scalation.minima

QuadraticSimplex

class QuadraticSimplex extends Error

This 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 nonlinear complementary slackness

See also

http://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 M-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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  3. final def ==(arg0: Any): Boolean

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  4. final def asInstanceOf[T0]: T0

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  5. def clone(): AnyRef

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    protected[java.lang]
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    @throws( ... )
  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. Neighter 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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  11. def finalize(): Unit

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  12. def flaw(method: String, message: String): Unit

    Show the flaw by printing the error message.

    Show the flaw by printing the error message.

    method

    the method where the error occurred

    message

    the error message

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  13. final def getClass(): Class[_]

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  14. def hashCode(): Int

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  15. final def isInstanceOf[T0]: Boolean

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  16. val k: Int

  17. val l: Int

  18. 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)

  19. final def ne(arg0: AnyRef): Boolean

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  20. final def notify(): Unit

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  21. final def notifyAll(): Unit

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  22. 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

  23. 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)

  24. def primal: VectorD

    Return the primal solution vector (x).

  25. 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 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

  26. def showTableau: Unit

    Show the current basis and tableau.

  27. 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.

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

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  29. def tableau: MatrixD

    Return the tableau (t).

  30. def toString(): String

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  31. final def wait(): Unit

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  32. final def wait(arg0: Long, arg1: Int): Unit

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  33. final def wait(arg0: Long): Unit

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  34. var x_B: Array[Int]

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

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