LP_SOLVE(1ES)

NAME
lp_solve    - Solve (mixed integer) linear programming problem.

SYNOPSIS
lp_solve [option]* "<" <input-file>

OPTIONS

-v  Verbose mode. Among other things, shows all the pivots.

-h  Help mode, prints the usage.

-d  Debug mode, all intermediate results are printed, and the branch-and-bound decisions in case of (mixed) integer problems.

-p  Only functional for pure LP problems. Print the values of the dual variables as well in the result. They are named r_1 until r_XXXXX unless specified by the user. Note that bounds (constraints on just one variable) are not considered real constraints, and are not given a row in the matrix, and are therefore not printed here.

-b <bound>
    Specify an upper (when minimizing) or lower (when maximizing) limit for the value of the objective function to the program. Only useful for (mixed) integer problems. If close enough, may speed up the calculations. The same result can be obtained by adding an extra constraint to the problem.

-c  When branching in MILP problems, take the ceiling of the selected non-integer variable first instead of the floor. This can influence the speed of MILP problems.

-e <value>
    Specify the accuracy with which it is checked whether the value of a variable is really integer. <value> must be between 0 and 0.5. Default value is 1e-6 and should be OK for most applications. Of course only useful for MILP problems.

-i  Print all intermediate valid solutions. Can give you useful solutions even if the total run time is too long. Only useful for (mixed) integer problems.

-s  Both rows and columns are scaled according to the geometric mean of the coefficients on them before solving. This might improve the numerical stability of your problem.

-I  Print info after reinverting.

-t  Trace pivot selection.

-mps
    Read from MPS file instead of lp file.

-degen
    Use random perturbations to reduce degeneracy, can increase numerical instability.

DESCRIPTION
The linear programming problem can be formulated as: Solve A.x >= V1, with V2.x maximal. A is a matrix, x a vector of (nonnegative) variables, V1 a vector called the right hand side, and V2 a vector specifying the objective function.
Any number of the variables may be specified to be of type integer.
This program solves problems of this kind. It is slightly more general than the above problem, in that every row of A (specifying one constraint) can have its own (in)equality, <=, >= or =. The result specifies values for all variables.
Uses a 'Simplex' algorithm and sparse matrix methods, for pure LP problems. If one or more of the variables is declared integer, the Simplex algorithm is iterated with a branch and bound algorithm, until the desired optimal solution is found.

INTERRUPT
In order to make it possible to get useful solutions for a mixed integer problem which takes too long to finish completely, the "interrupt" signal (kill -INT, ^C for many people) will not terminate the program when it is solving a MILP problem, but rather print the best valid solution found so far. This feature is not useful and thus disabled when you are solving a pure LP problem. If you see all zeros, it implies that no valid solution has been found yet. To really terminate the program, use a different signal, like KILL.
The "-i" option will print all intermediate valid solutions.

INPUT SYNTAX
The default input syntax is a set of algebraic expressions and "int" declarations in the following order:

<objective function>
<constraint>+
<declaration>*

where:

-   <objective function> is a linear combination of variables, ending with a semicolon, optionally preceded by "max: " or "min: " to indicate whether you want it to be minimized or maximized. The case is not important, "Max:" or "MAX:" will work as well. Maximization is the default.

-   <constraint> is an optional constraint name followed by a colon plus a linear combination of variables and constants, followed by a relational operator, followed again by a linear combination of variables and constants, ending with a semicolon. The relational operator can be any of the following: "<" "<=" "=" ">" ">=". There is no semantic difference between "<" and "<=" nor between ">" and ">=" (even for integer variables!).

-   <declaration> is of the form: "int" <var>+ ";" Commas are allowed between variables.

    So, the simplest linear problem consists of an objective function and 1 constraint.

EXAMPLE
The simple problem:

x1 >= 1
x2 >= 1
x1 + x2 >= 2
minimize x1 + x2 (= maximize -(x1 + x2)), with x1 integer

can be written as follows:

-x1 + -x2;
(or min: x1 + x2;)
x1 > 1;
x2 > 1;
x1 + x2 > 2;
int x1;

The correct result for (x1, x2) is of course (1, 1).
With the -mps option, lp_solve will accept MPS as input format.

BUGS
Specifying a constraint name for a bound (constraints on just single variables) does not have an effect: they are not stored inside the main matrix and are not assigned a dual variable.

-   The problem consists entirely of constraints on just single variables (so-called "bounds", like x < 1; ) and no constraint with more than 1 variable (like x + 3 y > 17; ). This leaves lp_solve with an empty problem matrix, as bounds are not stored in the main matrix. No real-life examples should be of this form, so I am not really chasing this problem.

-   Many people forget that lp_solve can only handle POSITIVE values for the variables. While reading MPS files it will however handle free or negative variables by replacing them with a variable pair <var>_neg and <var>_pos or -<var> respectively. It is up to the user to interpret the result of this transformation.

- Sometimes problems are numerically unstable, and the unavoidable rounding
    errors inside lp_solve will cause aborts. It is very hard to give general solutions to this problem, but try to keep all values in your problem in the order of magnitude of 1 by proper scaling. This is almost always better than using lp_solves built-in scaling (with -s). Almost parallel constraints are also not very good for numerical stability. Use "lp_solve -v" and observe the values of the pivots to see if there are any dangerously large or low numbers there.
    Building lp_solve with long doubles (see the Makefile) can help to increase numerical stability, but will also increase the run time considerably.
    You can consult the author as well if you encounter numerical problems, but please remember that it is very easy to formulate an infeasible LP problem, so be sure there is a solution.

SEE ALSO
The implementation of the simplex kernel was mainly based on:
W. Orchard-Hays: "Advanced Linear Programming Computing Techniques", McGraw-Hill 1968
The mixed integer branch and bound part was inspired by:
section 6.4 of "An Introduction to Linear Programming and Game Theory" by Paul R. Thie, second edition published by John Wiley and Sons in 1988.
This book refers to:
Dakin, R.J., "A Tree Search Algorithm for MILP Problems", Comput. J., 8 (1965) pp. 250-255

ACKNOWLEDGEMENTS
The work of Jeroen Dirks made the transition from the basic version 1.5 to the full version 2.0 possible. He contributed the procedural interface, a built-in MPS reader, and many fixes and enhancements to the code.

CONTRIBUTED BY
M.R.C.M. Berkelaar
Eindhoven University of Technology
Design Automation Section
P.O. Box 513
NL-5600 MB Eindhoven, The Netherlands
phone ...-31-40-473345
E-mail: michel@es.ele.tue.nl

STATUS
Use at own risk. Bug reports are welcome, as well as success stories.