## `linopt::Transparent::autostep` -- perform the next simplex step

#### Introduction

`linopt::Transparent::autostep(`tableau`)` perform the next step of the simplex algorithm for the given simplex tableau `tableau`.

#### Call(s)

`linopt::Transparent::autostep(tableau)`

#### Parameters

`tableau` | - | a simplex tableau of domain type `linopt::Transparent`

#### Returns

a simplex tableau of domain type `linopt::Transparent` or a set which contains the solution of the linear program.

#### Related Functions

`linopt::Transparent`, `linopt::Transparent::convert`, `linopt::Transparent::dual_prices`, `linopt::Transparent::phaseI_tableau`, `linopt::Transparent::result`, `linopt::Transparent::simplex`, `linopt::Transparent::suggest`, `linopt::Transparent::userstep`

#### Details

- `linopt::Transparent::autostep(`tableau`)` performs the next step of the simplex algorithm. This is the same step that `linopt::Transparent::suggest` would suggest for the given simplex tableau `tableau`.
- Normally `linopt::Transparent::autostep` returns the next simplex tableau. If the calculation of the simplex algorithm is finished `linopt::Transparent::autostep` returns a set containing a solution of the given linear program described by `tableau`.

#### Example 1

The ordinary simplex tableau of a given linear program is created:

    >> k := [[x + y >= 2], x, NonNegative]:
       t := linopt::Transparent(k)

                      +-                                   -+
                      |  "linopt", "restr", slk[1],  x,  y  |
                      |                                     |
                      |    "obj",     0,       0,    1,  0  |
                      |                                     |
                      |   slk[1],     -2,      1,   -1, -1  |
                      +-                                   -+

The next two steps of the simplex algorithm are executed for the given simplex tableau:

    >> linopt::Transparent::autostep(t);
       linopt::Transparent::autostep(%)

                      +-                                  -+
                      |  "linopt", "restr", slk[1], x,  y  |
                      |                                    |
                      |    "obj",     -2,      1,   0, -1  |
                      |                                    |
                      |      x,       2,      -1,   1,  1  |
                      +-                                  -+

                       +-                                 -+
                       |  "linopt", "restr", slk[1], x, y  |
                       |                                   |
                       |    "obj",     0,       0,   1, 0  |
                       |                                   |
                       |      y,       2,      -1,   1, 1  |
                       +-                                 -+

    >> delete k, t:

#### Example 2

The ordinary simplex tableau of a given linear program is created:

    >> k := [[x + y >= -1, x + y <= 3], x + 2*y, NonNegative]:
       t := linopt::Transparent(k)

                  +-                                           -+
                  |  "linopt", "restr", slk[1], slk[2],  x,  y  |
                  |                                             |
                  |    "obj",     0,       0,      0,    1,  2  |
                  |                                             |
                  |   slk[1],     1,       1,      0,   -1, -1  |
                  |                                             |
                  |   slk[2],     3,       0,      1,    1,  1  |
                  +-                                           -+

If the end of the simplex algorithm is reached, `linopt::Transparent::autostep` returns a solution of the given linear program:

    >> linopt::Transparent::suggest(t),
       linopt::Transparent::autostep(t)

                              OPTIMAL, {x = 0, y = 0}

    >> delete k, t:

#### Background

- References:

  Papadimitriou, Christos H; Steiglitz, Kenneth: Combinatorial Optimization; Algorithms and Complexity. Prentice-Hall, 1982.

  Nemhauser, George L; Wolsey, Laurence A: Integer and Combinatorial Optimization. New York, Wiley, 1988.

  Salkin, Harvey M; Mathur, Kamlesh: Foundations of Integer Programming. North-Holland, 1989.

  Neumann, Klaus; Morlock, Martin: Operations-Research. Munich, Hanser, 1993.

  Duerr, Walter; Kleibohm, Klaus: Operations Research; Lineare Modelle und ihre Anwendungen. Munich, Hanser, 1992.

  Suhl, Uwe H: MOPS - Mathematical OPtimization System. European Journal of Operational Research 72(1994)312-322. North-Holland, 1994.

  Suhl, Uwe H; Szymanski, Ralf: Supernode Processing of Mixed Integer Models. Boston, Kluwer Academic Publishers, 1994.

#### Changes

- new function
- The linopt library was completely rewritten in MuPAD version 2.0.

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