c

scalation.minima

SimplexBG

class SimplexBG extends MinimizerLP

The SimplexBG class solves Linear Programming (LP) problems using the Bartels-Golub (BG) Simplex Algorithm. Given a constraint matrix 'a', constant 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

The BG Simplex Algorithm performs LU Factorization/Decomposition of the basis-matrix ('ba' = 'B') rather than computing inverses ('b_inv'). It has benefits over the (Revised) Simplex Algorithm (less run-time, less memory, and much reduced chance of round off errors).

Linear Supertypes
MinimizerLP, Error, AnyRef, Any
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  1. SimplexBG
  2. MinimizerLP
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Instance Constructors

  1. new SimplexBG(a: MatrixD, b: VectorD, c: VectorD, x_B: Array[Int] = null)

    a

    the constraint matrix

    b

    the constant/limit vector

    c

    the cost/revenue vector

    x_B

    the initial basis (set of indices where x_i is in the basis)

Value Members

  1. 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

    Definition Classes
    MinimizerLP
  2. val checker: CheckLP
    Definition Classes
    SimplexBGMinimizerLP
  3. def dual: VectoD

    Return the dual solution vector (y).

  4. def entering(): Int

    Find the best variable x_l to enter the basis.

    Find the best variable x_l to enter the basis. Use Dantiz's Rule: index of max positive (cycling possible) z value. Return -1 to indicate no such column.

  5. final def flaw(method: String, message: String): Unit
    Definition Classes
    Error
  6. def infeasible: Boolean

    Determine whether the current solution (x = primal) is still primal feasible.

  7. 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 variable chosen to enter the basis

  8. def objF(x: VectoD): Double

    Return the optimal objective function value (f(x) = c x).

    Return the optimal objective function value (f(x) = c x).

    x

    the primal solution vector

    Definition Classes
    SimplexBGMinimizerLP
  9. def pivot(k: Int, l: Int): Unit

    Pivot by replacing x_k with x_l in the basis.

    Pivot by replacing x_k with x_l in the basis. Update b_inv (actually lu), b_ and c_.

    k

    the leaving variable

    l

    the entering variable

  10. def primal: VectoD

    Return the primal (basis only) solution vector (x).

  11. def primalFull(x: VectoD): VectorD

    Return the full primal solution vector (xx).

  12. def setBasis(j: Int = N-M, l: Int = M): Array[Int]

    There are M+N variables, N decision and M slack variables, of which, for each iteration, M are chosen for a Basic Feasible Solution (BFS).

    There are M+N variables, N decision and M slack variables, of which, for each iteration, M are chosen for a Basic Feasible Solution (BFS). The the variables not in the basis are set to zero. Setting j to N will start with the slack variables in the basis (only works if b >= 0).

    j

    the offset to start the basis

    l

    the size of the basis

  13. def showTableau(iter: Int): Unit

    Show the current BG tableau.

    Show the current BG tableau.

    iter

    the number of iterations do far

  14. def solve(): VectoD

    Solve a Linear Programming (LP) problem using the BG Simplex Algorithm.

    Solve a Linear Programming (LP) problem using the BG Simplex Algorithm. Iteratively pivot until there an optimal solution is found or it is determined that the solution is unbounded. Return the optimal vector x.

    Definition Classes
    SimplexBGMinimizerLP
  15. def toString(): String

    Convert the current BG tableau (basis, b_inv, b_, and c_) to a string.

    Convert the current BG tableau (basis, b_inv, b_, and c_) to a string.

    Definition Classes
    SimplexBG → AnyRef → Any
  16. def unbounded(u: VectoD): Boolean

    Check if u <= 0., the solution is unbounded.

    Check if u <= 0., the solution is unbounded.

    u

    the vector for leaving

  17. var x_B: Array[Int]