lp_solve - solve (mixed integer) linear programming problems

lp_solve [options] < input-file

-v[level]

Set verbosity:

   -v0: CRITICALSTOP
   -v1: CRITICAL
   -v2: SEVERE
   -v3: IMPORTANT (default)
   -v4: NORMAL
   -v5: DETAILED
   -v6: FULL

If level not provided (-v) then -v4 (NORMAL) is assumed.

-time

Print CPU time to parse input and to calculate result.

-Sdetail

Print solution. If detail omitted, then -S2 is used.

   -S0: Print nothing
   -S1: Only objective value
   -S2: Objective value + variables (default)
   -S3: Objective value + variables + constraints
   -S4: Objective value + variables + constraints + duals
   -S5: Objective value + variables + constraints + duals + lp model
   -S6: Objective value + variables + constraints + duals + lp model + lp scales

-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.

-min

Minimize the objective function. This is the default for MPS input. In lp_solve format you can specify minimization or maximization in the input file as well. The command line option overrides.

-max

Maximize the objective function. This is the default for lp_solve format input. In lp_solve format you can specify minimization or maximization in the input file as well. The command line option overrides.

-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.

-B

-B rule Specify branch-and-bound rule:

   -B0: Select Lowest indexed non-integer column (default)
   -B1: Select Random non-integer column
   -B2: Select Largest deviation from an integer value
   -B3: Select Best ???
   -B4: Select Median value deviation from an integer value
   -B5: Select Greedy ???

-e value

value is the tolerance of the test for 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 mode

Use automatic problem scaling:

	  -s:
	 -s0: Numerical range-based scaling
	 -s1: Geometric scaling
	 -s2: Curtis-reid scaling

This might improve the numerical stability of your problem.

-sp

Also do power scaling. This option must come AFTER -s mode option.

-sl

Also do Lagrange scaling. This option must come AFTER -s mode option.

-si

Also do Integer scaling. This option must come AFTER -s mode option.

-I

Print info after reinverting.

-t

Trace pivot selection.

-lp

Read from LP file (default).

-mps

Read from MPS file instead of lp file. For a short introduction to MPS see ftp://softlib.cs.rice.edu/pub/miplib/mps_format.

-parse_only

Parse input file but do not calculate (ie check).

-presolve

Presolve problem before optimizing.

-improvelevel

Iterative improvement level:

	 -improve0: none (default)
	 -improve1: FTRAN only
	 -improve2: BTRAN only
	 -improve3: FTRAN + BTRAN

-degen

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

-trej Trej

Set minimum pivot value to Trej.

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.

The "-i" option will print all intermediate valid solutions.

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.

-

A var must start with a letter (either upper or lower case), and may contain any number of additional letters, numerals, or characters from this list: _[]{}/.&#$%~'@^ So, the simplest linear problem consists of an objective function and 1 constraint.

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.

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.

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

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.

M.R.C.M. Berkelaar

Eindhoven University of Technology

Design Automation Section

P.O. Box 513

NL-5600 MB Eindhoven, The Netherlands

phone +31-40-2474792

E-mail: michel@es.ele.tue.nl

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