Value of the dual objective for a candidate pair of potentials

The dual turns a search over couplings into a search over two functions on the marginals. Any feasible pair gives a lower bound on the transport cost, so this is the optimum only when the potentials are. Feasibility against a cost matrix is not checked.

Usage

Ototk(a, b, f, g)

Arguments

a, b

Source and target marginals.

f, g

Kantorovich potentials.

Value

List with dual_val, estimate, n, m.

Details

Formula: <a, f> + <b, g>.

References

Villani, C. (2003). Topics in Optimal Transportation, AMS GSM 58, theorem 1.3.

Examples

Ototk(a = c(1, 2, 3, 4, 5, 6, 7, 8), b = 5L, f = c(1, 2, 3, 4, 5, 6, 7, 8),
  g = c(1, 2, 3, 4, 5, 6, 7, 8))
#> $dual_val
#> [1] 384
#> 
#> $estimate
#> [1] 384
#> 
#> $n
#> [1] 8
#> 
#> $m
#> [1] 1
#> 
#> $method
#> [1] "Kantorovich dual objective value"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"