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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- 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)
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- def comple(l: Int): Int
Return l's complementary variable.
Return l's complementary variable.
- l
whose complement
- def dual: VectorD
Return the dual solution vector (y).
- 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.
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- val k: Int
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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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- 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
- 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)
- def primal: VectorD
Return the primal solution vector 'x'.
- 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
- def showTableau(): Unit
Show the current basis and tableau.
- 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.
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- def tableau: MatrixD
Return the tableau 't'.
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