Quadratic programming

Minimize a convex quadratic subject to box bounds and/or linear equality/inequality constraints – TDA's mqp, mqpb, and mqpc (undocumented in tda.hlp; identified from qld's header, the Powell/Schittkowski active-set solver ZQPCVX shared with tda_mlsei1). The problem solved is $$\min\; d'x + \tfrac12 x'Cx \quad \mathrm{s.t.}\; A_e x = b_e,\; A_i x \ge b_i,\; l \le x \le u$$ tda_mqp is unconstrained (box bounds fixed at +-huge internally by TDA); tda_mqpb adds box bounds; tda_mqpc adds general equality/inequality constraints instead of bounds – no single TDA command accepts both bounds and general constraints. The suite checks all three commands against hand-solved instances: an unconstrained case, a box-clipped case, and a mixed equality+inequality case – see examples/coverage/mqpops2.cf.

Usage

tda_mqp(C, d, dir = tempfile("tda"), ...)

tda_mqpb(C, d, lower, upper, dir = tempfile("tda"), ...)

tda_mqpc(
  C,
  d,
  equalities = NULL,
  equalities_bounds = NULL,
  inequalities = NULL,
  inequalities_bounds = NULL,
  dir = tempfile("tda"),
  ...
)

Arguments

C

the symmetric objective matrix (n x n).

d

the linear objective term, length n.

dir

working directory.

...

passed to tda_run.

lower, upper

box bounds, each length n. tda_mqpb only.

equalities

an equality constraint matrix Ae x = equalities_bounds. tda_mqpc only.

equalities_bounds

the right-hand sides for equalities.

inequalities

an inequality constraint matrix Ai x >= inequalities_bounds. tda_mqpc only.

inequalities_bounds

the right-hand sides for inequalities.

Value

A list: x, the solution; value, the objective d'x + 0.5 x'Cx at x, computed directly in R since this command family prints no value of its own.

See also

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

Examples

# minimize x1^2 + x2^2 - 2x1 - 4x2 (unconstrained optimum (1, 2))
tda_mqp(diag(c(2, 2)), c(-2, -4))
#> $x
#> [1] 1 2
#> 
#> $value
#> [1] -5
#> 
tda_mqpb(diag(c(2, 2)), c(-2, -4), lower = c(0, 0), upper = c(0.5, 0.5))
#> $x
#> [1] 0.5 0.5
#> 
#> $value
#> [1] -2.5
#>