nag_opt_handle_solve_pennon (e04svc) is a solver from the NAG optimization modelling suite for problems such as, quadratic programming (QP), linear semidefinite programming (SDP) and semidefinite programming with bilinear matrix inequalities (BMI-SDP).

2 Specification

#include <nag.h>

#include <nage04.h>

void	nag_opt_handle_solve_pennon (void handle, Integer nvar, double x[], Integer nnzu, double u[], Integer nnzuc, double uc[], Integer nnzua, double ua[], double rinfo[], double stats[], Integer inform, NagError *fail)

3 Description

nag_opt_handle_solve_pennon (e04svc) serves as a solver for compatible problems stored as a handle. The handle points to an internal data structure which defines the problem and serves as means of communication for functions in the suite. First, the problem handle is initialized by nag_opt_handle_init (e04rac). Then some of the functions nag_opt_handle_set_linobj (e04rec), nag_opt_handle_set_quadobj (e04rfc), nag_opt_handle_set_simplebounds (e04rhc), nag_opt_handle_set_linconstr (e04rjc), nag_opt_handle_set_linmatineq (e04rnc) or nag_opt_handle_set_quadmatineq (e04rpc) may be used to formulate the objective function, (standard) constraints and matrix constraints of the problem. Once the problem is fully set, the handle may be passed to the solver. When the handle is not needed anymore, nag_opt_handle_free (e04rzc) should be called to destroy it and deallocate the memory held within. See nag_opt_handle_init (e04rac) for more details.

Problems which can be defined this way are, for example, (generally nonconvex) quadratic programming (QP)

\begin{array}{l} \underset{x \in ℝ^{n}}{minimize} & \frac{1}{2} x^{T} H x + c^{T} x & (a) \\ subject to & l_{B} \leq B x \leq u_{B} & (b) \\ l_{x} \leq x \leq u_{x}, & (c) \end{array}

(1)

linear semidefinite programming problems (SDP)

\begin{array}{l} \underset{x \in ℝ^{n}}{minimize} & c^{T} x & (a) \\ subject to & \sum_{i = 1}^{n} x_{i} A_{i}^{k} - A_{0}^{k} ⪰ 0, k = 1, \dots, m_{A} & (b) \\ l_{B} \leq B x \leq u_{B} & (c) \\ l_{x} \leq x \leq u_{x} & (d) \end{array}

(2)

or semidefinite programming problems with bilinear matrix inequalities (BMI-SDP)

\begin{array}{l} \underset{x \in ℝ^{n}}{minimize} & \frac{1}{2} x^{T} H x + c^{T} x & (a) \\ subject to & \sum_{i,j = 1}^{n} x_{i} x_{j} Q_{i j}^{k} + \sum_{i = 1}^{n} x_{i} A_{i}^{k} - A_{0}^{k} ⪰ 0, k = 1, \dots, m_{A} & (b) \\ l_{B} \leq B x \leq u_{B} & (c) \\ l_{x} \leq x \leq u_{x} . & (d) \end{array}

(3)

Here

c

l_{x}

and

u_{x}

are

n

-dimensional vectors,

H

is a symmetric

n

n

matrix,

l_{B}

u_{B}

are

m_{B}

-dimensional vectors,

B

is a general

m_{B}

n

rectangular matrix and

A_{i}^{k}

Q_{i j}^{k}

are symmetric matrices. The expression

S ⪰ 0

stands for a constraint on eigenvalues of a symmetric matrix

S

, namely, all the eigenvalues should be non-negative, i.e., the matrix should be positive semidefinite. See relevant functions of the suite for more details on the problem formulation.

The solver is based on a generalized Augmented Lagrangian method with a suitable choice of standard and matrix penalty functions. For a detailed description of the algorithm see Section 11. Under standard assumptions on the problem (Slater constraint qualification, boundedness of the objective function on the feasible set, see Stingl (2006) for details) the algorithm converges to a local solution. In case of convex problems such as linear SDP or convex QP, this is the global solution. The solver is suitable for both small dense and large-scale sparse problems.

The algorithm behaviour and solver strategy can be modified by various optional parameters (see Section 12) which can be set by nag_opt_handle_opt_set (e04zmc) and nag_opt_handle_opt_set_file (e04zpc) anytime between the initialization of the handle by nag_opt_handle_init (e04rac) and a call to the solver. Once the solver has finished, options may be modified for the next solve. The solver may be called repeatedly with various starting points and/or optional parameters.

There are several optional parameters with a multiple choice where the default choice is

AUTO

(for example,

Hessian Density

). This value means that the decision over the option is left to the solver based on the structure of the problem. Option getter nag_opt_handle_opt_get (e04znc) can be called to retrieve the choice of these options as well as on any other options.

Optional parameter

Task

may be used to switch the problem to maximization or to ignore the objective function and find only a feasible point.

Optional parameter

Monitor Frequency

may be used to turn on the monitor mode of the solver. The solver invoked in this mode pauses regularly even before the optimal point is found to allow monitoring the progress from the calling program. All the important error measures and statistics are available in the calling program which may terminate the solver early if desired (see argument inform).

3.1 Structure of the Lagrangian Multipliers

The algorithm works internally with estimates of both the decision variables, denoted by

x

, and the Lagrangian multipliers (dual variables) for standard and matrix constraints, denoted by

u

and

U

, respectively. You may provide initial estimates, request approximations during the run (the monitor mode turned on) and obtain the final values. The Lagrangian multipliers are split into two arrays, the multipliers

u

for (standard) constraints are stored in array u and multipliers

U

for matrix constraints in array ua. Both arrays need to conform to the structure of the constraints.

If the simple bounds were defined (nag_opt_handle_set_simplebounds (e04rhc) was successfully called), the first

2 n

elements of u belong to the corresponding Lagrangian multipliers, interleaving a multiplier for the lower and for the upper bound for each

x_{i}

. If any of the bounds were set to infinity, the corresponding Lagrangian multipliers are set to

0

and may be ignored.

Similarly, the following

2 m_{B}

elements of u belong to multipliers for the linear constraints, if formulated by nag_opt_handle_set_linconstr (e04rjc). The organization is the same, i.e., the multipliers for each constraint for the lower and upper bounds are alternated and zeroes are used for any missing (infinite bound) constraint.

A Lagrangian multiplier for a matrix constraint (one block) of dimension

d

d

is a dense symmetric matrix of the same dimension. All multipliers

U

are stored next to each other in array ua in the same order as the matrix constraints were defined by nag_opt_handle_set_linmatineq (e04rnc) and nag_opt_handle_set_quadmatineq (e04rpc). The lower triangle of each is stored in the packed column order (see Section 3.3.2 in the f07 Chapter Introduction). For example, if there are four matrix constraints of dimensions

7

3

1

1

, the dimension of array ua should be

36

. The first

28

elements

(d_{1} \times (d_{1} + 1) / 2)

belong to the packed lower triangle of

U_{1}

, followed by six elements of

U_{2}

and one element for each

U_{3}

and

U_{4}

. See for example Section 10 in nag_opt_sdp_read_sdpa (e04rdc).

3.2 Approximation of the Lagrangian Multipliers

By the nature of the algorithm, all inequality Lagrangian multiplier

u, U

are always kept positive during the computational process. This applies even to Lagrangian multipliers of inactive constraints at the solution. They will only be close to zero although they would normally be equal to zero exactly. This is one of the major differences between results from solvers based on the active set method (such as nag_opt_sparse_convex_qp_solve (e04nqc)) and others, such as, nag_opt_handle_solve_pennon (e04svc) or interior point methods. As a consequence, the initial estimate of the multipliers (if provided) might be adjusted by the solver to be sufficiently positive, also the estimates returned during the intermediate exits might only be a very crude approximation to their final values as they do not satisfy all the Karush–Kuhn–Tucker (KKT) conditions.

Another difference is that nag_opt_sparse_convex_qp_solve (e04nqc) merges multipliers for both lower and upper inequality into one element whose sign determines the inequality because there can be at most one active constraint and multiplier for the inactive is exact zero. Negative multipliers are associated with the upper bounds and positive with the lower bounds. On the other hand, E04SVF works with both multipliers at the same time so they are returned in two elements, one for the lower bound, the other for the upper bound (see Section 3.1). An equivalent result can be achieved by subtracting the upper bound multiplier from the lower one. This holds even when equalities are interpreted as two inequalities (see optional parameter

Transform Constraints

4 References

Ben–Tal A and Zibulevsky M (1997) Penalty/barrier multiplier methods for convex programming problems SIAM Journal on Optimization 7 347–366

Fujisawa K, Kojima M, Nakata K (1997) Exploiting sparsity in primal-dual interior-point method for semidefinite programming Math. Programming 79 235–253

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

Kočvara M and Stingl M (2003) PENNON – a code for convex nonlinear and semidefinite programming Optimization Methods and Software 18(3) 317–333

Kočvara M and Stingl M (2007) On the solution of large-scale SDP problems by the modified barrier method using iterative solvers Math. Programming (Series B) 109(2–3) 413–444

Mittelmann H D (2003) An independent benchmarking of SDP and SOCP solvers Math. Programming 95 407–430

Stingl M (2006) On the Solution of Nonlinear Semidefinite Programs by Augmented Lagrangian Methods, PhD thesis Institute of Applied Mathematics II, Friedrich–Alexander University of Erlangen–Nuremberg

5 Arguments

1: $handle$ – void *Input

On entry: the handle to the problem. It needs to be initialized by nag_opt_handle_init (e04rac) and the problem formulated by some of the functions nag_opt_handle_set_linobj (e04rec), nag_opt_handle_set_quadobj (e04rfc), nag_opt_handle_set_simplebounds (e04rhc), nag_opt_handle_set_linconstr (e04rjc), nag_opt_handle_set_linmatineq (e04rnc) and nag_opt_handle_set_quadmatineq (e04rpc). It must not be changed between the calls.

2: $nvar$ – IntegerInput

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 nag_opt_handle_init (e04rac).

3: $x [nvar]$ – doubleInput/Output

Note: intermediate stops take place only if

Monitor Frequency > 0

On entry: if

Initial X = 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: $nnzu$ – IntegerInput

On entry: the dimension of array u.

nnzu = 0

, u will not be referenced; otherwise it needs to match the dimension of constraints defined by nag_opt_handle_set_simplebounds (e04rhc) and nag_opt_handle_set_linconstr (e04rjc) as explained in Section 3.1.

Constraint:

nnzu \geq 0

5: $u [nnzu]$ – doubleInput/Output

Note: intermediate stops take place only if

Monitor Frequency > 0

nnzu > 0

, u holds Lagrangian multipliers (dual variables) for (standard) constraints, i.e., simple bounds defined by nag_opt_handle_set_simplebounds (e04rhc) and a set of

m_{B}

linear constraints defined by nag_opt_handle_set_linconstr (e04rjc). Either their initial estimates, intermediate approximations or final values, see Section 3.1.

nnzu = 0

, u will not be referenced and may be NULL.

On entry: if

Initial U = USER

(the default is

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: $nnzuc$ – IntegerInput

On entry: the dimension of array uc. If

nnzuc = 0

, uc will not be referenced. This argument is reserved for future releases of the NAG C Library which will allow definition of second order cone constraints. It needs to be set to

0

at the moment.

Constraint:

nnzuc = 0

7: $uc [nnzuc]$ – doubleInput/Output

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

8: $nnzua$ – IntegerInput

On entry: the dimension of array ua. If

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 nag_opt_handle_set_linmatineq (e04rnc) and nag_opt_handle_set_quadmatineq (e04rpc)) as explained in Section 3.1.

Constraint:

nnzua \geq 0

9: $ua [nnzua]$ – doubleInput/Output

Note: intermediate stops take place only if

Monitor Frequency > 0

nnzua > 0

, ua holds the Lagrangian multipliers for matrix constraints defined by nag_opt_handle_set_linmatineq (e04rnc) and nag_opt_handle_set_quadmatineq (e04rpc), see Section 3.1.

nnzua = 0

, ua will not be referenced and may be NULL.

On entry: if

Initial U = USER

(the default is

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: $rinfo [32]$ – doubleOutput

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:

$0$	objective function value $f (x)$
$1$	optimality (12)
$2$	feasibility (13)
$3$	complementarity (14)
$4$	minimum penalty
$5$	relative precision (11)
$6$	relative duality gap (10)
$7$	precision $\|f (x^{ℓ}) - f (x^{ℓ + 1})\|$
$8$	duality gap
$9$	minimum penalty for (standard) inequalities $p$
$10$	minimum penalty for matrix inequalities $P$
$11$	feasibility of equality constraints
$12$	feasibility of (standard) inequalities
$13$	feasibility of matrix inequalities
$14$	complementarity of equality constraints
$15$	complementarity of (standard) inequalities
$16$	complementarity of matrix inequalities
$17$ – $22$	DIMACS error measures (16) (only if turned on by $DIMACS Measures$ )
$23$ – $31$	reserved for future use

11: $stats [32]$ – doubleOutput

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

NO

), the measured time is returned in seconds.

$0$	number of the outer iterations
$1$	total number of the inner iterations
$2$	total number of the linesearch steps
$3$	number of evaluations of the augmented Lagrangian $F ()$ , (see (8))
$4$	number of evaluations of $\nabla F ()$
$5$	number of evaluations of $\nabla^{2} F ()$
$6$	reserved for future use
$7$	total running time of the solver
$8$	total running time of the solver without evaluations of the user's functions and monitoring stops
$9$	time spent in the inner iterations
$10$	time spent in Lagrangian multipliers updates
$11$	time spent in penalty parameters updates
$12$	time spent in matrix feasibility computation
$13$	time of evaluations of $F ()$
$14$	time of evaluations of $\nabla F ()$
$15$	time of evaluations of $\nabla^{2} F ()$
$16$	time of factorizations of the Newton system
$17$	time of factorizations of the matrix constraints
$18$ – $31$	reserved for future use

12: $inform$ – Integer *Input/Output

Note: intermediate stops take place only if

Monitor Frequency > 0

On initial entry: no effect.

On intermediate exit:

inform = 1

On intermediate re-entry: if set to

0

, solving the current problem is terminated and the function returns

fail . code =

NE_USER_STOP; otherwise the routine continues.

On final exit:

inform = 0

13: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_ALREADY_DEFINED: A different solver from the suite has already been used.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_DIM_MATCH: On entry, $nnzu = 〈value〉$ .
nnzu does not match the size of the Lagrangian multipliers for (standard) constraints.
The correct value is 0 for no (standard) constraints.

On entry, $nnzu = 〈value〉$ .
nnzu does not match the size of the Lagrangian multipliers for (standard) constraints.
The correct value is either 0 or $〈value〉$ .

On entry, $nnzua = 〈value〉$ .
nnzua does not match the size of the Lagrangian multipliers for matrix constraints.
The correct value is 0 for no matrix constraints.

On entry, $nnzua = 〈value〉$ .
nnzua does not match the size of the Lagrangian multipliers for matrix constraints.
The correct value is either 0 or $〈value〉$ .

On entry, $nnzuc = 〈value〉$ .
nnzuc does not match the size of the Lagrangian multipliers for second order cone constraints.
The correct value is 0 for no such constraints.
NE_FAILED_START: The current starting point is unusable.
The starting point $x^{0}$ , either provided by the user (if $Initial X = USER$ , the default) or the automatic estimate (if $Initial X = AUTOMATIC$ ), 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.
NE_HANDLE: 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 nag_opt_handle_init (e04rac) or it has been corrupted.
NE_INFEASIBLE: 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 $ε_{feas}$ ( $Stop Tolerance Feasibility$ ).
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_MAYBE_INFEASIBLE: 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.
NE_MAYBE_UNBOUNDED: 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.
NE_NO_IMPROVEMENT: 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.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REF_MATCH: On entry, $nvar = 〈value〉$ , expected $value = 〈value〉$ .
Constraint: nvar must match the value given during initialization of handle.
NE_SETUP_ERROR: This solver does not support general nonlinear objective and constraints.
NE_SUBPROBLEM: 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 $Inner Iteration Limit$ , a heuristic detected no progress in the inner iterations (if $Inner Stop Criteria = HEURISTIC$ , 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 $Inner Stop Tolerance$ and possibly $Init Value P$ or $Init Value Pmat$ 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 $P Update Speed$ might be of limited help.
NE_TOO_MANY_ITER: Outer iteration limit has been reached.
The requested accuracy is not achieved.
If $Outer Iteration Limit$ is left to the default, this error indicates numerical difficulties. Consider whether the stopping tolerances ( $Stop Tolerance 1$ , $Stop Tolerance 2$ , $Stop Tolerance Feasibility$ ) are set too low or optional parameters affecting the behaviour of the penalty updates ( $P Update Speed$ , $P Min$ or $Pmat Min$ ) 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.
NE_UNBOUNDED: The problem was found unbounded during preprocessing.
The objective function consists of an unrestricted ray and thus the problem does not have a solution.
NE_USER_STOP: User requested termination during a monitoring step.
NW_NOT_CONVERGED: 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 $Stop Criteria = SOFT$ (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 $Stop Criteria = STRICT$ will disallow this error but it is unlikely that the algorithm would reach a better solution.

7 Accuracy

The accuracy of the solution is driven by optional parameters

Stop Tolerance 1

Stop Tolerance 2

Stop Tolerance Feasibility

and

Stop Criteria

and in certain cases

DIMACS Measures

fail . code =

NE_NOERROR on the final exit, the returned point satisfies Karush–Kuhn–Tucker (KKT) conditions to the requested accuracy (under the default settings close to

\sqrt{ε}

) and thus it is a good estimate of a local solution. If

fail . code =

NW_NOT_CONVERGED, 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 11.2 and the description of the particular options.

8 Parallelism and Performance

nag_opt_handle_solve_pennon (e04svc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_opt_handle_solve_pennon (e04svc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

9.1 Description 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 send to two independent streams (files) which are set by optional parameters

Print File

and

Monitoring File

. Optional parameters

Print Level

Print Options

and

Monitoring Level

determine the exposed level of detail. This allows, for example, to generate a detailed log in a file while the condensed information is displayed on the screen.

By default (

Print File = 6

Print Level = 2

), four sections are printed to the standard output: a header, a list of options, an iteration log and a summary.

Header

The header contains statistics about the size of the problem as represented internally, i.e., it reflects any changes imposed by preprocessing and problem transformations (see, for example,

Presolve Block Detect

and

Transform Constraints

). The header may look like:

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]

It shows the total number of variables and how many of them were eliminated (e.g., fixed by a simple equality). A constraint with both a lower and an upper bound counts as 2 inequalities. Simple bounds are set by nag_opt_handle_set_simplebounds (e04rhc), matrix inequalities by nag_opt_handle_set_linmatineq (e04rnc) and nag_opt_handle_set_quadmatineq (e04rpc) and standard equalities and inequalities by nag_opt_handle_set_linconstr (e04rjc). Note that matrix constraints of dimension 1 are extracted and treated as (standard) inequalities as well. The header report concludes with the number of matrix constraints factorized as dense and sparse matrices, together with the largest dimension of the matrix inequalities.

Optional parameters list

The list shows all options of the solver, each displayed on one line. The line contains the option name, its current value and an indicator for how it was set. The options left at their defaults are noted by (d), the ones set by the user are noted by (U) and the options reset by the solver by (S). The solver will automatically set options which are set to

AUTO

or options which are not possible to satisfy in the given context (e.g., requesting

DIMACS Measures

for a nonlinear problem). Note that the output format is compatible with the file format expected by nag_opt_handle_opt_set_file (e04zpc). The output might look as follows:

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

Iteration log

Print Level = 2

, the status of each major iteration is condensed to one line. The line shows the major iteration number (

0

represents the starting point), the current objective value, KKT measures (optimality, feasibility and complementarity), minimal penalty and the number of inner iterations performed. Note that all these values are also available in

rinfo [0], \dots, rinfo [4]

and

stats [0]

. The output might look as follows:

-----------------------------------------------------------------
 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

Occasionally, a one letter flag is printed at the end of the line indicating that the inner subproblem was not solved to the required accuracy. The possibilities are M for maximum number of inner iterations, L for difficulties in the line search and ! when a heuristic stop took place. Repeated troubles in the subproblems may lead to

fail . code =

NE_SUBPROBLEM. The output below had

Inner Iteration Limit = 5

which was not enough in the first subproblem (first outer iteration).

 ----------------------------------------------------------------
  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

All KKT measures should normally converge to zero as the algorithm progresses and once the requested accuracy (

Stop Tolerance 2

) is achieved, the solver stops. However, the convergence is not necessarilly monotonic. The penalty parameters are decreased each major iteration which should improve overall the feasibility of the problem. This also increases the ill-conditioning which might lead to a higher number of inner iterations. A very high number of inner iterations usually signals numerical difficulties. See Section 11 for the algorithmic details.

Print Level > 2

, each major iteration produces significantly more detailed output comprising detailed error measures and output from every inner iteration. The output is self-explanatory so is not featured here in detail.

Summary

Once the solver finishes, a detailed summary is produced. An example is shown below:

--------------------------------------------------------------
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%)
--------------------------------------------------------------

It starts with the status line of the overall result which matches the fail value. It is followed by the final objective value and the error measures (including

DIMACS Measures

if turned on). Iteration counters, numbers of evaluations of the Augmented Lagrangian function and timing of the routine conclude the section. The timing of the algorithm is displayed only if

Stats Time

is set.

10 Example

Semidefinite Programming has many applications in several fields of mathematics, such as, combinatorial optimization, finance, statistics, control theory or structural optimization. However, these applications seldom come in the form of (2) or (3). Usually a reformulation is needed or even a relaxation is employed to achieve the desired formulation. This is also the case of the Lovász

ϑ

function computed in this example. See also Section 10 in nag_opt_handle_init (e04rac) for links to further examples in the suite.

The Lovász

ϑ

function (or also called

ϑ

number) of an undirected graph

G = (V, E)

is an important quantity in combinatorial optimization. It gives an upper bound to Shannon capacity of the graph

G

and is also related to the clique number and the chromatic number of the complement of

G

which are NP-hard problems.

The

ϑ

function can be expressed in various ways, here we use the following:

ϑ (G) = minimize \{λ_{\max} (H) ∣ H \in 𝕊^{n}, s_{i j} = 1 ​ if ​ i = j ​ or if ​ i j \notin E\}

where

n = |V|

and

𝕊^{n}

denotes the space of real symmetric

n

n

matrices. This eigenvalue optimization problem is easy to reformulate as an SDP problem by introducing an artificial variable

t

as follows:

\begin{array}{l} \underset{t, H}{minimize} & t \\ subject to & H ⪯ t I \\ H \in 𝕊^{n}, s_{i j} = 1 ​ if ​ i = j ​ or if ​ i j \notin E . \end{array}

Finally, this can be written as (2)) which is formulated in the example:

\begin{array}{l} \underset{t, x}{minimize} & t \\ subject to & t I + \sum_{i j \in E} x_{i j} E_{i j} - J ⪰ 0 \end{array}

where

J

is a matrix of all ones and

E_{i j}

is a matrix of all zeros except

(i, j)

and

(j, i)

The example also demonstrates how to set the optional parameters and how to retrieve them.

The data file stores the Petersen graph whose

ϑ

4

11 Algorithmic Details

This section contains a description of the algorithm used in nag_opt_handle_solve_pennon (e04svc) which is based on the implementation of the code called Pennon. For further details, see Kočvara and Stingl (2003), Stingl (2006) and Kočvara and Stingl (2007).

For simplicity, we will use the following problem formulation; its connection to (SDP) and (BMI-SDP) is easy to see:

\begin{matrix} \underset{x \in ℝ^{n}}{minimize} & f (x) \\ subject to & g_{k} (x) \geq 0, & k = 1, 2, \dots, m_{g} \\ h_{k} (x) = 0, & k = 1, 2, \dots, m_{h} \\ A_{k} (x) ⪰ 0, & k = 1, 2, \dots, m_{A}, \end{matrix}

(4)

where

f

g_{k}

h_{k}

are

C^{2}

functions from

ℝ^{n}

ℝ

and

A_{k}

is a

C^{2}

matrix function from

ℝ^{n}

𝕊^{m_{k}}

. Here

𝕊^{m}

denotes the space of real symmetric matrices

m \times m

and

S \in 𝕊^{m}

S ⪰ 0

stands for a constraint on eigenvalues of

S

, namely the matrix

S

should be positive semidefinite. Furthermore, we define the inner product on

𝕊^{m}

{〈A, B〉}_{𝕊^{m}} = trace (A B)

. The index

𝕊^{m}

will be omitted whenever the dimension is clear from the context. Finally, for

Φ : 𝕊^{m} \to 𝕊^{m}

and

X, Y \in 𝕊^{m}

D Φ (X; Y)

denotes the directional derivative of

Φ

with respect to

X

in direction

Y

11.1 Overview

The algorithm is based on a (generalized) augmented Lagrangian approach and on a suitable choice of smooth penalty/barrier functions

φ_{g} : ℝ \to ℝ

for (standard) inequality constraints and

φ_{A} : ℝ \to ℝ

for constraints on matrix eigenvalues. By means of

φ_{A}

we define a penalty/barrier function for matrix inequalities as follows.

Let

A \in 𝕊^{m}

have an eigenvalue decomposition

A = S^{T} Λ S

where

Λ = diag {(λ_{1}, λ_{2}, \dots, λ_{m})}^{T}

. We define matrix function

Φ_{P} : 𝕊^{m} \to 𝕊^{m}

for

P > 0

Φ_{P} : A \mapsto S^{T} (\begin{matrix} P φ_{A} (\frac{λ_{1}}{P}) & 0 & \dots & 0 \\ 0 & P φ_{A} (\frac{λ_{2}}{P}) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & P φ_{A} (\frac{λ_{m}}{P}) \end{matrix}) S .

(5)

Both

φ_{g}

and

φ_{A}

satisfy a number of assumptions (see Kočvara and Stingl (2003)) guaranteeing, in particular, that for any

p

P > 0

\begin{matrix} g_{k} (x) \geq 0 & \Leftrightarrow & p φ_{g} (g_{k} (x) / p) \geq 0, & k = 1, 2, \dots, m_{g}, \\ A_{k} (x) ⪰ 0 & \Leftrightarrow & Φ_{P} (A_{k} (x)) ⪰ 0, & k = 1, 2, \dots, m_{A} . \end{matrix}

(6)

Further in the text, we use simplified notation

φ_{p} (\cdot) = p φ_{g} (\cdot / p)

Thus for any

p

P > 0

, problem (4) has the same solution as the following augmented problem

\begin{matrix} \underset{x \in ℝ^{n}}{minimize} & f (x) \\ subject to & φ_{p} (g_{k} (x)) \geq 0, & k = 1, 2, \dots, m_{g} \\ h_{k} (x) = 0, & k = 1, 2, \dots, m_{h} \\ Φ_{P} (A_{k} (x)) ⪰ 0, & k = 1, 2, \dots, m_{A} . \end{matrix}

(7)

The Lagrangian of (7) can be viewed as a (generalized) augmented Lagrangian of (4):

\begin{matrix} F (x, u, v, U, p, P) & = & f (x) - \sum_{k = 1}^{m_{g}} u_{k} φ_{p} (g_{k} (x)) \\ + \sum_{k = 1}^{m_{h}} v_{k} h_{k} (x) \\ - \sum_{k = 1}^{m_{A}} 〈U_{k}, Φ_{P} (A_{k} (x))〉; \end{matrix}

(8)

where

u \in ℝ^{m_{g}}

v \in ℝ^{m_{h}}

and

U = (U_{1}, \dots, U_{m_{A}})

U_{k} \in 𝕊^{p_{k}}

k = 1, \dots, m_{A}

are Lagrange multipliers associated with the (standard) inequalities and equalities and the matrix inequality constraints, respectively.

The algorithm combines ideas of the (exterior) penalty and (interior) barrier methods with the augmented Lagrangian method, it can be defined as follows:

Algorithm 1 (Outer Loop)

Let

x^{0}

u^{0}

v^{0}

and

U^{0}

be given. Let

p^{0} > 0

P^{0} > 0

α^{0} > 0

. For

ℓ = 0, 1, \dots

repeat until a stopping criteria or maximum number of iterations is reached:

(i)

Find

x^{ℓ + 1}

v^{ℓ + 1}

satisfying

\begin{matrix} ‖\nabla_{x} F (x^{ℓ + 1}, u^{ℓ}, v^{ℓ + 1}, U^{ℓ}, p^{ℓ}, P^{ℓ})‖ \leq α^{ℓ} \\ ‖h (x^{ℓ + 1})‖ \leq α^{ℓ} \end{matrix}

(9)

(ii)

Update Lagrangian multipliers

\begin{matrix} U_{k}^{ℓ + 1} = D Φ_{P} (A_{k} (x^{ℓ + 1}); U_{k}^{ℓ}), & k = 1, 2, \dots, m_{A} \\ u_{k}^{ℓ + 1} = u_{k}^{ℓ} φ_{g}^{'} (g_{k} (x^{ℓ + 1}) / p^{ℓ}), & k = 1, 2, \dots, m_{g} \end{matrix}

(iii)

Update penalty parameters and inner problem stopping criteria

p^{ℓ + 1} < p^{ℓ}, P^{ℓ + 1} < P^{ℓ}, α^{ℓ + 1} \leq α^{ℓ} .

Step (i) of Algorithm 1, further referred as the inner problem, is the most time-consuming and thus the choice of the solver for (9) is critical for the overall efficiency of the method. See Section 11.4 below.

The inequality Lagrangian multipliers update in step (ii) is motivated by the fact that if

x^{ℓ + 1}

v^{ℓ + 1}

solve (9) exactly in iteration

ℓ

, we obtain

\nabla_{x} F (x^{ℓ + 1}, u^{ℓ + 1}, v^{ℓ + 1}, U^{ℓ + 1}, p^{ℓ}, P^{ℓ}) = 0 .

Details can be found, for example, in Stingl (2006).

In practise, numerical studies showed that it is not advantageous to do the full updates of multipliers

u

U

. Firstly, big changes in the multipliers may lead to a large number of iterations in subsequent solution of (9) and, secondly, the multipliers might become ill-conditioned after a few steps and the algorithm suffers from numerical instabilities. To overcome these difficulties, a restricted update is performed instead.

New Lagrangian multipliers for (standard) inequalities

u_{k}^{ℓ + 1}

, for

k = 1, 2, \dots, m_{g}

are limited not to violate the following bound

μ_{g} < \frac{u_{k}^{ℓ + 1}}{u_{k}^{ℓ}} < \frac{1}{μ_{g}}

for a given

0 < μ_{g} < 1

(see

U Update Restriction

A similar strategy is applied to the matrix multipliers

U_{k}^{ℓ + 1}

as well. For

0 < μ_{A} < 1

(see

Umat Update Restriction

) set

U_{k}^{new} = U_{k}^{ℓ + 1} + μ_{A} (U_{k}^{ℓ} - U_{k}^{ℓ + 1}) .

The penalty parameters

p, P

in step (iii) are updated by some constant factor dependent on the initial penalty parameters

p^{0}, P^{0}

and

P Update Speed

. The update process is stopped when

p_{\min}

and

P_{\min}

are reached (see

P Min

Pmat Min

Additional details about the multiplier and penalty update strategies, as well as local and global convergence properties under standard assumptions can be found in an extensive study Stingl (2006).

11.2 Stopping Criteria

Algorithm 1 is stopped when all the stopping criteria are satisfied to the requested accuracy, these are:

\begin{matrix} \frac{|f (x^{ℓ}) - F (x^{ℓ}, u^{ℓ}, v^{ℓ}, U^{ℓ}, p^{ℓ}, P^{ℓ})|}{1 + |f (x^{ℓ})|} \leq ε_{1}, ​ & (relative duality gap) \end{matrix}

(10)

\begin{matrix} \frac{|f (x^{ℓ}) - f (x^{ℓ - 1})|}{1 + |f (x^{ℓ})|} \leq ε_{1}, & (relative precision) \end{matrix}

(11)

and these based on Karush–Kuhn–Tucker (KKT) error measures, to keep the notation simple, formulation (4) is assumed and iteration index

ℓ

is dropped:

\begin{matrix} ‖\nabla f (x) - \sum_{k = 1}^{m_{g}} u_{k} \nabla g_{k} (x) + \sum_{k = 1}^{m_{h}} v_{k} \nabla h_{k} (x) - \sum_{k = 1}^{m_{A}} {[〈U_{k}, \frac{\partial}{\partial x_{i}} A_{k} (x)〉]}_{i = 1, \dots, n}‖ \leq ε_{2}, & (optimality) \end{matrix}

(12)

\begin{matrix} g_{k} (x) \geq - ε_{feas}, |h_{k} (x)| \leq ε_{feas}, A_{k} (x) ⪰ - ε_{feas} I for all ​ k, & (feasibility) \end{matrix}

(13)

\begin{matrix} |g_{k} (x) u_{k}| \leq ε_{2}, |h_{k} (x) v_{k}| \leq ε_{2}, |〈A_{k} (x), U_{k}〉| \leq ε_{2} . & (complementarity) \end{matrix}

(14)

Here

ε_{1}

ε_{2}

ε_{feas}

may be set in the option settings as

Stop Tolerance 1

Stop Tolerance 2

and

Stop Tolerance Feasibility

, respectively.

Note that if

Task = FEASIBLE POINT

, only the feasibility is taken into account.

There is an option for linear SDP problems to switch from stopping criteria based on the KKT conditions to

DIMACS Measures

, see Mittelmann (2003). This is the default choice. To keep the notation readable, these are defined here only for the following simpler formulation of linear SDP rather than (2):

\begin{array}{l} \underset{x \in ℝ^{n}}{minimize} & c^{T} x \\ subject to & A (x) = \sum_{i = 1}^{n} x_{i} A_{i} - A_{0} ⪰ 0 . \end{array}

(15)

In this case the algorithm stops when:

\begin{array}{l} {Derr}_{1} & = & \frac{‖A^{*} (U) - c‖}{1 + ‖c‖} \\ {Derr}_{2} & = & \max (0, \frac{- λ_{\min} (U)}{1 + ‖c‖}) \\ {Derr}_{4} & = & \max (0, \frac{- λ_{\min} (\sum_{i = 1}^{n} x_{i} A_{i} - A_{0})}{1 + ‖A_{0}‖}) \\ {Derr}_{5} & = & \frac{〈A_{0}, U〉 - c^{T} x}{1 + |〈A_{0}, U〉| + c^{T} x} \\ {Derr}_{6} & = & \frac{〈\sum_{i = 1}^{n} x_{i} A_{i} - A_{0}, U〉}{1 + |〈A_{0}, U〉| + |c^{T} x|} \end{array}

(16)

where

A^{*} (\cdot)

denote the adjoint operator to

A (\cdot)

{[A^{*} (U)]}_{i} = 〈A_{i}, U〉

They can be viewed as a scaled version of the KKT conditions.

{Derr}_{1}

represents the (scaled) norm of the gradient of the Lagrangian,

{Derr}_{2}

and

{Derr}_{4}

the dual and primal infeasibility, respectively, and

{Derr}_{5}

and

{Derr}_{6}

measure the duality gap and the complementary slackness. Note that in this solver

{Derr}_{2} = 0

by definition and

{Derr}_{3}

is automaticaly zero because the formulation involves slack variables which are not used here.

11.3 Choice of penalty functions $φ_{g}$ and $φ_{A}$

To treat the (standard) inequality constraints

g_{k} (x) \geq 0

, we use the penalty/barrier function proposed by Ben–Tal and Zibulevsky (1997):

φ_{g} (τ) = \{\begin{matrix} - τ + \frac{1}{2} τ^{2} & if ​ τ \leq \bar{τ} \\ - {(1 - \bar{τ})}^{2} \log (\frac{1 - 2 \bar{τ} + τ}{1 - \bar{τ}}) - \bar{τ} + \frac{1}{2} {\bar{τ}}^{2} & if ​ τ > \bar{τ}; \end{matrix}

with default

\bar{τ} = \frac{1}{2}

The choice of

φ_{A}

(and thus of

Φ_{P}

) is motivated by the complexity of the evaluation of

Φ_{P}

and its derivatives. If

φ_{A}

is defined as

φ_{A} (τ) = \frac{1}{1 + τ} - 1,

it is possible to avoid the explicit eigenvalue decomposition in (5) as it can be seen in the formulae below (note that index

k

is omitted):

\begin{matrix} Φ_{P} (A (x)) & = & P^{2} Z (x) - P I \\ \frac{\partial}{\partial x_{i}} Φ_{P} (A (x)) & = & - P^{2} Z (x) \frac{\partial A (x)}{\partial x_{i}} Z (x) \\ \frac{\partial^{2}}{\partial x_{i} \partial x_{j}} Φ_{P} (A (x)) & = & P^{2} Z (x) (\frac{\partial A (x)}{\partial x_{i}} Z (x) \frac{\partial A (x)}{\partial x_{j}} - \frac{\partial^{2} A (x)}{\partial x_{i} \partial x_{j}} + \frac{\partial A (x)}{\partial x_{j}} Z (x) \frac{\partial A (x)}{\partial x_{i}}) Z (x) \end{matrix}

(17)

where

Z (x) = {(A (x) + P I)}^{- 1} .

(18)

For details follow Kočvara and Stingl (2003). Note that, in particular, formula (17) requires nontrivial computational resources even if careful handling of the sparsity of partial derivatives of

A (x)

is implemented. nag_opt_handle_solve_pennon (e04svc) uses a set of strategies described in Fujisawa et al. (1997) adapted for parallel computation.

11.4 Solution of the inner problem

This section describes solving of the inner problem (step (i) of Algorithm 1). We attempt to find an approximate solution of the following system (in

x

and

v

) up to the given precision

α

\begin{matrix} \nabla_{x} F (x, u, v, U, p, P) & = & 0 \\ h (x) & = & 0 \end{matrix}

(19)

where the penalty parameters

p, P

, as well as the Lagrangian multipliers

u

and

U

are fixed.

A linesearch SQP framework is used due to its desirable convergence properties. It can be stated as follows.

Algorithm 2 (Inner Loop)

Let

x^{0}

v^{0}

be given (typically as the solution from the previous outer iteration),

p

P

u

U

and

α > 0

fixed. For

ℓ = 0, 1, \dots

(i)

Find a descent direction

d

by solving

(\begin{matrix} \nabla^{2} F (x^{ℓ}) & \nabla h (x^{ℓ}) \\ \nabla h {(x^{ℓ})}^{T} & 0 \end{matrix}) (\begin{matrix} d \\ d_{v} \end{matrix}) = - (\begin{matrix} \nabla F (x^{ℓ}) \\ h (x^{ℓ}) \end{matrix})

(20)

(ii)

Find a suitable step length

δ

and set

\begin{matrix} x^{ℓ + 1} = x^{ℓ} + δ d \\ v^{ℓ + 1} = v^{ℓ} + δ d_{v} \end{matrix}

(iii)

Stop if

Inner Iteration Limit

is reached or if

\begin{matrix} ‖\nabla_{x} F (x^{ℓ + 1}, u, v^{ℓ + 1}, U, p, P)‖ \leq α \\ ‖h (x^{ℓ + 1})‖ \leq α . \end{matrix}

System (20) is solved by the factorization routine MA97 (see Hogg and Scott (2011), in combination with an inertia correction strategy described in Stingl (2006). The step length selection is guided by

Linesearch Mode

If there are no equality constraints in the problem, the unconstrained minimization in Step (i) of Algorithm 1 simplifies to the modified Newton method with line-search (for details, see Kočvara and Stingl (2003)). Alternatively, the equality constraints

h_{k} (x) = 0

can be converted to two inequalities which would be treated with the remaining constraints (see

Transform Constraints

12 Optional Parameters

Several optional parameters in nag_opt_handle_solve_pennon (e04svc) define choices in the problem specification or the algorithm logic. In order to reduce the number of formal arguments of nag_opt_handle_solve_pennon (e04svc) 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 nag_opt_handle_opt_set (e04zmc) anytime between the initialization of the handle by nag_opt_handle_init (e04rac) and the call to the solver. Modification of the arguments 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

AUTO

), their value can be retrieved by nag_opt_handle_opt_get (e04znc). 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.

12.1 Description of the Optional Parameters

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 $ε$ is a generic notation for machine precision (see nag_machine_precision (X02AJC)).

All options accept the value

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 argument value given with this keyword will be ignored.

DIMACS Measures

a

Default

= CHECK

If the problem is a linear semidefinite programming problem, this argument specifies if DIMACS error measures (see Section 11.2) should be computed and/or checked. In other cases, this option reverts to

NO

automatically.

Constraint:

DIMACS Measures = COMPUTE

CHECK

NO

Hessian Density

a

Default

= AUTO

This optional parameter guides the solver on how the Hessian matrix of augmented Lagrangian

F (x, u, v, U, p, P)

should be built. Option

AUTO

leaves the decision to the solver and it is the recommended option. Setting it to DENSE bypasses the autodetection and the Hessian is always built as a dense matrix. Option SPARSE instructs the solver to use a sparse storage and factorization of the matrix if possible.

Constraint:

Hessian Density = AUTO

DENSE

SPARSE

Infinite Bound Size

r

Default

= 10^{20}

This defines the ‘infinite’ bound

bigbnd

in the definition of the problem constraints. Any upper bound greater than or equal to

bigbnd

will be regarded as

+ \infty

(and similarly any lower bound less than or equal to

- 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.

Constraint:

Infinite Bound Size \geq 1000

Initial P

a

Default

= AUTOMATIC

This optional parameter defines the choice of the penalty optional parameters

p^{0}

P^{0}

, see Algorithm 1.

$Initial P = AUTOMATIC$: The penalty optional parameters are chosen automatically as set by optional parameter $Init Value P$ , $Init Value Pmat$ and subject to automatic scaling. Note that $P^{0}$ might be increased so that the penalty function $Φ_{P} ()$ is defined for all matrix constraints at the starting point.
$Initial P = KEEP PREVIOUS$: The penalty optional parameters are kept from the previous run of the solver if possible. If not, this options reverts to $AUTOMATIC$ . Note that even if the matrix penalty optional parameters are the same as in the previous run, they are still subject to a possible increase so that the penalty function $Φ_{P} ()$ is well defined at the starting point.

Constraint:

Initial P = AUTOMATIC

KEEP PREVIOUS

Initial U

a

Default

= AUTOMATIC

This argument guides the solver on which initial Lagrangian multipliers are to be used.

$Initial U = AUTOMATIC$: The Lagrangian multipliers are chosen automatically as set by automatic scaling.
$Initial U = USER$: The values of arrays u and ua (if provided) are used as the initial Lagrangian multipliers subject to automatic adjustments. If one or the other array is not provided, the choice for missing data is as in $AUTOMATIC$ .
$Initial U = KEEP PREVIOUS$: The Lagrangian multipliers are kept from the previous run of the solver. If this option is set for the first run or optional parameters change the approach of the solver, the choice automatically reverts to $AUTOMATIC$ . This might be useful if the solver is hot started, for example, to achieve higher precision of the solution.

Constraint:

Initial U = AUTOMATIC

USER

KEEP PREVIOUS

Initial X

a

Default

= USER

This argument guides which starting point

x^{0}

is to be used.

$Initial X = AUTOMATIC$: The starting point is chosen automatically so that it satisfies simple bounds on the variables or as a zero vector. Input of argument x is ignored.
$Initial X = USER$: Initial values of argument x are used as a starting point.

Constraint:

Initial X = AUTOMATIC

USER

Init Value P

r

Default

= 1.0

This argument defines the value

p^{0}

, the initial penalty optional parameter for (standard) inequalities. A low value of the penalty causes the solution of the inner problem to be closer to the feasible region and thus to the desirable result. However, it also increases ill-conditioning of the system. It is not advisable to set the penalty too low unless a good starting point is provided.

Constraint:

\sqrt[4]{ε} \leq Init Value P \leq 10^{4}

Init Value Pmat

r

Default

= 1.0

The value of this option suggests

P^{0}

, the initial penalty optional parameter for matrix inequalities. It is similar to

Init Value P

(and the same advice applies), however,

P^{0}

gets increased automatically if the matrix constraints are more infeasible than the actual penalty optional parameter.

Constraint:

\sqrt[4]{ε} \leq Init Value Pmat \leq 10^{4}

Inner Iteration Limit

i

Default

= 100

The maximum number of the inner iterations (Newton steps) to be performed by Algorithm 2 in each outer iteration. Setting the option too low might lead to

fail . code =

NE_SUBPROBLEM. Values higher than

100

are unlikely to improve convergence.

Constraint:

Inner Iteration Limit > 0

Inner Stop Criteria

a

Default

= HEURISTIC

The precision

α

for the solution of the inner subproblem is determined in Algorithm 1 and under typical circumstances Algorithm 2 is expected to reach this precision within the given

Inner Iteration Limit

. If any problems are detected and

Inner Stop Criteria = HEURISTIC

, Algorithm 2 is allowed to stop before reaching the requested precision or the

Inner Iteration Limit

. This usually saves many unfruitful iterations and the solver may recover in the following iterations. If you suspect that the heuristic problem detection is not suitable for your problem, setting

Inner Stop Criteria = STRICT

disallows such behaviour.

Constraint:

Inner Stop Criteria = HEURISTIC

STRICT

Inner Stop Tolerance

r

Default

= 10^{- 2}

This option sets the required precision

α^{0}

for the first inner problem solved by Algorithm 2. The precison of the solution of the inner problem does not need to be very high in the first outer iterations and it is automatically adjusted through the outer iterations to reach the optimality limit

ε_{2}

in the last one.

Setting

α^{0}

too restrictive (too low) causes an increase of the number of inner iterations needed in the first outer iterations and might lead to

fail . code =

NE_SUBPROBLEM. In certain cases it might be helpful to use a more relaxed (higher)

α^{0}

and increase

P Update Speed

which should reduce the number of inner iterations needed at the beginning of the computation in exchange for a possibly higher number of the outer iterations.

Constraint:

ε < Inner Stop Tolerance \leq 10^{3}

Linesearch Mode

a

Default

= AUTO

This controls the step size selection in Algorithm 2. If

Linesearch Mode = FULLSTEP

(the default for linear problems), unit steps are taken where possible and the step shortening takes place only to avoid undefined regions for the matrix penalty function

Φ_{P} ()

(see (17)). This may be used for linear problems but it is not recommended for any nonlinear ones. If

Linesearch Mode = ARMIJO

, Armijo backtracking linesearch is used instead which is a fairly basic linesearch. If

Linesearch Mode = GOLDSTEIN

, a cubic safe guarded linesearch based on Goldstein condition is employed, this is the recommended (and default) choice for nonlinear problems.

Constraint:

Linesearch Mode = AUTO

FULLSTEP

ARMIJO

GOLDSTEIN

List

a

Default

= NO

This argument may be set to

YES

if you wish to turn on printing of each optional parameter specification as it is supplied.

Constraint:

List = YES

NO

Monitor Frequency

i

Default

= 0

Monitor Frequency > 0

, the solver returns to you at the end of every

i

th outer iteration. During these intermediate exits, the current point x and Lagrangian multipliers u, ua (if requested) are provided as well as the statistics and error measures (rinfo, stats). Argument inform helps to distinguish between intermediate and final exits and also allows immediate termination.

Monitor Frequency = 0

, the solver stops only once on the final point and no intermediate exits are made.

Constraint:

Monitor Frequency \geq 0

Monitoring File

i

Default

= - 1

(See Section 2.3.1.1 in How to Use the NAG Library and its Documentation for further information on NAG data types.)

i \geq 0

, the Nag_FileID number (as returned from nag_open_file (x04acc)) for the secondary (monitoring) output. If set to

- 1

, no secondary output is provided. The following information is output to the unit:

– a listing of the optional parameters;
– problem statistics, the iteration log and the final status as set by $Monitoring Level$ .

Constraint:

Monitoring File \geq - 1

Monitoring Level

i

Default

= 4

This argument 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

Constraint:

0 \leq Monitoring Level \leq 5

Outer Iteration Limit

i

Default

= 100

The maximum number of the outer iterations to be performed by Algorithm 1. If

Outer Iteration Limit = 0

, no iteration is performed, only quantities needed in the stopping criteria are computed and returned in rinfo. This might be useful in connection with

Initial X = USER

and

Initial U = USER

to check optimality of the given point. However, note that the rules for possible modifications of the starting point still apply, see u and ua. Setting the option too low might lead to

fail . code =

NE_TOO_MANY_ITER.

Constraint:

Outer Iteration Limit \geq 0

P Min

r

Default

= \sqrt{ε}

This controls

p_{\min}

, the lowest possible penalty value

p

used for (standard) inequalities. In general, very small values of the penalty optional parameters cause ill-conditioning which might lead to numerical difficulties. On the other hand, very high

p_{\min}

prevents the algorithm from reaching the requested accuracy on the feasibility. Under normal circumstances, the default value is recommended.

Constraint:

ε \leq P Min \leq 10^{- 2}

Pmat Min

r

Default

= \sqrt{ε}

This is an equivalent of

P Min

for the minimal matrix penalty optional parameter

P_{\min}

. The same advice applies.

Constraint:

ε \leq Pmat Min \leq 10^{- 2}

Preference

a

Default

= SPEED

This option affects how contributions from the matrix constraints (17) to the system Hessian matrix are computed. The default option of

Preference = SPEED

should be suitable in most cases. However, dealing with matrix constraints of a very high dimension may cause noticable memory overhead and switching to

Preference = MEMORY

may be required.

Constraint:

Preference = SPEED

MEMORY

Presolve Block Detect

a

Default

= YES

Presolve Block Detect = YES

, the matrix constraints are checked during preprocessoring to determine if they can be split into smaller independent ones, thus speeding up the solver.

Constraint:

Presolve Block Detect = YES

NO

Print File

i

Default

= 6

(See Section 2.3.1.1 in How to Use the NAG Library and its Documentation for further information on NAG data types.)

i \geq 0

, the Nag_FileID number (as returned from nag_open_file (x04acc), stdout as the default) for the primary output of the solver. If

Print File = - 1

, the primary output is completely turned off independently of other settings. The following information is output to the unit:

– a listing of optional parameters if set by $Print Options$ ;
– problem statistics, the iteration log and the final status from the solver as set by $Print Level$ .

Constraint:

Print File \geq - 1

Print Level

i

Default

= 2

This argument defines how detailed information should be printed by the solver to the primary output.

$i$	Output
$0$	No output from the solver
$1$	Only the final status and the objective value
$2$	Problem statistics, one line per outer iteration showing the progress of the solution, final status and statistics
$3$	As level 2 but detailed output of the outer iterations is provided and brief overview of the inner iterations
$4$ , $5$	As level 3 but details of the inner iterations are printed as well

Constraint:

0 \leq Print Level \leq 5

Print Options

a

Default

= YES

Print Options = YES

, a listing of optional parameters will be printed to the primary output.

Constraint:

Print Options = YES

NO

P Update Speed

i

Default

= 12

This option affects the rate at which the penalty optional parameters

p, P

are updated (Algorithm 1, step (iii)) and thus indirectly influences the overall number of outer iterations. Its value can be interpretted as the typical number of outer iterations needed to get from the initial penalty values

p^{0}

P^{0}

half-way to the

p_{\min}

and

P_{\min}

. Values smaller than

3

causes a very agressive penalty update strategy which might lead to the increased number of inner iterations and possibly to numerical difficulties. On the other hand, values higher than

15

produce a relatively conservative approach which leads to a higher number of the outer iterations.

If the solver encounters difficulties on your problem, a higher value might help. If your problem is working fine, setting a lower value might increase the speed.

Constraint:

1 \leq P Update Speed \leq 100

Stats Time

a

Default

= NO

This argument turns 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. It is possible to choose between CPU and wall clock time. Choice

YES

is equivalent to wall clock.

Constraint:

Stats Time = YES

NO

CPU

WALL CLOCK

Stop Criteria

a

Default

= SOFT

Stop Criteria = SOFT

, the solver is allowed to stop prematurely with a suboptimal solution,

fail . code =

NW_NOT_CONVERGED, if it predicts that a better estimate of the solution cannot be reached. This is the recommended option.

Constraint:

Stop Criteria = SOFT

STRICT

Stop Tolerance 1

r

Default

= \max (10^{- 6}, \sqrt{ε})

This option defines

ε_{1}

used as a tolerance for the relative duality gap (10) and the relative precision (11), see Section 11.2.

Constraint:

Stop Tolerance 1 > ε

Stop Tolerance 2

r

Default

= \max (10^{- 7}, \sqrt{ε})

This option sets the value

ε_{2}

which is used for optimality (12) and complementarity (14) tests from KKT conditions or if

DIMACS Measures = Check

for all DIMACS error measures instead. See Section 11.2.

Constraint:

Stop Tolerance 2 > ε

Stop Tolerance Feasibility

r

Default

= \max (10^{- 7}, \sqrt{ε})

This argument places an acceptance limit on the feasibility of the solution (13),

ε_{feas}

. See Section 11.2.

Constraint:

Stop Tolerance Feasibility > ε

Task

a

Default

= MINIMIZE

This argument specifies the required direction of the optimization. If

Task = FEASIBLE POINT

, 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 was set,

Task

reverts to

FEASIBLE POINT

automatically.

Constraint:

Task = MINIMIZE

MAXIMIZE

FEASIBLE POINT

Transform Constraints

a

Default

= AUTO

This argument controls how equality constraints are treated by the solver. If

Transform Constraints = EQUALITIES

, all equality constraints

h_{k} (x) = 0

from (4) are treated as two inequalities

h_{k} (x) \leq 0

and

h_{k} (x) \geq 0

, see Section 11.4. This is the default and the only option in this release for equality constrained problems.

Constraint:

Transform Constraints = AUTO

NO

EQUALITIES

U Update Restriction

r

Default

= 0.5

This defines the value

μ_{g}

giving the bounds on the updates of Lagrangian multipliers for (standard) inequalities between the outer iterations. Values close to

1

limit the changes of the multipliers and serve as a kind of smoothing, lower values allow more significant changes.

Based on numerical experience, big variation in the multipliers may lead to a large number of iterations in the subsequent step and might disturb the convergence due to ill-conditioning.

It might be worth experimenting with the value on your particular problem. Mid range values are recommended over the more extremal ones.

Constraint:

ε < U Update Restriction < 1

Umat Update Restriction

r

Default

= 0.3

This is an equivalent of

U Update Restriction

for matrix constraints, denoted as

μ_{A}

in Section 11.1. The advice above applies equally.

Constraint:

ε < Umat Update Restriction < 1

nag_opt_handle_solve_pennon (e04svc) (PDF version)

e04 Chapter Contents

e04 Chapter Introduction

NAG Library Manual

NAG Library Function Documentnag_opt_handle_solve_pennon (e04svc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

3.1 Structure of the Lagrangian Multipliers

3.2 Approximation of the Lagrangian Multipliers

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

9.1 Description of the Printed Output

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 φg and φA

11.4 Solution of the inner problem

12 Optional Parameters

12.1 Description of the Optional Parameters

NAG Library Function Document

nag_opt_handle_solve_pennon (e04svc)

11.3 Choice of penalty functions $φ_{g}$ and $φ_{A}$