// replaceBland uses the Bland rule to find the indices to swap if the minimum
// move is 0. The indices to be swapped are replace and minIdx (following the
// nomenclature in the main routine).
func replaceBland(A mat64.Matrix, ab *mat64.Dense, xb []float64, basicIdxs, nonBasicIdx []int, r, move []float64) (replace, minIdx int, err error) {
// Use the traditional bland rule, except don't replace a constraint which
// causes the new ab to be singular.
for i, v := range r {
if v > -blandNegTol {
continue
}
minIdx = i
err = computeMove(move, minIdx, A, ab, xb, nonBasicIdx)
if err != nil {
// Either unbounded or something went wrong.
return -1, -1, err
}
replace = floats.MinIdx(move)
if math.Abs(move[replace]) > blandZeroTol {
// Large enough that it shouldn't be a problem
return replace, minIdx, nil
}
// Find a zero index where replacement is non-singular.
biCopy := make([]int, len(basicIdxs))
for replace, v := range move {
if v > blandZeroTol {
continue
}
copy(biCopy, basicIdxs)
biCopy[replace] = nonBasicIdx[minIdx]
abTmp := extractColumns(A, biCopy)
// If the condition number is reasonable, use this index.
if mat64.Cond(abTmp, 1) < 1e16 {
return replace, minIdx, nil
}
}
}
return -1, -1, ErrBland
}
func simplex(initialBasic []int, c []float64, A mat64.Matrix, b []float64, tol float64) (float64, []float64, []int, error) {
err := verifyInputs(initialBasic, c, A, b)
if err != nil {
if err == ErrUnbounded {
return math.Inf(-1), nil, nil, ErrUnbounded
}
return math.NaN(), nil, nil, err
}
m, n := A.Dims()
// There is at least one optimal solution to the LP which is at the intersection
// to a set of constraint boundaries. For a standard form LP with m variables
// and n equality constraints, at least m-n elements of x must equal zero
// at optimality. The Simplex algorithm solves the standard-form LP by starting
// at an initial constraint vertex and successively moving to adjacent constraint
// vertices. At every vertex, the set of non-zero x values is the "basic
// feasible solution". The list of non-zero x's are maintained in basicIdxs,
// the respective columns of A are in ab, and the actual non-zero values of
// x are in xb.
//
// The LP is equality constrained such that A * x = b. This can be expanded
// to
// ab * xb + an * xn = b
// where ab are the columns of a in the basic set, and an are all of the
// other columns. Since each element of xn is zero by definition, this means
// that for all feasible solutions xb = ab^-1 * b.
//
// Before the simplex algorithm can start, an initial feasible solution must
// be found. If initialBasic is non-nil a feasible solution has been supplied.
// Otherwise the "Phase I" problem must be solved to find an initial feasible
// solution.
var basicIdxs []int // The indices of the non-zero x values.
var ab *mat64.Dense // The subset of columns of A listed in basicIdxs.
var xb []float64 // The non-zero elements of x. xb = ab^-1 b
if initialBasic != nil {
// InitialBasic supplied. Panic if incorrect length or infeasible.
if len(initialBasic) != m {
panic("lp: incorrect number of initial vectors")
}
ab = extractColumns(A, initialBasic)
xb, err = initializeFromBasic(ab, b)
if err != nil {
panic(err)
}
basicIdxs = make([]int, len(initialBasic))
copy(basicIdxs, initialBasic)
} else {
// No inital basis supplied. Solve the PhaseI problem.
basicIdxs, ab, xb, err = findInitialBasic(A, b)
if err != nil {
return math.NaN(), nil, nil, err
}
}
// basicIdxs contains the indexes for an initial feasible solution,
// ab contains the extracted columns of A, and xb contains the feasible
// solution. All x not in the basic set are 0 by construction.
// nonBasicIdx is the set of nonbasic variables.
nonBasicIdx := make([]int, 0, n-m)
inBasic := make(map[int]struct{})
for _, v := range basicIdxs {
inBasic[v] = struct{}{}
}
for i := 0; i < n; i++ {
_, ok := inBasic[i]
if !ok {
nonBasicIdx = append(nonBasicIdx, i)
}
}
// cb is the subset of c for the basic variables. an and cn
// are the equivalents to ab and cb but for the nonbasic variables.
cb := make([]float64, len(basicIdxs))
for i, idx := range basicIdxs {
cb[i] = c[idx]
}
cn := make([]float64, len(nonBasicIdx))
for i, idx := range nonBasicIdx {
cn[i] = c[idx]
}
an := extractColumns(A, nonBasicIdx)
bVec := mat64.NewVector(len(b), b)
cbVec := mat64.NewVector(len(cb), cb)
// Temporary data needed each iteration. (Described later)
r := make([]float64, n-m)
move := make([]float64, m)
// Solve the linear program starting from the initial feasible set. This is
// the "Phase 2" problem.
//
// Algorithm:
// 1) Compute the "reduced costs" for the non-basic variables. The reduced
// costs are the lagrange multipliers of the constraints.
// r = cn - an^T * ab^-T * cb
// 2) If all of the reduced costs are positive, no improvement is possible,
// and the solution is optimal (xn can only increase because of
// non-negativity constraints). Otherwise, the solution can be improved and
// one element will be exchanged in the basic set.
// 3) Choose the x_n with the most negative value of r. Call this value xe.
// This variable will be swapped into the basic set.
// 4) Increase xe until the next constraint boundary is met. This will happen
// when the first element in xb becomes 0. The distance xe can increase before
// a given element in xb becomes negative can be found from
// xb = Ab^-1 b - Ab^-1 An xn
// = Ab^-1 b - Ab^-1 Ae xe
// = bhat + d x_e
// xe = bhat_i / - d_i
// where Ae is the column of A corresponding to xe.
// The constraining basic index is the first index for which this is true,
// so remove the element which is min_i (bhat_i / -d_i), assuming d_i is negative.
// If no d_i is less than 0, then the problem is unbounded.
// 5) If the new xe is 0 (that is, bhat_i == 0), then this location is at
// the intersection of several constraints. Use the Bland rule instead
// of the rule in step 4 to avoid cycling.
for {
// Compute reduced costs -- r = cn - an^T ab^-T cb
var tmp mat64.Vector
err = tmp.SolveVec(ab.T(), cbVec)
if err != nil {
break
}
data := make([]float64, n-m)
tmp2 := mat64.NewVector(n-m, data)
tmp2.MulVec(an.T(), &tmp)
floats.SubTo(r, cn, data)
// Replace the most negative element in the simplex. If there are no
// negative entries then the optimal solution has been found.
minIdx := floats.MinIdx(r)
if r[minIdx] >= -tol {
break
}
for i, v := range r {
if math.Abs(v) < rRoundTol {
r[i] = 0
}
}
// Compute the moving distance.
err = computeMove(move, minIdx, A, ab, xb, nonBasicIdx)
if err != nil {
if err == ErrUnbounded {
return math.Inf(-1), nil, nil, ErrUnbounded
}
break
}
// Replace the basic index along the tightest constraint.
replace := floats.MinIdx(move)
if move[replace] <= 0 {
replace, minIdx, err = replaceBland(A, ab, xb, basicIdxs, nonBasicIdx, r, move)
if err != nil {
if err == ErrUnbounded {
return math.Inf(-1), nil, nil, ErrUnbounded
}
break
}
}
// Replace the constrained basicIdx with the newIdx.
basicIdxs[replace], nonBasicIdx[minIdx] = nonBasicIdx[minIdx], basicIdxs[replace]
cb[replace], cn[minIdx] = cn[minIdx], cb[replace]
tmpCol1 := mat64.Col(nil, replace, ab)
tmpCol2 := mat64.Col(nil, minIdx, an)
ab.SetCol(replace, tmpCol2)
an.SetCol(minIdx, tmpCol1)
// Compute the new xb.
xbVec := mat64.NewVector(len(xb), xb)
err = xbVec.SolveVec(ab, bVec)
if err != nil {
break
}
}
// Found the optimum successfully or died trying. The basic variables get
// their values, and the non-basic variables are all zero.
opt := floats.Dot(cb, xb)
xopt := make([]float64, n)
for i, v := range basicIdxs {
xopt[v] = xb[i]
}
return opt, xopt, basicIdxs, err
}
// findInitialBasic finds an initial basic solution, and returns the basic
// indices, ab, and xb.
func findInitialBasic(A mat64.Matrix, b []float64) ([]int, *mat64.Dense, []float64, error) {
m, n := A.Dims()
basicIdxs := findLinearlyIndependent(A)
if len(basicIdxs) != m {
return nil, nil, nil, ErrSingular
}
// It may be that this linearly independent basis is also a feasible set. If
// so, the Phase I problem can be avoided.
ab := extractColumns(A, basicIdxs)
xb, err := initializeFromBasic(ab, b)
if err == nil {
return basicIdxs, ab, xb, nil
}
// This set was not feasible. Instead the "Phase I" problem must be solved
// to find an initial feasible set of basis.
//
// Method: Construct an LP whose optimal solution is a feasible solution
// to the original LP.
// 1) Introduce an artificial variable x_{n+1}.
// 2) Let x_j be the most negative element of x_b (largest constraint violation).
// 3) Add the artificial variable to A with:
// a_{n+1} = b - \sum_{i in basicIdxs} a_i + a_j
// swap j with n+1 in the basicIdxs.
// 4) Define a new LP:
// minimize x_{n+1}
// subject to [A A_{n+1}][x_1 ... x_{n+1}] = b
// x, x_{n+1} >= 0
// 5) Solve this LP. If x_{n+1} != 0, then the problem is infeasible, otherwise
// the found basis can be used as an initial basis for phase II.
//
// The extra column in Step 3 is defined such that the vector of 1s is an
// initial feasible solution.
// Find the largest constraint violator.
// Compute a_{n+1} = b - \sum{i in basicIdxs}a_i + a_j. j is in basicIDx, so
// instead just subtract the basicIdx columns that are not minIDx.
minIdx := floats.MinIdx(xb)
aX1 := make([]float64, m)
copy(aX1, b)
col := make([]float64, m)
for i, v := range basicIdxs {
if i == minIdx {
continue
}
mat64.Col(col, v, A)
floats.Sub(aX1, col)
}
// Construct the new LP.
// aNew = [A, a_{n+1}]
// bNew = b
// cNew = 1 for x_{n+1}
aNew := mat64.NewDense(m, n+1, nil)
aNew.Copy(A)
aNew.SetCol(n, aX1)
basicIdxs[minIdx] = n // swap minIdx with n in the basic set.
c := make([]float64, n+1)
c[n] = 1
// Solve the Phase 2 linear program.
_, xOpt, newBasic, err := simplex(basicIdxs, c, aNew, b, 1e-10)
if err != nil {
return nil, nil, nil, errors.New(fmt.Sprintf("lp: error finding feasible basis: %s", err))
}
// If n+1 is part of the solution basis then the problem is infeasible. If
// not, then the problem is feasible and newBasic is an initial feasible
// solution.
if math.Abs(xOpt[n]) > phaseIZeroTol {
return nil, nil, nil, ErrInfeasible
}
// The value is zero. First, see if it's not in the basis (feasible solution).
basicIdx := -1
basicMap := make(map[int]struct{})
for i, v := range newBasic {
if v == n {
basicIdx = i
}
basicMap[v] = struct{}{}
xb[i] = xOpt[v]
}
if basicIdx == -1 {
// Not in the basis.
ab = extractColumns(A, newBasic)
return newBasic, ab, xb, nil
}
// The value is zero, but it's in the basis. See if swapping in another column
// finds a feasible solution.
for i := range xOpt {
if _, inBasic := basicMap[i]; inBasic {
continue
}
newBasic[basicIdx] = i
ab := extractColumns(A, newBasic)
xb, err := initializeFromBasic(ab, b)
if err == nil {
return newBasic, ab, xb, nil
}
}
return nil, nil, nil, ErrInfeasible
}
