Goldman-Tucker theorem

The Goldman-Tucker theorem establishes strict complementary slackness conditions for linear programs, if there is a finite-valued optimal solution, then,

$\exists x^*, \lambda^*$ with strict complementarity (only one of them zero) holding, i.e. not both of $x^*, \lambda^*$ are zero.

#incomplete

related to: Slater condition

References:

1. A. J. Goldman and A. W. Tucker, “4. Theory of Linear Programming,” in Linear Inequalities and Related Systems. (AM-38), Princeton University Press, 1957, pp. 53–98. doi: 10.1515/9781400881987-005.
2. https://mathoverflow.net/questions/303071/strict-complementary-slackness-for-semidefinite-programs-with-strong-duality
3. https://threesquirrelsdotblog.com/2023/06/06/strict-complementarity-of-linear-programs/