c

scalation.minima

QuadraticSimplex

class QuadraticSimplex extends Error

The QuadraticSimplex class solves Quadratic Programming (QP) problems using the Quadratic Simplex Algorithm. Given a constraint matrix 'a', constant vector 'b', cost matrix 'q' 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) = 1/2 x q x + c x subject to a x <= b, x >= 0

Creates an 'MM-by-NN' simplex tableau. This implementation is restricted to linear constraints 'a x <= b' and 'q' being a positive semi-definite matrix. Pivoting must now also handle non-linear complementary slackness.

See also

www.engineering.uiowa.edu/~dbricker/lp_stacks.html

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Instance Constructors

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

    a

    the M-by-N constraint matrix

    b

    the M-length constant/limit vector

    q

    the N-by-N cost/revenue matrix (second order component)

    c

    the N-length cost/revenue vector (first order component)

    x_B

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

Value Members

  1. final def !=(arg0: Any): Boolean
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  2. final def ##(): Int
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  6. def comple(l: Int): Int

    Return l's complementary variable.

    Return l's complementary variable.

    l

    whose complement

  7. def dual: VectorD

    Return the dual solution vector (y).

  8. def entering(): Int

    Find a variable 'x_l' to enter the basis.

    Find a variable 'x_l' to enter the basis. Determine the index of entering variable corresponding to column l. Neither the variable nor its complement may be in the current basis. Return -1 to indicate no such column.

  9. final def eq(arg0: AnyRef): Boolean
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  10. def equals(arg0: Any): Boolean
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  15. val k: Int
  16. val l: Int
  17. 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)

  18. final def ne(arg0: AnyRef): Boolean
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  19. final def notify(): Unit
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  20. final def notifyAll(): Unit
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  21. def objValue(x: VectorD): Double

    Return the optimal objective function value (f(x) = 1/2 x q x + c x).

    Return the optimal objective function value (f(x) = 1/2 x q x + c x).

    x

    the primal solution vector

  22. 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)

  23. def primal: VectorD

    Return the primal solution vector 'x'.

  24. def setBasis(j: Int = N, 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 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

  25. def showTableau(): Unit

    Show the current basis and tableau.

  26. def solve(): (VectorD, Double)

    Run the simplex algorithm starting from the initial BFS and iteratively find a non-basic variable to replace a variable in the current basis so long as the objective improves.

  27. final def synchronized[T0](arg0: ⇒ T0): T0
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  28. def tableau: MatrixD

    Return the tableau 't'.

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  32. final def wait(): Unit
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  33. var x_B: Array[Int]

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