class Simplex2P extends MinimizerLP
The Simplex2P class solves Linear Programming (LP) problems using a tableau based
Simplex Algorithm. Given a constraint matrix 'a', limit/RHS vector 'b' and
cost vector 'c', find values for the solution/decision vector 'x' that minimize
the objective function 'f(x)', while satisfying all of the constraints, i.e.,
minimize f(x) = c x subject to a x <= b, x >= 0
In case of 'a_i x >= b_i', use -b_i as an indicator of a ">=" constraint. The program will flip such negative b_i back to positive as well as use a surplus and artificial variable instead of the usual slack variable, i.e., a_i x <= b_i => a_i x + s_i = b_i // use slack variable s_i with coefficient 1 a_i x >= b_i => a_i x + s_i = b_i // use surplus variable s_i with coefficient -1 For each '>=' constraint, an artificial variable is introduced and put into the initial basis. These artificial variables must be removed from the basis during Phase I of the Two-Phase Simplex Algorithm. After this, or if there are no artificial variables, Phase II is used to find an optimal value for 'x' and the optimum value for 'f'.
Creates an 'MM-by-nn' simplex tableau with -- [0..M-1, 0..N-1] = a (constraint matrix) -- [0..M-1, N..M+N-1] = s (slack/surplus variable matrix) -- [0..M-1, M+N..nn-2] = r (artificial variable matrix) -- [0..M-1, nn-1] = b (limit/RHS vector) -- [M, 0..nn-2] = c (cost vector)
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Instance Constructors
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new
Simplex2P(a: MatrixD, b: VectorD, c: VectorD)
- a
the M-by-N constraint matrix
- b
the M-length limit/RHS vector (input b_i negative for surplus)
- c
the N-length cost vector
Value Members
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def
!=(arg0: Any): Boolean
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final
def
##(): Int
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val
EPSILON: Double
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final
def
asInstanceOf[T0]: T0
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def
check(x: VectoD, y: VectoD, f: Double): Boolean
Determine whether the current solution is correct.
Determine whether the current solution is correct.
- x
the primal solution vector x
- y
the dual solution vector y
- f
the minimum value of the objective function
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- MinimizerLP
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val
checker: CheckLP
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def
clone(): AnyRef
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def
dual: VectorD
Return the dual solution vector y (cost row (M) under the slack columns).
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def
entering(): Int
Find the best variable x_l to enter the basis.
Find the best variable x_l to enter the basis. Determine the index of entering variable corresponding to column l (e.g., using Dantiz's Rule or Bland's Rule). Return -1 to indicate no such column. 't(M).argmaxPos (jj)' // use Dantiz's rule (index of max positive, cycling possible) 't(M).firstPos (jj)' // use Bland's rule (index of first positive, FPE possible)
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eq(arg0: AnyRef): Boolean
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def
equals(arg0: Any): Boolean
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def
finalize(): Unit
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flaw(method: String, message: String): Unit
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def
getClass(): Class[_]
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def
hashCode(): Int
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def
infeasible: Boolean
Determine whether the current solution (x = primal) is still primal feasible.
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def
initBasis(): Unit
Initialize the basis to the slack and artificial variables.
Initialize the basis to the slack and artificial variables. Perform row operations to make cost row (t(M)) zero in artificial var columns. If b(i) is negative have a surplus and artificial variable, otherwise, just a slack variable.
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final
def
isInstanceOf[T0]: Boolean
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def
leaving(l: Int): Int
Find the best variable x_k to leave the basis given that x_l is entering.
Find the best variable x_k to leave the basis given that x_l is entering. Determine the index of the leaving variable corresponding to row k using the Min-Ratio Rule. Return -1 to indicate no such row.
- l
the entering variable (column)
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final
def
ne(arg0: AnyRef): Boolean
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final
def
notify(): Unit
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final
def
notifyAll(): Unit
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def
objF(x: VectoD): Double
Return the value of the objective function f(x) = c x.
Return the value of the objective function f(x) = c x.
- x
the coordinate values of the current point
- Definition Classes
- Simplex2P → MinimizerLP
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def
pivot(k: Int, l: Int): Unit
Pivot on entry (k, l) using Gauss-Jordan elimination to replace variable x_k with x_l in the basis.
Pivot on entry (k, l) using Gauss-Jordan elimination to replace variable x_k with x_l in the basis.
- k
the leaving variable (row)
- l
the entering variable (column)
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def
primal: VectorD
Return the primal solution vector x (only the basic variables are non-zero).
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def
removeArtificials(): Unit
Remove the artificial variables and reset the cost row (M) in the tableau.
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def
showTableau(iter: Int): Unit
Show the current tableau.
Show the current tableau.
- iter
the number of iterations do far
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def
solve(): VectorD
Solve the LP minimization problem using two phases if necessary.
Solve the LP minimization problem using two phases if necessary. Note: phase I is always minimization. Two phases are necessary if the number of artificial variables R > 0.
- Definition Classes
- Simplex2P → MinimizerLP
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def
solve_1(): VectorD
Run the simplex algorithm starting from an initial BFS and iteratively find a non-basic variable to replace a variable in the current basis so long as the objective function improves.
Run the simplex algorithm starting from an initial BFS and iteratively find a non-basic variable to replace a variable in the current basis so long as the objective function improves. Runs a single phase and return the optimal vector x.
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final
def
synchronized[T0](arg0: ⇒ T0): T0
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def
toString(): String
Convert the current tableau and basis to a string suitable for display.
Convert the current tableau and basis to a string suitable for display.
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final
def
wait(): Unit
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def
wait(arg0: Long, arg1: Int): Unit
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