## 81 simplex

### 81.1 Introduction to simplex

`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

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Categories: Numerical methods ·Optimization ·Share packages ·Package simplex

#### 81.1.1 Tests for simplex

There are some tests in the directory `share/simplex/Tests`.

#### 81.1.1.1 klee_minty

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));

#### 81.1.1.2 NETLIB

Some smaller problems from netlib (http://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 http://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

### 81.2 Functions and Variables for simplex

Option variable: epsilon_lp

  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`.

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  Categories: Package simplex

Function: linear_program (A, b, c)

  `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`.

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  Categories: Package simplex ·Numerical methods

Function: maximize_lp (obj, cond, [pos])

  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`.

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  Categories: Package simplex ·Numerical methods

Function: minimize_lp (obj, cond, [pos])

  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`.

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  Categories: Package simplex ·Numerical methods

Option variable: nonnegative_lp
Option variable: nonegative_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`.

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  Categories: Package simplex

Option variable: scale_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.

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  Categories: Package simplex

Variable: pivot_count_sx

  After `linear_program` returns, `pivot_count_sx` is the number of pivots in last computation.

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  Categories: Package simplex

Variable: pivot_max_sx

  `pivot_max_sx` is the maximum number of pivots allowed by `linear_program`.

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  Categories: Package simplex