Slater's condition

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In mathematics, Slater's condition (or Slater condition) is a sufficient condition for strong duality to hold for a convex optimization problem, named after Morton L. Slater.[1] Informally, Slater's condition states that the feasible region must have an interior point (see technical details below).

Slater's condition is a specific example of a constraint qualification.[2] In particular, if Slater's condition holds for the primal problem, then the duality gap is 0, and if the dual value is finite then it is attained.[3]

Formulation[edit]

Consider the optimization problem

where are convex functions. This is an instance of convex programming.

In words, Slater's condition for convex programming states that strong duality holds if there exists an such that is feasible (i.e. all constraints are satisfied and the nonlinear constraints are satisfied with strict inequalities).

Mathematically, Slater's condition states that strong duality holds if there exists an (where relint denotes the relative interior of the convex set ) such that

(the convex, nonlinear constraints)
[4]

Generalized Inequalities[edit]

Given the problem

where is convex and is -convex for each . Then Slater's condition says that if there exists an such that

and

then strong duality holds.[4]

References[edit]

  1. ^ Slater, Morton (1950). Lagrange Multipliers Revisited (PDF). Cowles Commission Discussion Paper No. 403 (Report). Reprinted in Giorgi, Giorgio; Kjeldsen, Tinne Hoff, eds. (2014). Traces and Emergence of Nonlinear Programming. Basel: Birkhäuser. pp. 293–306. ISBN 978-3-0348-0438-7.
  2. ^ Takayama, Akira (1985). Mathematical Economics. New York: Cambridge University Press. pp. 66–76. ISBN 0-521-25707-7.
  3. ^ Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2nd ed.). Springer. ISBN 0-387-29570-4.
  4. ^ a b Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 3, 2011.