess function.
The solver terminated with not enough degrees of freedom.
$\mathbf{errorid}=23$
The solver terminated due to an invalid problem definition.
$\mathbf{errorid}=23$
The solver terminated due to an invalid option.
$\mathbf{errorid}=51$
The solver detected an infeasible problem.
$\mathbf{errorid}=54$
The solver terminated due to diverging iterates.
$\mathbf{errorid}=10601$
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ must be a vector of size $\u27e8\mathit{\text{value}}\u27e9$ array. Supplied argument has $\u27e8\mathit{\text{value}}\u27e9$ dimensions.
$\mathbf{errorid}=10601$
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ must be a vector of size $\u27e8\mathit{\text{value}}\u27e9$ array. Supplied argument was a vector of size $\u27e8\mathit{\text{value}}\u27e9$.
$\mathbf{errorid}=10601$
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ must be a vector of size $\u27e8\mathit{\text{value}}\u27e9$ array. The size for the supplied array could not be ascertained.
$\mathbf{errorid}=10602$
On entry, the raw data component of $\u27e8\mathit{\text{value}}\u27e9$ is null.
$\mathbf{errorid}=10603$
On entry, unable to ascertain a value for $\u27e8\mathit{\text{value}}\u27e9$.
$\mathbf{errorid}=10605$
On entry, the communication class $\u27e8\mathit{\text{value}}\u27e9$ has not been initialized correctly.
$\mathbf{errorid}=10703$
An exception was thrown during IO (writing).
$\mathbf{errorid}=-99$
An unexpected error has been triggered by this routine.
$\mathbf{errorid}=-399$
Your licence key may have expired or may not have been installed correctly.
The memory address for an array in a callback has changed.
7Accuracy
The accuracy of the solution is driven by optional parameter Stop Tolerance 1.
If $\mathbf{errorid}={\mathbf{0}}$ on the final exit, the returned point satisfies Karush–Kuhn–Tucker (KKT) conditions to the requested accuracy (under the default settings close to $\sqrt{\epsilon}$ where $\epsilon $ is the machine precision) and thus it is a good estimate of a local solution. If $\mathbf{errorid}={\mathbf{50}}$, some of the convergence conditions were not fully satisfied but the point still seems to be a reasonable estimate and should be usable. Please refer to Section 11.1 and the description of the particular options.
8Parallelism and Performance
Please see the description for the underlying computational routine in this section of the
FL Interface documentation.
9Further Comments
9.1Description of the Printed Output
The solver can print information to give an overview of the problem and of the progress of the computation. The output may be sent to two independent streams (files) which are set by optional parameters Print File and Monitoring File. Optional parameters Print Level and Monitoring Level determine the exposed level of detail. This allows, for example, the generation of a detailed log in a file while the condensed information is displayed on the screen. This section also describes what kind of information is made available to the monitoring function monit via rinfo and stats.
There are four sections printed to the primary output with the default settings (level $2$): a derivative check, a header, an iteration log and a summary. At higher levels more information will be printed, including any internal IPOPT options that have been changed from their default values.
Derivative Check
If Verify Derivatives is set, then information will appear about any errors detected in the user-supplied derivative functions objgrd, congrd or hess. It may look like this:
Starting derivative checker for first derivatives.
* grad_f[ 1] = -2.000000E+00 ~ 2.455000E+01 [ 1.081E+00]
* jac_g [ 1, 4] = 4.700969E+01 v ~ 5.200968E+01 [ 9.614E-02]
Starting derivative checker for second derivatives.
* obj_hess[ 1, 1] = 1.881000E+03 v ~ 1.882000E+03 [ 5.314E-04]
* 1-th constr_hess[ 1, 3] = 2.988964E+00 v ~ -1.103543E-02 [ 3.000E+00]
Derivative checker detected 3 error(s).
The first line indicates that the value for the partial derivative of the objective with respect to the first variable as returned by objgrd (the first one printed) differs sufficiently from a finite difference estimation derived from objfun (the second one printed). The number in square brackets is the relative difference between these two numbers.
The second line reports on a discrepancy for the partial derivative of the first constraint with respect to the fourth variable. If the indicator v is absent, the discrepancy refers to a component that had not been included in the sparsity structure, in which case the nonzero structure of the derivatives should be corrected. Mistakes in the first derivatives should be corrected before attempting to correct mistakes in the second derivatives.
The third line reports on a discrepancy in a second derivative of the objective function, differentiated with respect to the first variable, twice.
The fourth line reports on a discrepancy in a second derivative of the first constraint, differentiated with respect to the first and third variables.
Header
If ${\mathbf{Print\; Level}}\ge 1$, the header will contain statistics about the size of the problem how the solver sees it, i.e., it reflects any changes imposed by preprocessing and problem transformations. The header may look like:
Number of nonzeros in equality constraint Jacobian...: 4
Number of nonzeros in inequality constraint Jacobian.: 8
Number of nonzeros in Lagrangian Hessian.............: 10
Total number of variables............................: 4
variables with only lower bounds: 4
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 1
Total number of inequality constraints...............: 2
inequality constraints with only lower bounds: 2
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
If ${\mathbf{Print\; Level}}=2$, the status of each iteration is condensed to one line. The line shows:
iter
The current iteration count. This includes regular iterations and iterations during the restoration phase. If the algorithm is in the restoration phase, the letter r will be appended to the iteration number. The iteration number $0$ represents the starting point. This quantity is also available as ${\mathbf{stats}}\left(0\right)$ of monit.
objective
The unscaled objective value at the current point (given the current NLP scaling). During the restoration phase, this value remains the unscaled objective value for the original problem. This quantity is also available as ${\mathbf{rinfo}}\left(0\right)$ of monit.
inf_pr
The unscaled constraint violation at the current point (given the current NLP scaling). This quantity is the infinity-norm (max) of the (unscaled) constraints ${g}_{i}$. During the restoration phase, this value remains the constraint violation of the original problem at the current point. This quantity is also available as ${\mathbf{rinfo}}\left(1\right)$ of monit.
inf_du
The scaled dual infeasibility at the current point (given the current NLP scaling). This quantity measure the infinity-norm (max) of the internal dual infeasibility, ${\lambda}_{i}$ of Eq. (4a) in the implementation paper Wächter and Biegler (2006), including inequality constraints reformulated using slack variables and NLP scaling. During the restoration phase, this is the value of the dual infeasibility for the restoration phase problem. This quantity is also available as ${\mathbf{rinfo}}\left(2\right)$ of monit.
lg(mu)
$log10$ of the value of the barrier parameter $\mu $. $\mu $ itself is also available as ${\mathbf{rinfo}}\left(3\right)$ of monit.
||d||
The infinity norm (max) of the primal step (for the original variables x and the internal slack variables s). During the restoration phase, this value includes the values of additional variables, $\stackrel{-}{p}$ and $\stackrel{-}{n}$ (see Eq. (30) in Wächter and Biegler (2006)). This quantity is also available as ${\mathbf{rinfo}}\left(4\right)$ of monit.
lg(rg)
$log10$ of the value of the regularization term for the Hessian of the Lagrangian in the augmented system (${\delta}_{w}$ of Eq. (26) and Section 3.1 in Wächter and Biegler (2006)). A dash (–) indicates that no regularization was done. The regularization term itself is also available as ${\mathbf{rinfo}}\left(5\right)$ of monit.
alpha_du
The step size for the dual variables (${\alpha}_{k}^{z}$ of Eq. (14c) in Wächter and Biegler (2006)). This quantity is also available as ${\mathbf{rinfo}}\left(6\right)$ of monit.
alpha_pr
The step size for the primal variables (${\alpha}_{k}$ of Eq. (14a) in Wächter and Biegler (2006)). This quantity is also available as ${\mathbf{rinfo}}\left(7\right)$ of monit. The number is usually followed by a character for additional diagnostic information regarding the step acceptance criterion.
f
f-type iteration in the filter method without second-order correction
F
f-type iteration in the filter method with second-order correction
h
h-type iteration in the filter method without second-order correction
H
h-type iteration in the filter method with second-order correction
k
penalty value unchanged in merit function method without second-order correction
K
penalty value unchanged in merit function method with second-order correction
n
penalty value updated in merit function method without second-order correction
N
penalty value updated in merit function method with second-order correction
R
Restoration phase just started
w
in watchdog procedure
s
step accepted in soft restoration phase
t/T
tiny step accepted without line search
r
some previous iterate restored
ls
The number of backtracking line search steps (does not include second-order correction steps). This quantity is also available as ${\mathbf{stats}}\left(1\right)$ of monit.
Note that the step acceptance mechanisms in IPOPT consider the barrier objective function (5) which is usually different from the value reported in the objective column. Similarly, for the purposes of the step acceptance, the constraint violation is measured for the internal problem formulation, which includes slack variables for inequality constraints and potentially NLP scaling of the constraint functions. This value, too, is usually different from the value reported in inf_pr. As a consequence, a new iterate might have worse values both for the objective function and the constraint violation as reported in the iteration output, seemingly contradicting globalization procedure.
Note that all these values are also available in ${\mathbf{rinfo}}\left(0\right),\dots ,{\mathbf{rinfo}}\left(7\right)$ and ${\mathbf{stats}}\left(0\right),\dots ,{\mathbf{stats}}\left(1\right)$of the monitoring function monit.
If ${\mathbf{Print\; Level}}>2$, each iteration produces significantly more detailed output comprising detailed error measures and output from internal operations. The output is reasonably self-explanatory so it is not featured here in detail.
Summary
Once the solver finishes, a detailed summary is produced if ${\mathbf{Print\; Level}}\ge 1$. An example is shown below:
Number of Iterations....: 6
(scaled) (unscaled)
Objective...............: 7.8692659500479623E-01 6.2324586324379867E+00
Dual infeasibility......: 7.9744615766675617E-10 6.3157735687207093E-09
Constraint violation....: 8.3555384833289281E-12 8.3555384833289281E-12
Complementarity.........: 0.0000000000000000E+00 0.0000000000000000E+00
Overall NLP error.......: 7.9744615766675617E-10 6.3157735687207093E-09
Number of objective function evaluations = 7
Number of objective gradient evaluations = 7
Number of equality constraint evaluations = 7
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 7
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 6
Total CPU secs in IPOPT (w/o function evaluations) = 0.724
Total CPU secs in NLP function evaluations = 0.343
EXIT: Optimal Solution Found.
It starts with the total number of iterations the algorithm went through. Then, five quantities are printed, all evaluated at the termination point: the value of the objective function, the dual infeasibility, the constraint violation, the complementarity and the NLP error.
This is followed by some statistics on the number of calls to user-supplied functions and CPU time taken in user-supplied functions and the main algorithm. Lastly, status at exit is indicated by a short message. Detailed timings of the algorithm are displayed only if Stats Time is set.
9.2Additional Licensor
Parts of the code for handle_solve_ipopt are distributed according to terms imposed by another licensor. Please refer to Library Licensors for further details.
10Example
This example is based on Problem 73 in Hock and Schittkowski (1981) and involves the minimization of the linear function
handle_solve_ipopt is an implementation of IPOPT (see Wächter and Biegler (2006)) that is fully supported and maintained by NAG. It uses Harwell packages MA97 for the underlying sparse linear algebra factorization and MC68 approximate minimum degree or METIS algorithm for the ordering. Any issues relating to handle_solve_ipopt should be directed to NAG who assume all responsibility for the handle_solve_ipopt function and its implementation.
Range constraints of the form $l\le c\left(x\right)\le u$ can be expressed in this formulation by introducing slack variables ${x}_{s}\ge 0$, ${x}_{t}\ge 0$ (increasing $n$ by $2$) and defining new equality constraints $g(x,{x}_{s})\equiv c\left(x\right)-l-{x}_{s}=0$ and $g(x,{x}_{t})\equiv u-c\left(x\right)-{x}_{t}=0$.
with the homotopy parameter $\mu $, which is driven to zero (see e.g., Byrd et al. (1997) and Gould et al. (2001)). Here, $X\u2254\mathrm{diag}\left(x\right)$ for a vector $x$ (similarly $z\u2254\mathrm{diag}\left(z\right)$, etc.), and $e$ stands for the vector of all ones for appropriate dimension, while $\lambda \in {\mathbb{R}}^{m}$ and $z\in {\mathbb{R}}^{n}$ correspond to the Lagrange multipliers for the equality constraints (3) and the bound constraints (4), respectively.
Note, that the equations (7), (8) and (9) for $\mu =0$ together with ‘$x$, $z\ge 0$’ are the Karush–Kuhn–Tucker (KKT) conditions for the original problem (2), (3) and (4). Those are the first-order optimality conditions for (2), (3) and (4) if constraint qualifications are satisfied (Conn et al. (2000)).
Starting from an initial point supplied in x, handle_solve_ipopt computes an approximate solution to the barrier problem (5) and (6) for a fixed value of $\mu $ (by default, $0.1$), then decreases the barrier parameter, and continues the solution of the next barrier problem from the approximate solution of the previous one.
A sophisticated overall termination criterion for the algorithm is used to overcome potential difficulties when the Lagrange multipliers become large. This can happen, for example, when the gradients of the active constraints are nearly linear dependent. The termination criterion is described in detail by Wächter and Biegler (2006) (also see below Section 11.1).
11.1Stopping Criteria
Using the individual parts of the primal-dual equations (7), (8) and (9), we define the optimality error for the barrier problem as
with scaling parameters ${s}_{d}$, ${s}_{c}\ge 1$ defined below (not to be confused with NLP scaling factors described in Section 11.2). By ${E}_{0}(x,\lambda ,z)$ we denote (10) with $\mu =0$; this measures the optimality error for the original problem (2), (3) and (4). The overall algorithm terminates if an approximate solution $({\stackrel{~}{x}}_{*},{\stackrel{~}{\lambda}}_{*},{\stackrel{~}{z}}_{*})$ (including multiplier estimates) satisfying
is found, where ${\epsilon}_{\mathit{tol}}>0$ is the user-supplied error tolerance in optional parameter Stop Tolerance 1.
Even if the original problem is well scaled, the multipliers $\lambda $ and $z$ might become very large, for example, when the gradients of the active constraints are (nearly) linearly dependent at a solution of (2), (3) and (4). In this case, the algorithm might encounter numerical difficulties satisfying the unscaled primal-dual equations (7), (8) and (9) to a tight tolerance. In order to adapt the termination criteria to handle such circumstances, we choose the scaling factors
in (10). In this way, a component of the optimality error is scaled, whenever the average value of the multipliers becomes larger than a fixed number ${s}_{\mathrm{max}}\ge 1$ (${s}_{\mathrm{max}}=100$ in our implementation). Also note, in the case that the multipliers diverge, ${E}_{0}(x,\lambda ,z)$ can only become small, if a Fritz John point for (2), (3) and (4) is approached, or if the primal variables diverge as well.
11.2Scaling the NLP
Ideally, the formulated problem should be scaled so that, near the solution, all function gradients (objective and constraints), when nonzero, are of a similar order of a magnitude. handle_solve_ipopt will compute automatic NLP scaling factors for the objective and constraint functions (but not the decision variables) and apply them if large imbalances of scale are detected. This rescaling is only computed at the starting point. References to scaled or unscaled objective or constraints in Section 9.1 and Section 11 should be understood in this context.
12Optional Parameters
Several optional parameters in handle_solve_ipopt define choices in the problem specification or the algorithm logic. In order to reduce the number of formal arguments of handle_solve_ipopt these optional parameters have associated default values that are appropriate for most problems. Therefore, you need only specify those optional parameters whose values are to be different from their default values.
The remainder of this section can be skipped if you wish to use the default values for all optional parameters.
The optional parameters can be changed by calling handle_opt_set anytime between the initialization of the handle and the call to the solver. Modification of the optional parameters during intermediate monitoring stops is not allowed. Once the solver finishes, the optional parameters can be altered again for the next solve.
If any options are set by the solver (typically those with the choice of $\mathrm{AUTO}$), their value can be retrieved by handle_opt_get. If the solver is called again, any such arguments are reset to their default values and the decision is made again.
The following is a list of the optional parameters available. A full description of each optional parameter is provided in Section 12.1.
For each option, we give a summary line, a description of the optional parameter and details of constraints.
The summary line contains:
the keywords, where the minimum abbreviation of each keyword is underlined;
a parameter value,
where the letters $a$, $i$ and $r$ denote options that take character, integer and real values respectively.
the default value, where the symbol $\epsilon $ is a generic notation for machine precision (see precision).
All options accept the value $\mathrm{DEFAULT}$ to return single options to their default states.
Keywords and character values are case and white space insensitive.
Defaults
This special keyword may be used to reset all optional parameters to their default values. Any value given with this keyword will be ignored.
Hessian Mode
$a$
Default $=\mathrm{AUTO}$
This parameter specifies whether the Hessian will be user-supplied (in hx) or approximated by handle_solve_ipopt using a limited-memory quasi-Newton L-BFGS method. In the $\mathrm{AUTO}$ setting, if no Hessian structure has been registered in the problem with a call to handle_set_nlnhess, and there are explicitly nonlinear user-supplied functions, then the Hessian will be approximated. Otherwise hess will be called if and only if any of handle_set_nlnobj and handle_set_nlnconstr have been used to define the problem. Approximating the Hessian is likely to require more iterations to achieve convergence but will reduce the time spent in user-supplied functions.
Constraint: ${\mathbf{Hessian\; Mode}}=\mathrm{AUTO}$, $\mathrm{EXACT}$ or $\mathrm{APPROXIMATE}$.
Infinite Bound Size
$r$
Default $\text{}={10}^{20}$
This defines the ‘infinite’ bound $\mathit{bigbnd}$ in the definition of the problem constraints. Any upper bound greater than or equal to $\mathit{bigbnd}$ will
be regarded as $+\infty $ (and similarly any lower bound less than or equal to $-\mathit{bigbnd}$ will be regarded as $-\infty $). Note that a modification of this optional parameter does not influence constraints which have already been defined; only the constraints formulated after the change will be affected.
It also serves as a limit for the objective function to be considered unbounded ($\mathbf{errorid}={\mathbf{54}}$).
If $i\ge 0$, the
unit number
for the secondary (monitoring) output. If set to $-1$, no secondary output is provided. The information output to this unit is controlled by Monitoring Level.
Constraint: ${\mathbf{Monitoring\; File}}\ge -1$.
Monitoring Level
$i$
Default $=4$
This parameter sets the amount of information detail that will be printed by the solver to the secondary output. The meaning of the levels is the same as with Print Level.
This parameter specifies the ordering to be used by the internal sparse linear algebra solver. It affects the number of nonzeros in the factorized matrix and thus influences the cost per iteration.
${\mathbf{Matrix\; Ordering}}=\mathrm{AUTO}$
A heuristic is used to choose automatically between METIS and AMD orderings.
${\mathbf{Matrix\; Ordering}}=\mathrm{BEST}$
Both AMD and METIS orderings are computed at the beginning of the solve and the one with the fewest nonzeros in the factorized matrix is selected.
${\mathbf{Matrix\; Ordering}}=\mathrm{AMD}$
An approximate minimum degree (AMD) ordering is used.
${\mathbf{Matrix\; Ordering}}=\mathrm{METIS}$
METIS ordering is used.
Constraint: ${\mathbf{Matrix\; Ordering}}=\mathrm{AUTO}$, $\mathrm{BEST}$, $\mathrm{AMD}$ or $\mathrm{METIS}$.
Outer Iteration Limit
$i$
Default $\text{}=100$
The maximum number of iterations to be performed by handle_solve_ipopt. Setting the option too low might lead to $\mathbf{errorid}={\mathbf{22}}$.
If $i\ge 0$, the
unit number
for the primary output of the solver. If ${\mathbf{Print\; File}}=-1$, the primary output is completely turned off independently of other settings. The default value is the advisory message unit number as defined by x04abf (no CPP interface) at the time of the optional parameters initialization, e.g., at the initialization of the handle. The information output to this unit is controlled by Print Level.
Constraint: ${\mathbf{Print\; File}}\ge -1$.
Print Level
$i$
Default $=2$
This parameter defines how detailed information should be printed by the solver to the primary output.
$i$
Output
$0$
No output from the solver
$1$
Additionally, derivative check information, the Header and Summary.
$2$
Additionally, the Iteration log.
$3$, $4$
Additionally, details of each iteration with scalar quantities printed.
$5$
Additionally, individual components of arrays are printed resulting in large output.
Constraint: $0\le {\mathbf{Print\; Level}}\le 5$.
Stats Time
$a$
Default $=\mathrm{NO}$
This parameter allows you to turn on timings of various parts of the algorithm to give a better overview of where most of the time is spent. This might be helpful for a choice of different solving approaches.
Constraint: ${\mathbf{Stats\; Time}}=\mathrm{YES}$ or $\mathrm{NO}$.
This parameter specifies the required direction of the optimization. If ${\mathbf{Task}}=\mathrm{FEASIBLEPOINT}$, the objective function (if set) is ignored and the algorithm stops as soon as a feasible point is found with respect to the given tolerance. If no objective function is set, Task reverts to $\mathrm{FEASIBLEPOINT}$ automatically.
Constraint: ${\mathbf{Task}}=\mathrm{MINIMIZE}$, $\mathrm{MAXIMIZE}$ or $\mathrm{FEASIBLE\; POINT}$.
Time Limit
$r$
Default $\text{}={10}^{6}$
A limit to the number of seconds that the solver can use to solve one problem. If during the convergence check this limit is exceeded, the solver will terminate with a corresponding error message.
Constraint: ${\mathbf{Time\; Limit}}>0$.
Verify Derivatives
$a$
Default $=\mathrm{NO}$
This parameter specifies whether the function should perform numerical checks on the consistency of the user-supplied functions. It is recommended that such checks are enabled when first developing the formulation of the problem, however, the derivative check results in a significant increase of the number of the function evaluations and thus it shouldn't be used in production code.
Constraint: ${\mathbf{Verify\; Derivatives}}=\mathrm{YES}$ or $\mathrm{NO}$.