e04mtc (handle_solve_lp_ipm) : NAG Library CL Interface, Mark 29

Note: this function uses optional parameters to 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.

1 Purpose

e04mtc is a solver from the NAG optimization modelling suite for large-scale Linear Programming (LP) problems based on an interior point method (IPM).

2 Specification

The function may be called by the names: e04mtc or nag_opt_handle_solve_lp_ipm.

3 Description

e04mtc solves a large-scale linear optimization problem in the following form:

\begin{array}{l} \underset{x \in ℝ^{n}}{minimize} & c^{T} x & (a) \\ subject to & l_{A} \leq A x \leq u_{A} & (b) \\ l_{x} \leq x \leq u_{x}, & (c) \end{array}

(1)

where

n

is the number of decision variables and

m

is the number of linear constraints. Here

c

x

l_{x}

u_{x}

are

n

-dimensional vectors,

A

is an

m \times n

sparse matrix and

l_{A}

u_{A}

are

m

-dimensional vectors.

e04mtc implements two algorithmic variants of the interior point method for solving linear optimization problems: the infeasible Primal-Dual interior point method and homogeneous Self-Dual interior point method. In general, the Self-Dual algorithm has a slightly higher price per iteration, however, it is able to declare infeasibility or unboundness of the problem, whereas the Primal-Dual relies, in this case, on heuristics. For a detailed description of both algorithms see Section 11. The algorithm is chosen by the

LPIPM Algorithm

, the default is Primal-Dual.

e04mtc solves linear programming problems stored as a handle. The handle points to an internal data structure which defines the problem and serves as a means of communication for functions in the NAG optimization modelling suite. First, the problem handle is initialized by calling e04rac. Then some of the functions e04rec, e04rfc, e04rhc or e04rjc may be called to formulate the objective function, bounds of the variables, and the block of linear constraints, respectively. Once the problem is fully set, the handle may be passed to the solver. When the handle is not needed anymore, e04rzc should be called to destroy it and deallocate the memory held within. See Section 4.1 in the E04 Chapter Introduction for more details about the NAG optimization modelling suite.

The solver method can be modified by various optional parameters (see Section 12) which can be set by e04zmc and e04zpc anytime between the initialization of the handle by 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 optional parameters.

The optional parameter

Task

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

Several options may have significant impact on the performance of the solver. Even if the defaults were chosen to suit the majority of problems, it is recommended to experiment in order to find the most suitable set of options for a particular problem, see Sections 11 and 12 for further details.

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), denoted by

u

. The multipliers

u

are stored in the array u and conform to the structure of the constraints.

If the simple bounds have been defined (e04rhc was successfully called), the first

2 n

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

elements of u belong to multipliers for the linear constraints (if e04rjc has been successfully called). The organization is the same, i.e., the multipliers for each constraint for the lower and upper bounds are alternated and zeros are used for any missing (infinite bound) constraints.

Some solvers merge multipliers for both lower and upper inequality into one element whose sign determines the inequality. Negative multipliers are associated with the upper bounds and positive with the lower bounds. An equivalent result can be achieved with this storage scheme by subtracting the upper bound multiplier from the lower one. This is also consistent with equality constraints.

4 References

Andersen E D, Gondzio J, Mészáros C and Xu X (1996) Implementation of interior point methods for large scale linear programming HEC/Université de Genève

Colombo M and Gondzio J (2008) Further development of multiple centrality correctors for interior point methods Computational Optimization and Algorithms 41(3) 277–305

Goldfard D and Scheinberg K (2004) A product-form Cholesky factorization method for handling dense columns in interior point methods for linear programming Mathematical Programming 99(1) 1–34

Gondzio J (1996) Multiple centrality corrections in a primal-dual method for linear programming Computational Optimization and Algorithms 6(2) 137–156

Gondzio J (2012) Interior point methods 25 years later European Journal of Operations Research 218(3) 587–601

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

HSL (2011) A collection of Fortran codes for large-scale scientific computation http://www.hsl.rl.ac.uk/

Karypis G and Kumar V (1998) A fast and high quality multilevel scheme for partitioning irregular graphs SIAM J. Sci. Comput. 20(1) 359–392

Mészáros C (1996) The efficient implementation of interior point methods for linear programming and their applications PhD Thesis Eötvös Loránd University of Science, Budapest

Nocedal J and Wright S J (2006) Numerical Optimization (2nd Edition) Springer Series in Operations Research, Springer, New York

Wright S W (1997) Primal-dual interior point methods SIAM, Philadelphia

Xu X, Hung P-F and Ye Y (1996) A simplified homogeneous and self-dual linear programming algorithm and its implementation Annals of Operations Research 62(1) 151–171

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

The accuracy of the solution is determined by optional parameters

LPIPM Stop Tolerance

and

LPIPM Stop Tolerance 2

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 the solution. If

fail . code =

NW_NOT_CONVERGED, some of the convergence conditions were not fully satisfied but the point is a reasonable estimate and still usable. Please refer to Section 11.3 and the description of the particular options.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

e04mtc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

e04mtc 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 sent to two independent streams (files) which are set by optional parameters

Print File

and

Monitoring File

. Optional parameters

Print Level

Print Solution

and

Print Options

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

By default (

Print File = 6

Print Level = 2

), six sections are printed to the standard output:

Header
Optional parameters list
Problem statistics
Iteration log
Summary
Solution

The iteration log varies depending on which algorithm has been selected to solve the problem (Primal-Dual or Self-Dual).

The header is a message indicating the start of the solver. It should look like:

----------------------------------------------
 E04MT, Interior point method for LP problems
----------------------------------------------

Optional parameters list

The list shows all options of the solver, each displayed on one line. The output contains the option name, its current value and an indicator for how it was set. The options unchanged from the default setting are noted by ‘d’, options you set are noted by ‘U’, and options reset by the solver are noted by ‘S’. Note that the output format is compatible with the file format expected by e04zpc. The output might look as follows:

     Lp Presolve                   =                 Yes     * d
     Lpipm Algorithm               =         Primal-dual     * d
     Lpipm Centrality Correctors   =                  -6     * U
     Lpipm Iteration Limit         =                 100     * d

Problem statistics

Print Level \geq 2

, statistics on the original and the presolved problems are printed. More detailed statistics as well as a list of the presolve operations are also printed for

Print Level

3

or above, for example:

 Problem Statistics
   No of variables                  7
     free (unconstrained)           0
     bounded                        7
   No of lin. constraints           7
     nonzeroes                     41
   Objective function          Linear
 
 Presolved Problem Measures
   No of variables                 13
     free (unconstrained)           0
   No of lin. constraints           7
     nonzeroes                     47

Print Level \geq 2

, the solver prints the status of each iteration.

Primal-Dual algorithm

Print Level = 2

, the output shows the iteration number (

0

represents the starting point), the current primal and dual objective value, KKT measures (optimality, feasibility and complementarity), average complementarity error

μ

, number of centrality correction steps (MCC) performed and a column for additional information (I). Note that all these values are also available in rinfo and stats. The output might look as follows:

----------------------------------------------------------------------------
it|    pobj    |    dobj    |  optim  |  feas   |  compl  |   mu   | mcc | I
----------------------------------------------------------------------------
 0  2.02532E+03 -7.37272E+02  7.71E+00  4.91E+00  2.16E+00  4.4E+03
 1 -2.13398E+01 -1.62136E+04  5.82E-02  2.09E-01  2.12E+01  4.7E+02   2
 2 -1.09237E+02 -2.81254E+03  3.12E-03  4.84E-15  4.84E-01  5.3E+01   0
 3 -2.10923E+02 -6.07429E+02  4.61E-04  4.99E-14  3.70E-02  7.8E+00   0

Print Level = 3

, the solver also prints for each iteration the primal and dual steps as well as the maximum error of the backsolves performed. The output takes the following form:

----------------------------------------------------------------------------------------------------------
it|    pobj    |    dobj    |  optim  |  feas   |  compl  |   mu   |  pstep  |  dstep  |  errbs  | mcc | I
----------------------------------------------------------------------------------------------------------
 0  2.02532E+03 -7.37272E+02  7.71E+00  4.91E+00  2.16E+00  4.4E+03
 1 -2.13398E+01 -1.62136E+04  5.82E-02  2.09E-01  2.12E+01  4.7E+02  9.57E-01  9.92E-01  1.54E-12   2
 2 -1.09237E+02 -2.81254E+03  3.12E-03  4.84E-15  4.84E-01  5.3E+01  1.00E+00  9.46E-01  3.02E-12   0
 3 -2.10923E+02 -6.07429E+02  4.61E-04  4.99E-14  3.70E-02  7.8E+00  1.00E+00  8.53E-01  7.66E-11   0

Print Level = 2

, the output shows the iteration number (

0

represents the starting point), the current primal and dual objective value, convergence measures (primal infeasibility (12), dual infeasibility (13) and duality gap (14)), the value of the additional variable

τ

and the number of centrality correction steps performed. The output might look as follows:

----------------------------------------------------------------------------
it|    pobj    |    dobj    |  p.inf  |  d.inf  |  d.gap  |   tau  | mcc | I
----------------------------------------------------------------------------
 0  1.01907E+01  0.00000E+00  6.98E+02  1.12E+01  1.35E+01  1.0E+00
 1  5.80391E+00 -2.39478E+00  4.98E-04  3.11E-02  2.58E-02  3.5E-01   0
 2  7.09323E+00  3.62789E+00  1.89E-04  1.18E-02  9.79E-03  1.4E-01   0
 3 -1.33628E+01  5.87563E+00  1.94E-05  1.21E-03  1.01E-03  3.3E-02   0

Print Level = 3

, the solver also prints for each iteration

ρ_{A}

(15), the value of the variable

κ

, the step size as well as the maximum error of the backsolves performed. The output might look as follows:

------------------------------------------------------------------------------------------------------------------
it|    pobj    |    dobj    |  p.inf  |  d.inf  |  d.gap  |  rhoa  |   tau  |  kappa |   step  |  errbs  | mcc | I
------------------------------------------------------------------------------------------------------------------
 0  1.01907E+01  0.00000E+00  6.98E+02  1.12E+01  1.35E+01  1.0E+01
 1  5.80391E+00 -2.39478E+00  4.98E-04  3.11E-02  2.58E-02  2.4E+00  3.5E-01  1.0E+00  6.52E-01  3.43E-13   0
 2  7.09323E+00  3.62789E+00  1.89E-04  1.18E-02  9.79E-03  7.5E-01  1.4E-01  9.8E-01  6.21E-01  2.85E-13   0
 3 -1.33628E+01  5.87563E+00  1.94E-05  1.21E-03  1.01E-03  2.8E+00  3.3E-02  7.9E-01  8.97E-01  9.31E-13   0

Occasionally, when numerical instabilities are too big, the solver will restart the iteration and switch to an augmented system formulation. In such cases the letters RS will be printed in the information column (I).

Print Level > 3

, for both the Primal-Dual and the Self-Dual algorithms, each iteration produces more information that expands over several lines. This additional information contains:

The method used (normal equation, augmented system);
The centring parameter $σ$ ;
The total number of iterative refinements performed;
The number of iterative refinements performed in the centrality correction steps;
The number of factorizations performed at the current iteration;
The type of factorization performed (Cholesky, Bunch–Parlett);
The value of the perturbation added to the diagonal in the normal equation formulation or on the zero block in the augmented system formulation;
The total time spent in the iteration if $Stats Time$ is not set to NO.

The output might look as follows:

----------- Details of Iteration   1 ------------
method                            Normal Equation
sigma                                    6.72E-02
iterative refinements                           0
iterative refinements wmcc                      0
factorizations                                  1
matrix type                              Cholesky
diagonal perturbation                    0.00E+00
time iteration                           0.02 sec

Once the solver finishes, a detailed summary is produced:

-------------------------------------------------
Status: converged, an optimal solution found
-------------------------------------------------
Final primal objective value        -4.647531E+02
Final dual objective value          -4.647531E+02
Absolute primal infeasibility        1.605940E-15
Relative primal infeasibility        1.210247E-16
Absolute dual infeasibility          4.272004E-12
Relative dual infeasibility          6.068111E-15
Absolute complementarity gap         9.387105E-07
Relative complementarity gap         2.507021E-10
Iterations                                      7

It starts with the status line of the overall result which matches the fail value and is followed by the final primal and dual objective values as well as the error measures and iteration count.

Optionally, if

Stats Time

is set, the timings of the different parts of the algorithm are displayed. It might look as follows:

Timing
  Total time                            28.78 sec
  Presolver                              0.07 sec   (  0.2%)
  Core                                  28.71 sec   ( 99.8%)
    Initialization                       1.67 sec   (  5.8%)
    Factorization                       16.18 sec   ( 56.5%)
    Compute directions                   5.29 sec   ( 18.5%)
    Weighted MCC                         5.50 sec   ( 19.2%)
  Iterative refinement                   0.24 sec   (  0.8%)

Print Solution = X

, the values of the primal variables and their bounds on the primary and secondary outputs. It might look as follows:

Primal variables:
  idx   Lower bound        Value      Upper bound
    1  -1.00000E-02   -1.00000E-02    1.00000E-02
    2  -1.00000E-01   -1.00000E-01    1.50000E-01
    3  -1.00000E-02    3.00000E-02    3.00000E-02
    4  -4.00000E-02    2.00000E-02    2.00000E-02
    5  -1.00000E-01   -6.74853E-02    5.00000E-02
    6  -1.00000E-02   -2.28013E-03         inf
    7  -1.00000E-02   -2.34528E-04         inf

Print Solution = YES

ALL

, the values of the dual variables are also printed. It should look as follows:

Box bounds dual variables:
  idx   Lower bound        Value      Upper bound        Value
    1  -1.00000E-02    3.30098E-01    1.00000E-02    0.00000E+00
    2  -1.00000E-01    1.43844E-02    1.50000E-01    0.00000E+00
    3  -1.00000E-02    0.00000E+00    3.00000E-02    9.09967E-02
    4  -4.00000E-02    0.00000E+00    2.00000E-02    7.66124E-02
    5  -1.00000E-01    4.92258E-12    5.00000E-02    0.00000E+00
    6  -1.00000E-02    2.42274E-11         inf       0.00000E+00
    7  -1.00000E-02    4.83752E-12         inf       0.00000E+00

Linear constraints dual variables:
  idx   Lower bound        Value      Upper bound        Value
    1  -1.30000E-01    0.00000E+00   -1.30000E-01    1.43111E+00
    2       -inf       0.00000E+00   -4.90000E-03    4.07810E-10
    3       -inf       0.00000E+00   -6.40000E-03    5.64870E-10
    4       -inf       0.00000E+00   -3.70000E-03    1.25984E-10
    5       -inf       0.00000E+00   -1.20000E-03    1.87338E-11
    6  -9.92000E-02    1.50098E+00         inf       0.00000E+00
    7  -3.00000E-03    1.51661E+00    2.00000E-03    0.00000E+00

9.2 Internal Changes

Internal changes have been made to this function as follows:

At Mark 28.7:
Updates that in general provide overall performance enhancements.

For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.

10 Example

This example demonstrates how to use e04mtc to solve a small LP problem with the two algorithms implemented (Primal-Dual and Self-Dual). The solver is called twice on the same handle with different values of optional parameters.

We solve the following linear programming problem:

minimize

- 0.02 x_{1} - 0.2 x_{2} - 0.2 x_{3} - 0.2 x_{4} - 0.2 x_{5} + 0.04 x_{6} + 0.04 x_{7}

subject to the bounds

\begin{matrix} - 0.01 \leq x_{1} \leq 0.01 \\ - 0.1 \leq x_{2} \leq 0.15 \\ - 0.01 \leq x_{3} \leq 0.03 \\ - 0.04 \leq x_{4} \leq 0.02 \\ - 0.1 \leq x_{5} \leq 0.05 \\ - 0.01 \leq x_{6} \\ - 0.01 \leq x_{7} \end{matrix}

and the general constraints

\begin{matrix} x_{1} & + & x_{2} & + & x_{3} & + & x_{4} & + & x_{5} & + & x_{6} & + & x_{7} & = & - 0.13 \\ 0.15 x_{1} & + & 0.04 x_{2} & + & 0.02 x_{3} & + & 0.04 x_{4} & + & 0.02 x_{5} & + & 0.01 x_{6} & + & 0.03 x_{7} & \leq & - 0.0049 \\ 0.03 x_{1} & + & 0.05 x_{2} & + & 0.08 x_{3} & + & 0.02 x_{4} & + & 0.06 x_{5} & + & 0.01 x_{6} & \leq & - 0.0064 \\ 0.02 x_{1} & + & 0.04 x_{2} & + & 0.01 x_{3} & + & 0.02 x_{4} & + & 0.02 x_{5} & \leq & - 0.0037 \\ 0.02 x_{1} & + & 0.03 x_{2} & + & 0.01 x_{5} & \leq & - 0.0012 \\ - 0.0992 & \leq & 0.70 x_{1} & + & 0.75 x_{2} & + & 0.80 x_{3} & + & 0.75 x_{4} & + & 0.80 x_{5} & + & 0.97 x_{6} \\ - 0.003 & \leq & 0.02 x_{1} & + & 0.06 x_{2} & + & 0.08 x_{3} & + & 0.12 x_{4} & + & 0.02 x_{5} & + & 0.01 x_{6} & + & 0.97 x_{7} & \leq & 0.002 . \end{matrix}

10.1 Program Text

Program Text (e04mtce.c)

10.2 Program Data

Program Data (e04mtce.d)

10.3 Program Results

Program Results (e04mtce.r)

11 Algorithmic Details

This section contains the description of the underlying algorithms used in e04mtc, which implements the infeasible Primal-Dual and homogeneous Self-Dual methods. For further details, see Andersen et al. (1996), Gondzio (2012), Mészáros (1996) and Wright (1997).

For simplicity, we consider the following primal linear programming formulation

\begin{array}{l} \underset{x \in ℝ^{n}}{minimize} & c^{T} x & (a) \\ subject to & A x = b & (b) \\ x \geq 0 & (c) \end{array}

(2)

where

c

x \in ℝ^{n}

b \in ℝ^{m}

and

A \in ℝ^{m \times n}

with full row rank. The dual formulation for (2) is given by

\begin{array}{l} \underset{y \in ℝ^{m}, z \in ℝ^{n}}{maximize} & b^{T} y & (a) \\ subject to & A^{T} y + z = c & (b) \\ z \geq 0 & (c) \\ y ​ (free) & (d) \end{array}

(3)

where

y

and

z

denote the dual variables. Solutions of the primal (2) and dual (3) problem are connected by the strong duality theory (see for example, Nocedal and Wright (2006)) and are characterized by the first order optimality conditions, the so-called Karush–Kuhn–Tucker (KKT) conditions, which are stated as follows:

\begin{matrix} A^{T} y + z & = & c & (dual feasibility, optimality) \\ A x & = & b & (primal feasibility) \\ X Z e & = & 0 & (complementarity) \\ (x, z) & \geq & 0 \end{matrix}

(4)

where we define

X

and

Z

as the diagonal matrices with the elements

x_{i}

and

z_{i}

, respectively, and

e = {(1, 1, \dots, 1)}^{T}

as the

n

-vector of ones.

The underlying algorithm applies an iterative method to find an optimal solution

(x^{*}, y^{*}, z^{*})

of the system (4) employing variants of Newton's method and modifying the search directions and step lengths so that the non-negative constraints are preserved at every iteration.

11.1 The Infeasible-interior-point Primal-Dual Algorithm

A direct solution of the nonlinear system of equations (4) by Newton's method is impractical as it exhibits slow progress towards the solution. Instead, a sequence of the following perturbed KKT conditions are solved so that the complementarity is driven to zero through the iteration sequence

\begin{matrix} A x & = & b \\ A^{T} y + z & = & c \\ X Z e & = & σ μ e \\ (x, z) & > & 0 . \end{matrix}

(5)

Here

μ

is the average complementarity error at the current iteration

μ = x^{T} z / n

, called duality measure, and

σ \in (0, 1)

, called centring parameter, is the reduction factor that we wish to achieve in the duality measure.

Each iteration of the Primal-Dual algorithm makes one step of Newton's method applied to the perturbed first order optimality conditions (5) with a given

σ

and

μ

. In particular, a Newton search direction is computed by solving a system of linear equations and a length of the step

α

is determined so that the bounds

(x, z) > 0

are not violated. The residual of (4) and

μ

define stopping criteria and the algorithm terminates once they are reduced to the requested accuracy, see Section 11.3.

Given an

x

z \in ℝ_{+}^{n}

and

y \in ℝ^{m}

, Newton's direction is obtained by solving the following system:

(\begin{matrix} A & 0 & 0 \\ 0 & A^{T} & I \\ Z & 0 & X \end{matrix}) (\begin{matrix} Δ x \\ Δ s \\ Δ z \end{matrix}) = (\begin{matrix} r_{b} \\ r_{c} \\ μ e - X Z e \end{matrix}),

(6)

where

\begin{matrix} r_{b} = b - A x, & (Primal infeasibilities) \\ r_{c} = c - A^{T} y - z & (Dual infeasibilities) \end{matrix}

denote the violations of the primal and the dual constraints, respectively. Primal and dual infeasibilities,

r_{b}

and

r_{c}

, are reduced at the same rate

(1 - α)

, given a step size

α \in (0, 1)

. The Primal-Dual algorithm does not require feasibility of the solutions during the optimization process. Feasibility is attained during the process as optimality is approached.

Once the system is solved,

Δ x

and

Δ z

are used to compute the maximum step sizes in primal space

(α_{P})

and dual space (

α_{D}

) such that the non-negativity of variables is preserved

\begin{array}{l} α_{P} = \min {1, ρ \cdot \max {α \geq 0 : x^{k} + α Δ x^{k} \geq 0}} \\ α_{D} = \min {1, ρ \cdot \max {α \geq 0 : z^{k} + α Δ z^{k} \geq 0}} \end{array},

where

0 < ρ < 1

is a reduction parameter close to

1

, typically

ρ \in (0.9, 1)

. The next iterate is updated as follows:

\begin{array}{l} x^{k + 1} & \leftarrow & x^{k} + α_{P} Δ x^{k} \\ (y^{k + 1}, z^{k + 1}) & \leftarrow & (y^{k}, z^{k}) + α_{D} (Δ y^{k}, Δ z^{k}) \end{array}

Finally, the barrier parameter

μ

is decreased by a given factor and the process is repeated until the stopping criteria (see Section 11.3) or maximum number of iterations is reached.

11.1.1 The Barrier Method

Note that there is also another way to obtain the perturbed KKT conditions (5). They can be derived starting from the primal formulation (2) and replacing the non-negativity constraints

x \geq 0

by a logarithmic barrier term

\sum_{j = 1}^{n} \log x_{j}

with a barrier parameter

τ > 0

. This approach leads to the primal logarithmic barrier problem defined as

\begin{array}{l} \underset{x \in ℝ^{n}}{minimize} & c^{T} x - τ \sum_{j = 1}^{n} \log x_{j} \\ subject to & A x = b . \end{array}

(7)

The Lagrangian formulation for (7) is

L (x, y, τ) = c^{T} x - τ \sum_{j = 1}^{n} \log x_{j} - y^{T} (A x - b),

and the first order optimality conditions for problem (7) are

\begin{matrix} \nabla_{x} L & = & c - τ X^{−1} e - A^{T} y = 0 \\ \nabla_{y} L & = & b - A x = 0, \\ x & > & 0 (Barrier avoids x = 0) . \end{matrix}

Finally, by introducing

Z = τ X^{−1}

and

z = Z e

the same system of equations (5) is formulated.

11.2 Homogeneous Self-Dual Algorithm

A homogeneous Self-Dual (HSD) embedding of the primal linear programming and its dual was proposed in Xu et al. (1996). As its name suggest, the HSD and its dual are equivalent. Self-Dual formulations embed the original problem (2) in a larger linear programming problem such that the latter is primal and dual feasible, with known feasible points, and from which solution we can extract optimal solutions or certificates of infeasibility of the original problem.

We define the homogeneous and Self-Dual linear feasibility (HLF) model as follows:

\begin{array}{l} \underset{x, z \in ℝ^{n}, y \in ℝ^{m}, τ, κ \in ℝ}{minimize} & 0 \\ subject to & A x - b τ = 0 \\ A^{T} y + z - c τ = 0 \\ - c^{T} x + b^{T} y - κ = 0 \\ x, τ, z, κ \geq 0, y ​ (free) \end{array},

(8)

where

τ

and

κ

are two additional variables. The model (8) is a Self-Dual linear programming problem with zero right-hand side and a zero objective vector. If

(\hat{x}, \hat{τ}, \hat{y}, \hat{z}, \hat{κ})

is a strictly complementarity solution for (8), then if

\hat{τ} > 0

, the linear programming problem has an optimal solution given by

(x^{*}, y^{*}, z^{*}) = (\hat{x}, \hat{y}, \hat{z}) / \hat{τ},

and the duality gap is given by

c^{T} x^{*} - b^{T} y^{*} = \hat{κ} / \hat{τ} = 0

. The homogeneous algorithm is an application of the Primal-Dual method for the computation of a strictly complementarity solution to (8).

Homogeneous and Self-Dual interior point methods have several advantages besides an inherent ability to detect infeasibility (which improves the detection of divergence in Primal-Dual algorithms). The HSD model includes the ease of finding a suitable starting point and it is generally more robust in the presence of free variables. However, some disadvantages need to be noted: HSD is larger than the original problem. In particular, it increases the number of linear equations solved per iteration by one, requiring an extra backsolve step, which make it slightly slower than the Primal-Dual algorithm. Moreover, numerical experiments indicate that the required number of iterations on feasible problems might be slightly increased.

11.3 Stopping Criteria

11.3.1 Convergence – Optimal Termination

11.3.1.1 Primal-Dual algorithm

The Primal-Dual algorithm is stopped when the first order optimality conditions are satisfied to the requested accuracy. These conditions are relative primal and dual feasibility and duality gap, defined as:

relative dual feasibility

$\frac{‖ A^{T} y + z - c ‖}{1 + ‖ c ‖} \leq ε_{1}$ (9)
relative primal feasibility

$\frac{‖ A x - b ‖}{1 + ‖ b ‖} \leq ε_{1}$ (10)
relative duality gap

$\frac{μ}{1 + | c^{T} x |} \leq ε_{1}$ (11)

Furthermore, final absolute primal and dual infeasibilities, and duality gap are also returned. Here

ε_{1}

may be set using

LPIPM Stop Tolerance

and the norm denotes the 2-norm.

11.3.1.2 Self-Dual algorithm

Similar to the Primal-Dual algorithm, the homogeneous Self-Dual algorithm is stopped when the following measures are satisfied to the requested accuracy.

relative primal feasibility

$\frac{‖ b τ - A x ‖}{\max (1, ‖ b τ_{0} - A x_{0} ‖)} \leq ε_{1}$ (12)
relative dual feasibility

$\frac{‖ τ c - A^{T} y - z ‖}{\max (1, ‖ c τ_{0} - A^{T} y_{0} - z_{0} ‖)} \leq ε_{1}$ (13)
relative duality gap

$\frac{‖ κ + c^{T} x - b^{T} y ‖}{\max (1, | κ_{0} + c^{T} x_{0} - b^{T} y_{0} |)} \leq ε_{1}$ (14)

which measure the relative reduction in the primal, dual and gap infeasibility, respectively. In addition, an extra measure is considered to quantify the accuracy in the objective function, which is given by

ρ_{A} = \frac{| c^{T} x - b^{T} y |}{τ + | b^{T} y |} \leq ε_{2} .

(15)

Here

ε_{1}

and

ε_{2}

may be set using

LPIPM Stop Tolerance

and

LPIPM Stop Tolerance 2

, respectively.

Premature termination is triggered if the current iteration exhibits fast convergence and the optimality measures lie within a small range. In particular, the Self-Dual algorithm is stopped if the above termination conditions are met within a small factor and

τ > 1000 κ

. This measure is tracked after the first

10

iterations.

11.3.2 Infeasibility/Unboundedness Detection

11.3.2.1 Primal-Dual algorithm

The Primal-Dual algorithm detects infeasible problems fairly reliably by using a set of heuristics. When several of these heuristics classify the problem as infeasible throughout a sufficient number of iterations, the algorithm is stopped.

Note that in order to obtain a certificate of infeasibility, the use of homogeneous Self-Dual algorithm is recommended, see Section 11.3.2.2.

11.3.2.2 Self-Dual algorithm

The problem is concluded to be primal or dual infeasible if one of the following conditions hold:

1.Both the relative primal (12) and dual (13) feasibility of the HLF model (8) are satisfied and the value of $τ$ satisfies

$τ \leq ε_{2} \max (1, κ)$
2.or if the following inequalities hold

$μ \leq ε_{2} μ_{0}$

$τ \leq ε_{2} \min (1, κ) .$

Then the problem is declared dual infeasible if

c^{T} x < 0

or primal infeasible otherwise.

11.3.3 Suboptimal Solution

The solver stops prematurely and reports suboptimal solution when it predicts that the current estimate of the solution will not be improved in subsequent iterations. In most cases the returned solution should be acceptable.

11.4 Solving the KKT System

The solution of the Newton system of equations (6) is the most computationally costly operation. In practice, system (6) is reduced to the augmented system by eliminating

Δ z

from the last block of equations as follows:

(\begin{matrix} - D^{2} & A^{T} \\ A & 0 \end{matrix}) (\begin{matrix} Δ x \\ Δ y \end{matrix}) = (\begin{matrix} r \\ h \end{matrix}),

(16)

where

\begin{matrix} D^{2} & = & Z^{−1} X \\ r & = & r c - X^{−1} (μ e - X Z e) \\ h & = & r_{b} \end{matrix}

and

Δ z = {- X}^{−1} (μ e - X Z e) - X^{−1} Z Δ x

. This is a system of

m + n

variables and constraints, symmetric and indefinite. Submatrix

D

is diagonal and positive definite.

The system (16) can be reduced further by eliminating

Δ x

, to a positive definite system usually called normal equations defined as

(A D^{2} A^{T}) Δ y = A D^{2} r + h

(17)

and

\begin{matrix} Δ z & = & - r c - A^{T} Δ y \\ Δ x & = & {- Z}^{−1} (μ e - X Z e) - X Z^{−1} Δ z . \end{matrix}

Typically, formulation (17) is preferred for many problems as the system matrix can be factorized by a sparse Cholesky. However, this brings some well-known disadvantages: Ill-conditioning of the system is often observed during the final stages of the algorithm, and free (unbounded) variables require certain modifications. If matrix

A

contains dense columns (columns with relatively many nonzeros), then

A D^{2} A^{T}

has many nonzeros, which in turn makes the factorization expensive. On the other hand, solving the augmented system is usually slower, but it normally avoids the fill-in caused by dense columns and can handle free variables directly in the formulation.

e04mtc can detect and handle dense columns effectively: depending on the number and the density of the ‘dense’ columns, the solver may either choose to directly use an augmented system formulation or to treat these columns separately in a product-form Cholesky factorization as described in Goldfard and Scheinberg (2004). It is also possible to manually override the automatic choice via the optional parameter

LPIPM System Formulation

and let the solver use a normal equations or an augmented system formulation.

Badly scaled optimal solutions may present numerical challenges, therefore, iterative refinement using mixed-precision is employed for reducing the roundoff errors produced during the solution of the system. When the condition number of the system

A D^{2} A^{T}

prevents the satisfactory use of iterative refinement, e04mtc switches automatically to an augmented system formulation, reporting RS (Restart) in the last column of the iteration log (I). Furthermore, e04mtc provides several scaling techniques to adjust the numerical characteristics of the problem data, see

LPIPM Scaling

Finally, factorization of the system matrix can degrade sparsity, so the resulting fill-in can be large, therefore, several ordering techniques are included to minimize it. e04mtc uses Harwell packages MA97 (see Hogg and Scott (2011) and HSL (2011)) for the underlying sparse linear algebra factorization and MC68 approximate minimum degree algorithm, and METIS (Karypis and Kumar (1998)) nested dissection algorithm for the ordering.

11.5 Weighted Multiple Centrality Correctors

As previously stated, the factorization of the system at every iteration usually accounts for most of the computation time, therefore, it is always desirable to reuse the factors if possible and to reduce the total number of iterations. An efficient computational method is obtained by splitting the computation of the Newton direction into two steps, namely the affine-scaling direction and its correction step, called Mehrotra's predictor-corrector. However, Mehrotra's predictor-corrector technique aims to correct the affine-direction in a full step, which is considered an aggressive approach.

e04mtc implements a high-order method, called weighted multiple centrality correctors (WMCC), see Gondzio (1996) and Colombo and Gondzio (2008). This technique attempts to correct the affine-direction recursively as long as the step sizes increase at least by a fraction of a given aspiration level, up to a maximum number of times. The heuristic to determine the maximum number of WMCC is based on the ratio between the cost of the factorization and that of backsolving and, therefore, may lead to non-repeatable results. To avoid an undesired behaviour, you can fix the maximum number of WMCC with optional parameter

LPIPM Centrality Correctors

11.6 Further Details

e04mtc includes an advance preprocessing phase (called presolve) to reduce the dimensions of the problem before passing it to the solver. The reduction in problem size generally improves the behaviour of the solver, shortening the total computation time. In addition, infeasibility may also be detected during preprocessing. The default behaviour of the presolve can be modified by

LP Presolve

The initial point

x_{0}

is always computed using heuristics. Effective warm-starting strategies for interior point methods are still subject to intensive academic research, and it is not available in this release.

12 Optional Parameters

Several optional parameters in e04mtc define choices in the problem specification or the algorithm logic. In order to reduce the number of formal arguments of e04mtc 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 e04zmc 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

AUTO

), their value can be retrieved by 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:

All options accept the value

DEFAULT

to return single options to their default states.

Keywords and character values are case and white space insensitive.

This special keyword may be used to reset all optional parameters to their default values. Any value given with this keyword will be ignored.

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

This parameter allows you to reduce the level of presolving of the problem or turn it off completely. If the presolver is turned off, the solver will try to handle the problem you have given. In such a case, the presence of fixed variables or linear dependencies in the constraint matrix can cause numerical instabilities to occur. In normal circumstances, it is recommended to use the full presolve which is the default.

Constraint:

LP Presolve = FULL

BASIC

NO

As described in Section 11, e04mtc implements the infeasible Primal-Dual algorithm, see Section 11.1, and the homogeneous Self-Dual algorithm, see Section 11.2. This parameter controls which one to use.

Constraint:

LPIPM Algorithm = PRIMAL-DUAL

PD

SELF-DUAL

SD

This parameter controls the number of centrality correctors (see Section 11.5) used at each iteration. Each corrector step attempts to improve the current iterate for the price of additional solve(s) of the factorized system matrix in order to reduce the total number of iterations. Therefore, it trades the additional solves of the system with the number of factorizations. The more expensive the factorization is with respect to the solve, the more corrector steps should be allowed.

i > 0

, the maximum number of corrector steps will be computed by timing heuristics (the ratio between the times of the factorization and the solve in the first iteration) but will not be greater than

i

. The number computed by the heuristic can be recovered after the solve or during a monitoring step in stats. This might cause non-repeatable results.

i < 0

, the maximum number of corrector steps will be set to

| i |

If it is set to

0

, no additional centrality correctors will be used and the algorithm reverts to Mehrotra's predictor-corrector.

The maximum number of iterations to be performed by e04mtc. Setting the option too low might lead to

fail . code =

NE_TOO_MANY_ITER.

Constraint:

LPIPM Iteration Limit \geq 1

This parameter controls the maximum number of iterative refinement iterations (see Section 11.4) used at each main iteration when

LPIPM System Formulation = Normal Equations

. When solving the Normal Equations linear system for numerically challenging problems, mixed-precision iterative refinement may be used until the roundoff errors are reduced to an acceptable level or until the number of refinements reached its maximum value set by this parameter.

Constraint:

LPIPM Max Iterative Refinement \geq 0

This parameter controls the type of scaling to be applied on the constraint matrix

A

before solving the problem. More precisely, the scaling procedure will try to find diagonal matrices

D_{1}

and

D_{2}

such that the values in

D_{1} A D_{2}

are of a similar order of magnitude. The solver is less likely to run into numerical difficulties when the constraint matrix is well scaled.

Constraint:

LPIPM Scaling = ARITHMETIC

GEOMETRIC

NONE

This parameter defines the frequency of how often function monit is called. If

i > 0

, the solver calls monit at the end of every

i

th iteration. If it is set to

0

, the function is not called at all.

Constraint:

LPIPM Monitor Frequency \geq 0

This parameter sets the value

ε_{1}

which is the tolerance for the convergence measures in the stopping criteria, see Section 11.3.

Constraint:

LPIPM Stop Tolerance > ε

This parameter sets the additional tolerance

ε_{2}

used in the stopping criteria for the Self-Dual algorithm, see Section 11.3.

Constraint:

LPIPM Stop Tolerance 2 > ε

As described in Section 11.4, e04mtc can internally work either with the normal equations formulation (17) or with the augmented system (16). A brief discussion of advantages and disadvantages is presented in Section 11.4. Option

AUTO

leaves the decision to the solver based on the structure of the constraints and it is the recommended option. This will typically lead to the normal equations formulation unless there are many dense columns or the system is significantly cheaper to factorize as the augmented system. Note that in some cases even if

LPIPM System Formulation = NORMAL EQUATIONS

the solver might switch the formulation through the computation to the augmented system due to numerical instabilities or computational cost.

Constraint:

LPIPM System Formulation = AUTO

AUGMENTED SYSTEM

AS

NORMAL EQUATIONS

NE

(See Section 3.1.1 in the Introduction to the NAG Library CL Interface for further information on NAG data types.)

i \geq 0

, the Nag_FileID number (as returned from 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 if set by $Print Options$ ;
–problem statistics, the iteration log and the final status as set by $Monitoring Level$ ;
–the solution if set by $Print Solution$ .

Constraint:

Monitoring File \geq −1

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

Constraint:

0 \leq Monitoring Level \leq 5

(See Section 3.1.1 in the Introduction to the NAG Library CL Interface for further information on NAG data types.)

i \geq 0

, the Nag_FileID number (as returned from 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 $Print Level$ ;
–the solution if set by $Print Solution$ .

Constraint:

Print File \geq −1

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

$i$	Output
$0$	No output from the solver
$1$	Only the final status and the primal and dual objective value
$2$	Problem statistics, one line per iteration showing the progress of the solution with respect to the convergence measures, final status and statistics
$3$	As level $2$ but each iteration line is longer, including step lengths and errors
$4, 5$	As level $3$ but further details of each iteration are presented

$0$	Number of iterations.
$1$	Total number of centrality correction steps performed.
$2$	Total number of iterative refinements performed.
$3$	Value of the perturbation added to the diagonal in the normal equation formulation or on the zero block in the augmented system formulation.
$4$	Total number of factorizations performed.
$5$	Total time spent in the solver.
$6$	Time spent in the presolve phase.
$7$	Time spent in the last iteration.
$8$	Total time spent factorizing the system matrix.
$9$	Total time spent backsolving the system matrix.
$10$	Total time spent in the multiple centrality correctors phase.
$11$	Time spent in the initialization phase.
$12$	Number of nonzeros in the system matrix.
$13$	Number of nonzeros in the system matrix factor.
$14$	Maximum error of the backsolve.
$15$	Number of columns in $A$ considered dense by the solver.
$16$	Maximum number of centrality corrector steps.
$17$ – $99$	Reserved for future use.

NAG CL Interface
e04mtc (handle_solve_lp_ipm)

▸▿ Contents

1 Purpose

2 Specification

3 Description

3.1 Structure 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

9.2 Internal Changes

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

11 Algorithmic Details

11.1 The Infeasible-interior-point Primal-Dual Algorithm

11.1.1 The Barrier Method

11.2 Homogeneous Self-Dual Algorithm

11.3 Stopping Criteria

11.3.1 Convergence – Optimal Termination

11.3.1.1 Primal-Dual algorithm

11.3.1.2 Self-Dual algorithm

11.3.2 Infeasibility/Unboundedness Detection

11.3.2.1 Primal-Dual algorithm

11.3.2.2 Self-Dual algorithm

11.3.3 Suboptimal Solution

11.4 Solving the KKT System

11.5 Weighted Multiple Centrality Correctors

11.6 Further Details

12 Optional Parameters

12.1 Description of the Optional Parameters

NAG CL Interfacee04mtc (handle_​solve_​lp_​ipm)

▸▿ Contents

1 Purpose

2 Specification

3 Description

3.1 Structure 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

9.2 Internal Changes

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

11 Algorithmic Details

11.1 The Infeasible-interior-point Primal-Dual Algorithm

11.1.1 The Barrier Method

11.2 Homogeneous Self-Dual Algorithm

11.3 Stopping Criteria

11.3.1 Convergence – Optimal Termination

11.3.1.1 Primal-Dual algorithm

11.3.1.2 Self-Dual algorithm

11.3.2 Infeasibility/Unboundedness Detection

11.3.2.1 Primal-Dual algorithm

11.3.2.2 Self-Dual algorithm

11.3.3 Suboptimal Solution

11.4 Solving the KKT System

11.5 Weighted Multiple Centrality Correctors

11.6 Further Details

12 Optional Parameters

12.1 Description of the Optional Parameters

NAG CL Interface
e04mtc (handle_solve_lp_ipm)