This documentation consists of four parts.

- Introduction

We will start with the question why at all do we need such tools. What can go wrong when we solve a linear program? What can a verification tool do for us in these cases? - Theory

After that we look at the mathematics behind the computations done in Lurupa. What happens in the algorithms and how can we make sure that the computed bounds are rigorously, including rounding errors. - Usage

The usage of Lurupa has to be divided into two parts.- Stand-alone usage

In this section we will investigate the usage as a stand-alone command line tool. - API usage

Here we look at the usage through the API.

- Stand-alone usage
- License terms

Finally the terms under which you are allowed to use Lurupa.

While the infeasibilty of this problem can be easily seen, many linear programming solvers like CPLEX 9.000, lp_solve 5.5, and MATLAB R14SP3's linprog will regard this problem as feasible. With the default accuracy they will happily return with an optimal solution not even displaying a warning.

A similar construction for an unbounded problem,

fools CPLEX and lp_solve into returning 0 as the optimal solution for the variables and thus as the optimal function value.

Given, these examples are very simple. They are just used to illustrate that even in these dimensions problems may arrise. Adjusting the accuracy of the solvers would enable them to return the correct results. One may ask if 0.999 is different from 1? Is 0.999999 different? What about 0.999999999999? If not the solvers are correct. The values used above, however, are recognized by the solvers as being different from 1.

Going to higher dimensions the problem becomes more subtle. The dependencies between the constraints cannot be seen easily. Let us look at a problem from the Netlib lp library [7] , a free collection of insdustrial and academical linear programs. With 56 constraints and 97 variables *adlittle* is one of its smaller members. While being in fact feasible, *adlittle* suffers from ill--posedness. Perturbing the right hand side of the equality constraints by subtracting a tiny multiple of the 96th column of the equation matrix renders the linear program infeasible. Running this problem through CPLEX and lp_solve does again return a solution without any warnings.

Now we will have a look at what Lurupa returns when confronted with these problems. Using lp_solve to compute approximate solutions as a start off, Lurupa returns for the primal infeasible problem the optimal value interval . While the lower bound does not help us much here, the upper bound tells us, that the problem is at least very close to infeasibility as Lurupa could not find verified feasible points. What this means is explained in the next section. Computing bounds for the second, unbounded problem Lurupa returns the interval . Here we can pinpoint the unboundedness in the infinite lower bound. Using the final example as input, Lurupa computes the bounds (we need the `resetbas`

option of the solver module in this case to deal with the bad numerics). Again the upper bound indicates the problem being at least very close to infeasibility.

On the other hand let us have a look at what Lurupa does when confronted with a feasible lp. Taking *adlittle* again, now with the original coefficients, Lurupa returns the interval for the optimal value. This does not only give a verified interval containing the optimal value, but also verifies that feasible solutions in fact exist. This problem cannot be infeasible or unbounded, as the lower and upper bounds prove the existence of dual and primal feasible points, respectively.

Results of Lurupa being applied to the whole selection of Netlib problems is contained in [5] .

Let us have a look at a simple example. We use a decimal floating point arithmetic with only 2 digits mantissa and compute

Here means the floating point evaluation of . The 0.1 is completely annihilated by rounding the result to 2 digits. Repeating this using an interval arithmetic gives us the following

We arrive at a guaranteed enclosure, 10.1 is certainly between 10 and 11.

But how do we get there? The equation gives a hint how to arrive at this result. The interval enclosure of the expression is formed by the lower bound and the upper bound . These are computed using directed rounding towards minus infinity and plus infinity, respectively.

Having arrived at intervals we now have to continue our computations using these. The objective stays just the same. Compute an interval that is guaranteed to contain the correct result using directed rounding. As we do not know which number in the intervals is the correct one, we have to take all into account. If we have to divide the result of the previous example by , this means

The lower bound is the result of rounding 10 over 3 downwards, and the upper bound stems from rounding 11 over 2 upwards.

For a more thorough introduction to interval arithmetic have a look at Kearfott [4] and Hayes [2] and the references therein. The interval computations homepage [6] is also a good starting point.

we use a standard linear programming solver to compute an approximate solution . We make no assumptions about the accuracy of the computed approximations. A good approximation, however, usually results in a sharper enclosure of the optimal value.

The next step is to enforce the simple bounds . We achieve this by setting the components of that violate their corresponding bound to the violated bound. If or , we set or , respectively.

Now we use an interval linear system solver to compute a rigorous enclosure of the solution set of . This step transforms the point vector into an interval vector , which is verified to contain at least one point satisfying the equations.

Finally we just have to check if the inequalities and the simple bounds are satisfied by and thus by all points contained in the interval vector. If this is the case, contains at least one point that satisfies all constraints. Taking the maximum of the objective function over the box , that is over the objective value of all points in the interval vector, we arrive at an objective value that is verified to be larger than the one of a feasible point. This in turn is larger than the optimal value, which leaves us with an upper bound on the optimal value.

If does not satisfy the constraints, the iteration starts. We perturb the linear program by lowering the right hand sides of the inequalities and tightening the simple bounds . Using this perturbed linear program as an input to the linear programming solver, we compute a new approximate solution . With the above procedure we try again to derive an interval vector verified to contain a feasible point of the *original* linear program.

If the linear programming solver does not return an approximate solution in any stage of the iteration, due to numerical failure for example, the only upper bound we can derive is plus infinity. The same holds if the interval linear system solver cannot compute an enclosure of the solution set of the equations. As the equations do not change during the iteration, this can be ruled out after the first iteration.

The lower bound on the optimal value can be derived in a similar way from the dual linear program

Here, however, we can make use of the special structure of the dual, thus reducing the computational complexity. The details are contained in [3] .

- Stand-alone usage using Lurupa over the command line
- API usage including Lurupa as a library into a larger framework

In each case an lp solver specific module is needed to translate the generic calls from Lurupa into calls to the solver API and to transfer data between Lurupa and the solver.

`lurupa`

is controlled through various options. Mandatory are the options to select an lp and a solver module. `-lp file`

- This option specifies the file containing the lp to be processed. The file has to be in a format that can be handled by the solver module.
`-sm file`

- This option specifies the solver module and thus the lp solver that is used to compute the approximate solutions.

The most basic call, using a solver module `Module`

, assumend to be located in the local directory, would thus be

lurupa -sm Module -lp lp.mps

`lp.mps`

. This call, however, would just call the solver through the module and solve the lp. To compute the bounds we have to specify which bounds we want. This is done via the bound parameters `-lb`

- This option specifies, that the lower bound for the optimal value shall be computed.
`-ub`

- This option specifies, that the upper bound for the optimal value shall be computed.

`lp.mps`

is therefor done with lurupa -sm Module -lp lp.mps -lb -ub

It's important to note that while the computed bounds are rigorous the command line client does not provide for rigorous printing of the bounds. This means that the values printed for the bounds are rounded to display precision.

Several more options are available to alter Lurupa's behaviour. These can be divided into options affecting the input, the algorithm, and the output.

**Options affecting the input**

`-i d`

- This option specifies, that the cost vector and the constraint matrices and right hand sides are inflated to intervals with the given midpoint and a radius of
`d`

. The parameters get multiplied by the interval .

**Options affecting the algorithm**

`-alpha d`

- This option changes the algorithm parameter to
`d`

. For details on the parameter refer to [3] . `-eta d`

- This option changes the algorithm parameter to
`d`

. For details on the parameter refer to [3] . `-inflate`

- This option tells Lurupa to inflate perturbed lps in case of infeasibility. Due to the exponential deflation used in the algorithm the gap between lps that can not be verified and lps that are considered infeasible by the solver might be missed. If this option is specified Lurupa tries to inflate the feasible region of perturbed, infeasible lps to reach this gap. Using this option increases the number of algorithm iterations.

**Options affecting the output**

`-csv file`

- This option tells Lurupa to append a line containing the comma seperated results to
`file`

. If not present, the extension`.csv`

is appended to the filename. `-latex file`

- This option tells Lurupa to append a line contating the results to
`file`

, formatted to be used in a LaTeX table. If not present, the extension`.tex`

is appended to the filename. `-t`

- This option causes all messages printed by Lurupa to be prepended with the current time.
`-vn`

- This option selects the verbosity level of the messages printed by Lurupa. The verbosity increases with
`n`

. Possible values are:`-v0: No messages`

- Print no messages from Lurupa. The command line client just prints the final results.
`-v1: Errors`

- Add error messages from Lurupa to
`-v0`

. `-v2: Warnings`

- Add warnings.
`-v3: Brief`

- Add brief program flow messages like the iteration number and whether we are currently approximately solving a perturbed problem or verifying the feasibility of a solution.
`-v4: Normal`

- Add more program flow messages like whether we are perturbing or approximately solving a perturbed lp, and intermediate results like the approximate optimal value of perturbed lps.
`-v5: Verbose`

- Add detailed information like changes of the deflation parameters.

`-v2`

. `-write_vm style`

- This option tells Lurupa to save intermediate vectors and matrices to disk. The format depends on
`style`

. Specifying`octave`

stores all these in readable by Octave, this means all in one file with meta information about the dimension and the name of the vector or matrix. The file is named after the lp. Specifying`matlab`

stores these in one file per vector or matrix. These files are put into a directory named after the model.

Remain the options that print informational messages.

`-h, -?, --help`

- These options tell Lurupa to print usage information about the command line client containing the supported parameters along with all parameters supported by the solver module if specified.
`-V`

- This option tells Lurupa to print version information about the command line client, the core, and the solver module if specified. This also contains values like the set algorithm parameters.
`-Vb`

- This option tells Lurupa to print a brief version of the information printed by
`-V`

.

As mentioned the solver module might support additional options. These are specific to the solver module. They have the common prefix `-sm,`

with additional parameters appended with commas. A typical option would be `-sm,timeout,3600`

, which might tell the solver to time out after 3600 seconds.

**lp_solve 3.2** No further options

**lp_solve 4.0.1.0 and 4.0.1.11**

`-sm,timeout,i`

- This option sets the timeout in seconds to
`i`

. Setting this calls lp_solve's`set_timeout`

, causing lp_solve to stop solving an lp if it takes longer than`i`

seconds. `-sm,vn`

- This option selects the verbosity level of the messages printed by lp_solve. The verbosity increases with
`n`

. Possible values are (taken from the lp_solve documentation):`v0: NEUTRAL`

- Only some specific debug messages in debug print routines are reported.
`v1: CRITICAL`

- Only critical messages are reported. Hard errors like instability, out of memory, ...
`v2: SEVERE`

- Only severe messages are reported. Errors.
`v3: IMPORTANT`

- Only important messages are reported. Warnings and Errors.
`v4: NORMAL`

- Normal messages are reported.
`v5: DETAILED`

- Detailed messages are reported. Like model size, continuing B&B improvements, ...
`v6: FULL`

- All messages are reported. Useful for debugging purposes and small models.

`v3`

.

**lp_solve 5.5**

`-sm,resetbas`

- This option causes the solver module to re-solve lps in the case of numerical failure with an all-slack basis. While this increases the runtime it may help with numerically difficult lps.
`-sm,s`

- This option enables the default scaling of lp_solve (Numerical range-based scaling).
`-sm,timeout,i`

- This option sets the timeout in seconds to
`i`

. Setting this calls lp_solve's`set_timeout`

, causing lp_solve to stop solving an lp if it takes longer than`i`

seconds. `-sm,vn`

- This option selects the verbosity level of the messages printed by lp_solve. The verbosity increases with
`n`

. Possible values are (taken from the lp_solve documentation):`v0: NEUTRAL`

- Only some specific debug messages in debug print routines are reported.
`v1: CRITICAL`

- Only critical messages are reported. Hard errors like instability, out of memory, ...
`v2: SEVERE`

- Only severe messages are reported. Errors.
`v3: IMPORTANT`

- Only important messages are reported. Warnings and Errors.
`v4: NORMAL`

- Normal messages are reported.
`v5: DETAILED`

- Detailed messages are reported. Like model size, continuing B&B improvements, ...
`v6: FULL`

- All messages are reported. Useful for debugging purposes and small models.

`v3`

. `-sm,wmps`

- This option causes the solver module to save the perturbed lps in mps format. The filename is constructed of the lp name and the current iteration number.

**C++**

To access the API we need to include the two header files globals.h and Lurupa.h. Now we need to create an instance of the Lurupa class. All interaction with Lurupa is done via this object. As in the stand-alone case we need a solver module, which selects the lp solver to be used. Selecting the lp to be solved, we arrive at the following code that will appear more a less like this in every API usage of Lurupa.

#include <lurupa/globals.h> #include <lurupa/Lurupa.h> [...] Lurupa lurupa; bool module_set = lurupa.set_module("path/to/module"); bool lp_read = lurupa.read_lp("path/to/lp", 0);

`read_lp`

reads the lp as it is and does not inflate it to an interval problem.Now we can compute an approximate solution and after that the rigorous bounds on the optimal value.

double optimal_value; Solver_status status; bool lp_solved = lurupa.solve_lp(optimal_value, status); double lower_bound, upper_bound; int lower_iterations, upper_iterations; Bound_status st_lower = lurupa.lower_bound(lower_bound, lower_iterations); Bound_status st_upper = lurupa.upper_bound(upper_bound, upper_iterations);

This is all that is necessary to compute rigorous bounds on the optimal value. For details on how to set the algorithm parameters, how to change output and the like refer to the reference of the class Lurupa.

**C**

There is a C wrapper for the library, which maps all calls to an internal Lurupa object. To use it we have to include the cLurupa.h header. The names of the routines are chosen after the names of the corresponding member routines of Lurupa with a prefix of `clu`

. The previous C++ example looks like the following using the C wrapper.

#include <lurupa/cLurupa.h> [...] BOOL module_set, lp_read, lp_solved; double optimal_value, lower_bound, upper_bound; BOUND_STATUS st_lower, st_upper; int lower_iterations, upper_iterations; clu_init(); module_set = clu_set_module("path/to/module"); lp_read = clu_read_lp("path/to/lp", 0); lp_solved = clu_solve_lp(&optimal_value, &status); st_lower = clu_lower_bound(&lower_bound, &lower_iterations); st_upper = clu_upper_bound(&upper_bound, &upper_iterations);

Lurupa is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 2.1 of the License, or (at your option) any later version.

This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details.

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