Documentation

# `linopt`::`Transparent::dual_prices`

Get the dual solution belonging to the given tableau

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## Syntax

```linopt::Transparent::dual_prices(`tableau`)
```

## Description

`linopt::Transparent::dual_prices(tableau)` returns the dual solution of the linear optimization problem given by `tableau`.

This procedure returns the dual solution belonging to the given tableau in form of a set of lists containing two elements, the first one is a restriction and the second one is the value belonging to the slack variable connected to the restriction in the dual solution.

## Examples

### Example 1

Here it is demonstrated that the dual solution of the final tableau is similar to the second element of the result of `linopt::minimize` using the option `DualPrices`:

First we compute the final tableau of the simplex algorithm:

```k := [{x <= 2, y <= 2, x + 2*y >= 4}, - x + y, NonNegative]: t := linopt::Transparent(k): t := linopt::Transparent::simplex(t)```

Now we compute the solutions:

```linopt::Transparent::dual_prices(t); linopt::minimize(k, DualPrices)[2]```

`delete k, t:`

### Example 2

We compute the dual solution of another linear program:

```k := [{x <= 2, y <= 2, x + 2*y >= 4}, -x + y, NonNegative]: t := linopt::Transparent(k); linopt::Transparent::dual_prices(t)```

`delete k, t:`

## Parameters

 `tableau` A simplex tableau of domain type `linopt::Transparent`

## Return Values

Set of lists, each containing 2 elements.

## 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.