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Next: Functions and Variables for simplex, Previous: Package simplex, Up: Package simplex [Contents][Index]
simplex is a package for linear optimization using the simplex algorithm.
Example:
(%i1) load("simplex")$
(%i2) minimize_lp(x+y, [3*x+2*y>2, x+4*y>3]);
9 7 1
(%o2) [--, [y = --, x = -]]
10 10 5
There are some tests in the directory share/simplex/Tests.
The function klee_minty produces input for linear_program, for which
exponential time for solving is required without scaling.
Example:
load("klee_minty")$
apply(linear_program, klee_minty(6));
A better approach:
epsilon_sx : 0$ scale_sx : true$ apply(linear_program, klee_minty(10));
Some smaller problems from netlib (https://www.netlib.org/lp/data/)
test suite are converted to a format, readable by Maxima. Problems are
adlittle, afiro, kb2, sc50a and
share2b. Each problem has three input files in CSV format for
matrix A and vectors b and c.
Example:
A : read_matrix("adlittle_A.csv", 'csv)$
b : read_list("adlittle_b.csv", 'csv)$
c : read_list("adlittle_c.csv", 'csv)$
linear_program(A, b, c)$
%[2];
=> 225494.9631623802
Results:
PROBLEM MINIMUM SCALING adlittle +2.2549496316E+05 false afiro -4.6475314286E+02 false kb2 -1.7499001299E+03 true sc50a -6.4575077059E+01 false share2b -4.1573518187E+02 false
The Netlib website https://www.netlib.org/lp/data/readme lists the values as
PROBLEM MINIMUM adlittle +2.2549496316E+05 afiro -4.6475314286E+02 kb2 -1.7499001299E+03 sc50a -6.4575077059E+01 share2b -4.1573224074E+02
Previous: Introduction to simplex, Up: Package simplex [Contents][Index]
Default value: 10^-8
Epsilon used for numerical computations in linear_program; it is
set to 0 in linear_program when all inputs are rational.
Example:
(%i1) load("simplex")$
(%i2) minimize_lp(-x, [1e-9*x + y <= 1], [x,y]);
Warning: linear_program(A,b,c): non-rat inputs found, epsilon_lp= 1.0e-8
Warning: Solution may be incorrect.
(%o2) Problem not bounded!
(%i3) minimize_lp(-x, [10^-9*x + y <= 1], [x,y]);
(%o3) [- 1000000000, [y = 0, x = 1000000000]]
(%i4) minimize_lp(-x, [1e-9*x + y <= 1], [x,y]), epsilon_lp=0;
(%o4) [- 9.999999999999999e+8, [y = 0, x = 9.999999999999999e+8]]
See also: linear_program, ratnump.
linear_program is an implementation of the simplex algorithm.
linear_program(A, b, c) computes a vector x for which
c.x is minimum possible among vectors for which A.x = b
and x >= 0. Argument A is a matrix and arguments b
and c are lists.
linear_program returns a list which contains the minimizing
vector x and the minimum value c.x. If the problem is not
bounded, it returns "Problem not bounded!" and if the problem is not
feasible, it returns "Problem not feasible!".
To use this function first load the simplex package with
load("simplex");.
Example:
(%i2) A: matrix([1,1,-1,0], [2,-3,0,-1], [4,-5,0,0])$
(%i3) b: [1,1,6]$
(%i4) c: [1,-2,0,0]$
(%i5) linear_program(A, b, c);
13 19 3
(%o5) [[--, 4, --, 0], - -]
2 2 2
See also: minimize_lp, scale_lp, and epsilon_lp.
Maximizes linear objective function obj subject to some linear
constraints cond. See minimize_lp for detailed
description of arguments and return value.
See also: minimize_lp.
Minimizes a linear objective function obj subject to some linear
constraints cond. cond a list of linear equations or
inequalities. In strict inequalities > is replaced by >=
and < by <=. The optional argument pos is a list
of decision variables which are assumed to be positive.
If the minimum exists, minimize_lp returns a list which
contains the minimum value of the objective function and a list of
decision variable values for which the minimum is attained. If the
problem is not bounded, minimize_lp returns "Problem not
bounded!" and if the problem is not feasible, it returns "Problem not
feasible!".
The decision variables are not assumed to be non-negative by default. If
all decision variables are non-negative, set nonnegative_lp to
true or include all in the optional argument pos. If
only some of decision variables are positive, list them in the optional
argument pos (note that this is more efficient than adding
constraints).
minimize_lp uses the simplex algorithm which is implemented in
maxima linear_program function.
To use this function first load the simplex package with
load("simplex");.
Examples:
(%i1) minimize_lp(x+y, [3*x+y=0, x+2*y>2]);
4 6 2
(%o1) [-, [y = -, x = - -]]
5 5 5
(%i2) minimize_lp(x+y, [3*x+y>0, x+2*y>2]), nonnegative_lp=true;
(%o2) [1, [y = 1, x = 0]]
(%i3) minimize_lp(x+y, [3*x+y>0, x+2*y>2], all);
(%o3) [1, [y = 1, x = 0]]
(%i4) minimize_lp(x+y, [3*x+y=0, x+2*y>2]), nonnegative_lp=true;
(%o4) Problem not feasible!
(%i5) minimize_lp(x+y, [3*x+y>0]);
(%o5) Problem not bounded!
There is also a limited ability to solve linear programs with symbolic constants.
(%i1) declare(c,constant)$
(%i2) maximize_lp(x+y, [y<=-x/c+3, y<=-x+4], [x, y]), epsilon_lp=0;
Is (c-1)*c positive, negative or zero?
p;
Is c*(2*c-1) positive, negative or zero?
p;
Is c positive or negative?
p;
Is c-1 positive, negative or zero?
p;
Is 2*c-1 positive, negative or zero?
p;
Is 3*c-4 positive, negative or zero?
p;
1 1
(%o2) [4, [x = -----, y = 3 - ---------]]
1 1
1 - - (1 - -) c
c c
(%i1) (assume(c>4/3), declare(c,constant))$
(%i2) maximize_lp(x+y, [y<=-x/c+3, y<=-x+4], [x, y]), epsilon_lp=0;
1 1
(%o2) [4, [x = -----, y = 3 - ---------]]
1 1
1 - - (1 - -) c
c c
See also: maximize_lp, nonnegative_lp, epsilon_lp.
Default value: false
If nonnegative_lp is true all decision variables to
minimize_lp and maximize_lp are assumed to be non-negative.
nonegative_lp is a deprecated alias.
See also: minimize_lp.
Default value: false
When scale_lp is true,
linear_program scales its input so that the maximum absolute value in each row or column is 1.
After linear_program returns,
pivot_count_sx is the number of pivots in last computation.
pivot_max_sx is the maximum number of pivots allowed by linear_program.
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