Constrained least squares

Least squares with optional linear equality constraints, linear inequality constraints, and non-negativity – TDA's mls, mlse, mlsi, mlsei, mlsei1, and mnls, each as its thin wrapper sharing one implementation. The problem solved is $$\min\; \|Ax - b\|_2 \quad \mathrm{s.t.}\; A_e x = b_e,\; A_i x \ge b_i,\; (x \ge 0)$$ These are TDA's documented semantics (tda.hlp, entries ##mls/##mlse/##mlsi/##mlsei/##mnls): an earlier working note had assumed mlse/mlsi took their constraint counts, which the source does not support – tda_mlse and tda_mls turn out to make the identical call (confirmed empirically: an overdetermined, inconsistent instance returns the plain least-squares fit rather than an error), and tda_mlsi treats its entire input as inequality rows. tda_mlsei and tda_mlsei1 solve the same general equality+least-squares+inequality problem via two different TDA algorithms (Lawson-Hanson vs. an undocumented active-set QP, confirmed to agree to solver tolerance on two independent instances, not merely assumed from the similar name). tda_mnls is the only one of the six with non-negativity, applied to every variable (TDA's k split point fixed at 0) and cannot be combined with inequalities (mnls has no general inequality block of its own).

Usage

tda_mls(A, b, dir = tempfile("tda"), ...)

tda_mlse(A, b, dir = tempfile("tda"), ...)

tda_mlsi(inequalities, inequalities_bounds, dir = tempfile("tda"), ...)

tda_mlsei(
  A = NULL,
  b = NULL,
  equalities = NULL,
  equalities_bounds = NULL,
  inequalities = NULL,
  inequalities_bounds = NULL,
  dir = tempfile("tda"),
  ...
)

tda_mlsei1(
  A = NULL,
  b = NULL,
  equalities = NULL,
  equalities_bounds = NULL,
  inequalities = NULL,
  inequalities_bounds = NULL,
  dir = tempfile("tda"),
  ...
)

tda_mnls(
  A,
  b,
  equalities = NULL,
  equalities_bounds = NULL,
  dir = tempfile("tda"),
  ...
)

Arguments

A

the least-squares data matrix (ma x n). Required for tda_mls/tda_mlse/tda_mnls; optional for tda_mlsei/tda_mlsei1 (NULL for a pure feasibility problem, together with equalities and/or inequalities); not used by tda_mlsi.

b

the target vector for A x ~ b, length ma.

dir

working directory.

...

passed to tda_run.

inequalities

an inequality constraint matrix Ai x >= inequalities_bounds. tda_mlsi (required, its only input) and tda_mlsei/tda_mlsei1 (optional).

inequalities_bounds

the right-hand sides for inequalities.

equalities

an equality constraint matrix Ae x = equalities_bounds. tda_mlsei/ tda_mlsei1/tda_mnls only.

equalities_bounds

the right-hand sides for equalities.

Value

A list: x, the solution; residual, the Euclidean norm of A %*% x - b (NA when A is absent), computed directly in R from A, x and b rather than trusted from TDA's text, since tda_mlsei1 prints no residual at all; rank, the reported rank of A (NA for tda_mnls and tda_mlsei1, neither of which report one).

Details

tda_mls, tda_mlse and tda_mlsi also accept the manual's single-matrix form, mls(S, B) with S = [X, y]: when the right-hand side is not given, the last column of the first argument is split off as y.

See also

Other optimization: tda_boolean_min(), tda_mlp(), tda_mlpi(), tda_mqap(), tda_mqp()

Examples

# unconstrained: fit x1, x2 to two noisy targets
tda_mls(diag(2), c(2.1, 2.9))
#> $x
#> [1] 2.1 2.9
#> 
#> $residual
#> [1] 0
#> 
#> $rank
#> [1] 2
#> 
# the manual's augmented form S = [X, y] works too
tda_mls(cbind(diag(2), c(2.1, 2.9)))
#> $x
#> [1] 2.1 2.9
#> 
#> $residual
#> [1] 0
#> 
#> $rank
#> [1] 2
#> 
# x1 + x2 = 3, x1 close to 2, x2 close to 3
tda_mlsei(diag(2), c(2, 3), equalities = rbind(c(1, 1)),
         equalities_bounds = 3)
#> $x
#> [1] 1 2
#> 
#> $residual
#> [1] 1.414214
#> 
#> $rank
#> [1] 1
#>