NAG FL Interface
e04svf (handle_​solve_​pennon)

1 Purpose

2 Specification

3 Description

3.1 Structure of the Lagrangian Multipliers

3.2 Approximation of the Lagrangian Multipliers

4 References

5 Arguments

1: $\mathbf{handle}$ – Type (c_ptr) Input

On entry: the handle to the problem. It needs to be initialized by e04raf and the problem formulated by some of the routines e04ref, e04rff, e04rhf, e04rjf, e04rnf and e04rpf. It must not be changed between calls to the NAG optimization modelling suite.

2: $\mathbf{nvar}$ – Integer Input

On entry: $n$, the number of decision variables $x$ in the problem. It must be unchanged from the value set during the initialization of the handle by e04raf.

3: $\mathbf{x}\left(\mathbf{nvar}\right)$ – Real (Kind=nag_wp) array Input/Output

Note: intermediate stops take place only if $\mathbf{Monitor\; Frequency}>0$.

On entry: if $\mathbf{Initial\; X}=\mathrm{USER}$ (the default), $x^0$, the initial estimate of the variables $x$, otherwise x need not be set.

On intermediate exit: the value of the variables $x$ at the end of the current outer iteration.

On intermediate re-entry: the input is ignored.

On final exit: the final value of the variables $x$.

4: $\mathbf{nnzu}$ – Integer Input

On entry: the dimension of array u.

If $\mathbf{nnzu}=0$, u will not be referenced; otherwise it needs to match the dimension of constraints defined by e04rhf and e04rjf as explained in Section 3.1.

Constraint: $\mathbf{nnzu}\ge 0$.

5: $\mathbf{u}\left(\mathbf{nnzu}\right)$ – Real (Kind=nag_wp) array Input/Output

Note: intermediate stops take place only if $\mathbf{Monitor\; Frequency}>0$.

if $\mathbf{nnzu}>0$, u holds Lagrangian multipliers (dual variables) for (standard) constraints, i.e., simple bounds defined by e04rhf and a set of $m_B$ linear constraints defined by e04rjf. Either their initial estimates, intermediate approximations or final values, see Section 3.1.

if $\mathbf{nnzu}=0$, u will not be referenced.

On entry: if $\mathbf{Initial\; U}=\mathrm{USER}$ (the default is $\mathrm{AUTOMATIC}$), $u^0$, the initial estimate of the Lagrangian multipliers $u$, otherwise u need not be set.

On intermediate exit: the estimate of the multipliers $u$ at the end of the current outer iteration.

On intermediate re-entry: the input is ignored.

On exit: the final value of multipliers $u$.

6: $\mathbf{nnzuc}$ – Integer Input

On entry: the dimension of array uc. If $\mathbf{nnzuc}=0$, uc will not be referenced. This argument is reserved for future releases of the NAG Library which will allow definition of second-order cone constraints. It needs to be set to $0$ at the moment.

Constraint: $\mathbf{nnzuc}=0$.

7: $\mathbf{uc}\left(\mathbf{nnzuc}\right)$ – Real (Kind=nag_wp) array Input/Output

uc is reserved for future releases of the NAG Library which will allow definition of second-order cone constraints. It is not referenced at the moment.

8: $\mathbf{nnzua}$ – Integer Input

On entry: the dimension of array ua. If $\mathbf{nnzua}=0$, ua will not be referenced; otherwise it needs to match the total number of nonzeros in all matrix Lagrangian multipliers (constraints defined by e04rnf and e04rpf) as explained in Section 3.1.

Constraint: $\mathbf{nnzua}\ge 0$.

9: $\mathbf{ua}\left(\mathbf{nnzua}\right)$ – Real (Kind=nag_wp) array Input/Output

Note: intermediate stops take place only if $\mathbf{Monitor\; Frequency}>0$.

If $\mathbf{nnzua}>0$, ua holds the Lagrangian multipliers for matrix constraints defined by e04rnf and e04rpf, see Section 3.1.

If $\mathbf{nnzua}=0$, ua will not be referenced.

On entry: if $\mathbf{Initial\; U}=\mathrm{USER}$ (the default is $\mathrm{AUTOMATIC}$), $U^0$, the initial estimate of the matrix Lagrangian multipliers $U$, otherwise ua need not be set.

On intermediate exit: the estimate of the matrix multipliers $U$ at the end of the outer iteration.

On intermediate re-entry: the input is ignored.

On final exit: the final estimate of the multipliers $U$.

10: $\mathbf{rinfo}\left(32\right)$ – Real (Kind=nag_wp) array Output

On intermediate or final entry: error measures and various indicators (see Section 11 for details) at the end of the current (or final) outer iteration as given in the table below:

$1$	objective function value $f\left(x\right)$
$2$	optimality (12)
$3$	feasibility (13)
$4$	complementarity (14)
$5$	minimum penalty
$6$	relative precision (11)
$7$	relative duality gap (10)
$8$	precision $\left|f\left(x^{\ell }\right)-f\left(x^{\ell +1}\right)\right|$
$9$	duality gap
$10$	minimum penalty for (standard) inequalities $p$
$11$	minimum penalty for matrix inequalities $P$
$12$	feasibility of equality constraints
$13$	feasibility of (standard) inequalities
$14$	feasibility of matrix inequalities
$15$	complementarity of equality constraints
$16$	complementarity of (standard) inequalities
$17$	complementarity of matrix inequalities
$18$–$23$	DIMACS error measures (16) (only if turned on by DIMACS Measures)
$24$–$32$	reserved for future use

11: $\mathbf{stats}\left(32\right)$ – Real (Kind=nag_wp) array Output

On intermediate or final exit: solver statistics at the end of the current (or final) outer iteration as given in the table below. Note that time statistics is provided only if Stats Time is set (the default is $\mathrm{NO}$), the measured time is returned in seconds.

$1$	Number of the outer iterations.
$2$	Total number of the inner iterations.
$3$	Total number of the linesearch steps.
$4$	Number of evaluations of the augmented Lagrangian $F\left(\right)$, (see (8)).
$5$	Number of evaluations of $\nabla F\left(\right)$.
$6$	Number of evaluations of ${\nabla }^{2}F\left(\right)$.
$7$	Reserved for future use.
$8$	Total running time of the solver.
$9$	Total running time of the solver without evaluations of the user's functions and monitoring stops.
$10$	Time spent in the inner iterations.
$11$	Time spent in Lagrangian multipliers updates.
$12$	Time spent in penalty parameters updates.
$13$	Time spent in matrix feasibility computation.
$14$	Time of evaluations of $F\left(\right)$.
$15$	Time of evaluations of $\nabla F\left(\right)$.
$16$	Time of evaluations of ${\nabla }^{2}F\left(\right)$.
$17$	Time of factorizations of the Newton system.
$18$	Time of factorizations of the matrix constraints.
$19$–$32$	reserved for future use.

12: $\mathbf{inform}$ – Integer Input/Output

Note: intermediate stops take place only if $\mathbf{Monitor\; Frequency}>0$.

On initial entry: no effect.

On intermediate exit: $\mathbf{inform}=1$.

On intermediate re-entry: if set to $0$, solving the current problem is terminated and the routine returns $\mathbf{ifail}=\mathbf{20}$; otherwise the routine continues.

On final exit: $\mathbf{inform}=0$.

13: $\mathbf{ifail}$ – Integer Input/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if $\mathbf{ifail}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ is used it is essential to test the value of ifail on exit.

On exit: $\mathbf{ifail}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

$\mathbf{ifail}=1$

The supplied handle does not define a valid handle to the data structure for the NAG optimization modelling suite. It has not been initialized by e04raf or it has been corrupted.

$\mathbf{ifail}=2$

This solver does not support fixed variables.

This solver does not support the model defined in the handle.

$\mathbf{ifail}=3$

A different solver from the suite has already been used.

$\mathbf{ifail}=4$

On entry, $\mathbf{nvar}=〈\mathit{\text{value}}〉$, expected $\mathrm{value}=〈\mathit{\text{value}}〉$.
Constraint: nvar must match the value given during initialization of handle.

$\mathbf{ifail}=5$

On entry, $\mathbf{nnzu}=〈\mathit{\text{value}}〉$.
nnzu does not match the size of the Lagrangian multipliers for (standard) constraints.
$\mathbf{nnzu}=0$ or $〈\mathit{\text{value}}〉$.

On entry, $\mathbf{nnzu}=〈\mathit{\text{value}}〉$.
nnzu does not match the size of the Lagrangian multipliers for (standard) constraints.
$\mathbf{nnzu}=0$ when there are no (standard) constraints.

On entry, $\mathbf{nnzua}=〈\mathit{\text{value}}〉$.
nnzua does not match the size of the Lagrangian multipliers for matrix constraints.
$\mathbf{nnzua}=0$ or $〈\mathit{\text{value}}〉$.

On entry, $\mathbf{nnzua}=〈\mathit{\text{value}}〉$.
nnzua does not match the size of the Lagrangian multipliers for matrix constraints.
$\mathbf{nnzua}=0$ when there are no matrix constraints.

On entry, $\mathbf{nnzuc}=〈\mathit{\text{value}}〉$.
nnzuc does not match the size of the Lagrangian multipliers for second-order cone constraints.
$\mathbf{nnzuc}=0$ when there are no second-order cone constraints.

$\mathbf{ifail}=20$

User requested termination during a monitoring step.

$\mathbf{ifail}=21$

The current starting point is unusable.

The starting point $x^0$, either provided by you (if $\mathbf{Initial\; X}=\mathrm{USER}$, the default) or the automatic estimate (if $\mathbf{Initial\; X}=\mathrm{AUTOMATIC}$), must not be extremely infeasible in the matrix constraints (infeasibility of order ${10}^6$ and higher) and all the functions used in the problem formulation must be evaluatable.

In the unlikely case this error is triggered, it is necessary to provide a better estimate of the initial values.

$\mathbf{ifail}=22$

Outer iteration limit has been reached.
The requested accuracy is not achieved.

If $\mathbf{Outer\; Iteration\; Limit}$ is left to the default, this error indicates numerical difficulties. Consider whether the stopping tolerances ($\mathbf{Stop\; Tolerance\; 1}$, $\mathbf{Stop\; Tolerance\; 2}$, $\mathbf{Stop\; Tolerance\; Feasibility}$) are set too low or optional parameters affecting the behaviour of the penalty updates ($\mathbf{P\; Update\; Speed}$, $\mathbf{P\; Min}$ or $\mathbf{Pmat\; Min}$) have been modified inadvisedly. The iteration log should reveal more about the misbehaviour. Providing a different starting point might be of help in certain situations.

$\mathbf{ifail}=23$

The inner subproblem could not be solved to the required accuracy.
Inner iteration limit has been reached.

The inner subproblem could not be solved to the required accuracy.
Limited progress in the inner subproblem triggered a stop (heuristic inner stop criteria).

The inner subproblem could not be solved to the required accuracy.
Line search or another internal component failed.

A problem with the convergence of the inner subproblem is typically a sign of numerical difficulties of the whole algorithm. The inner subproblem might be stopped before reaching the required accuracy because of the $\mathbf{Inner\; Iteration\; Limit}$, a heuristic detected no progress in the inner iterations (if $\mathbf{Inner\; Stop\; Criteria}=\mathrm{HEURISTIC}$, default) or if an internal component failed (for example, line search was unable to find a suitable step). The algorithm tries to recover, however, it might give up after several attempts with one of these error messages. If it occurs in the very early iterations, consider increasing $\mathbf{Inner\; Stop\; Tolerance}$ and possibly $\mathbf{Init\; Value\; P}$ or $\mathbf{Init\; Value\; Pmat}$ which should ease the first iterations. An occurrence in later iterations indicates numerical difficulties typically due to scaling and/or ill-conditioning or the problem is close to infeasible. Reducing the tolerance on the stopping criteria or increasing $\mathbf{P\; Update\; Speed}$ might be of limited help.

$\mathbf{ifail}=24$

Unable to make progress, the algorithm was stopped.

This error is returned if the solver cannot decrease the duality gap over a range of iterations. This can be due to the scaling of the problem or the problem might be close to primal or dual infeasibility.

$\mathbf{ifail}=50$

The algorithm converged to a suboptimal solution.
The full accuracy was not achieved. The solution should still be usable.

This error may be reported only if $\mathbf{Stop\; Criteria}=\mathrm{SOFT}$ (default). The solver predicted that it is unable to reach a better estimate of the solution. However, the error measures indicate that the point is a reasonable approximation. Typically, only the norm of the gradient of the Lagrangian (optimality) does not fully satisfy the requested tolerance whereas the others are well below the tolerance.

Setting $\mathbf{Stop\; Criteria}=\mathrm{STRICT}$ will disallow this error but it is unlikely that the algorithm would reach a better solution.

$\mathbf{ifail}=51$

The problem was found to be infeasible during preprocessing.

One or more of the constraints (or its part after preprocessing) violates the constraints by more than ${\epsilon }_{\mathrm{feas}}$ ($\mathbf{Stop\; Tolerance\; Feasibility}$).

$\mathbf{ifail}=52$

The problem was found unbounded during preprocessing.

The objective function consists of an unrestricted ray and thus the problem does not have a solution.

$\mathbf{ifail}=53$

The problem seems to be infeasible, the algorithm was stopped.

Whilst the algorithm cannot definitively detect that the problem is infeasible, several indirect indicators suggest that it might be the case.

$\mathbf{ifail}=54$

The problem seems to be unbounded, the algorithm was stopped.

Whilst the algorithm cannot definitively detect that the problem is unbounded, several indirect indicators (such as a rapid decrease in the objective function and a lack of convergence in the inner subproblem) suggest that this might be the case. A good scaling of the objective function is always highly recommended to avoid situations when unusual behavior triggers falsely this error exit.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

9.1 Description of the Printed Output

E04SV, NLP-SDP Solver (Pennon)
---------------------------------------------------------------------------
Number of variables             2                 [eliminated            0]
                           simple  linear  nonlin
(Standard) inequalities         3       0       0
(Standard) equalities                   0       0
Matrix inequalities                     1       1 [dense    2, sparse    0]
                                                  [max dimension         3]

Outer Iteration Limit         =                  20 * U
Stop Tolerance 1              =         1.00000E-06 * d
Stop Tolerance 2              =         1.00000E-07 * d
Hessian Density               =               Dense * S

-----------------------------------------------------------------
 it |  objective |  optim  |   feas  |  compl  | pen min  | inner
-----------------------------------------------------------------
  0   0.00000E+00  7.34E+00  1.23E-01  4.41E+01  1.00E+00   0
  1  -3.01998E-01  2.54E-03  0.00E+00  1.89E+00  1.00E+00   6
  2  -2.53008E+00  1.06E-03  1.30E-01  3.22E-01  3.17E-01   8
  3  -2.08172E+00  6.52E-03  1.85E-02  4.54E-02  1.01E-01   7
  4  -2.01060E+00  6.47E-03  4.10E-03  1.02E-02  3.19E-02   3

 ----------------------------------------------------------------
  it |  objective |  optim  |   feas  |  compl  | pen min | inner
 ----------------------------------------------------------------
   0  0.00000E+00  1.46E+03  5.01E+01  1.46E+03  6.40E+01   0
   1  3.78981E+02  3.86E+01  0.00E+00  1.21E+04  6.40E+01   5 M
   2  9.11724E+02  1.46E-02  0.00E+00  9.24E+02  4.45E+01   5

--------------------------------------------------------------
Status: converged, an optimal solution found
--------------------------------------------------------------
Final objective value                2.300000E+01
Relative precision                   5.873755E-09
Optimality                           1.756062E-10
Feasibility                          9.048738E-08
Complementarity                      1.855566E-08
DIMACS error 1                       8.780308E-11
DIMACS error 2                       0.000000E+00
DIMACS error 3                       0.000000E+00
DIMACS error 4                       4.524369E-08
DIMACS error 5                       4.065998E-10
DIMACS error 6                       3.948012E-10
Iteration counts
  Outer iterations                             13
  Inner iterations                             82
  Linesearch steps                             95
Evaluation counts
  Augm. Lagr. values                           96
  Augm. Lagr. gradient                         96
  Augm. Lagr. hessian                          82
Timing
  Total time                    0 h  0 min  3 sec
    Evaluations + monitoring             0.04 sec
    Solver itself                        3.09 sec
  Inner minimization step                2.72 sec   ( 87.1%)
    Augm. Lagr. value                    0.28 sec   (  9.0%)
    Augm. Lagr. gradient                 0.67 sec   ( 21.6%)
    Augm. Lagr. hessian                  1.11 sec   ( 35.4%)
    system matr. factor.                 0.64 sec   ( 20.5%)
    const. matr. factor.                 0.40 sec   ( 12.8%)
  Multiplier update                      0.01 sec   (  0.3%)
  Penalty update                         0.02 sec   (  0.5%)
  Feasibility check                      0.15 sec   (  4.7%)
--------------------------------------------------------------

9.2 Internal Changes

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

11 Algorithmic Details

11.1 Overview

11.2 Stopping Criteria

11.3 Choice of penalty functions ${\phi }_g$ and ${\phi }_A$

11.4 Solution of the inner problem

12 Optional Parameters

• Defaults
• DIMACS Measures
• Hessian Density
• Infinite Bound Size
• Initial P
• Initial U
• Initial X
• Init Value P
• Init Value Pmat
• Inner Iteration Limit
• Inner Stop Criteria
• Inner Stop Tolerance
• Linesearch Mode
• List
• Monitor Frequency
• Monitoring File
• Monitoring Level
• Outer Iteration Limit
• Pmat Min
• P Min
• Preference
• Presolve Block Detect
• Print File
• Print Level
• Print Options
• P Update Speed
• Stats Time
• Stop Criteria
• Stop Tolerance 1
• Stop Tolerance 2
• Stop Tolerance Feasibility
• Task
• Transform Constraints
• Umat Update Restriction
• U Update Restriction

12.1 Description of the Optional Parameters

• 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 x02ajf).