The KKT System

  • Robert J. Vanderbei
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 37)


The most time-consuming aspect of each iteration of the path-following method is solving the system of equations that defines the step direction vectors Δx, Δy, Δw, and Δz:
$$A\Delta x + \Delta w = \rho $$
$${A^T}\Delta y - \Delta z = \sigma $$
$$Z\Delta x + X\Delta z = \mu e - XZe$$
$$W\Delta y + Y\Delta w = \mu e - YWe.$$


Normal Equation Dense Column Positive Semidefinite Step Direction Positive Semidefinite Matrice 
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  1. The KKT system for general inequality constrained optimization problems was derived by Kuhn & Tucker (1951). It was later discovered that W. Karush had proven the same result in his 1939 master’s thesis at the University of Chicago (Karush 1939 ). John (1948) was also an early contributor to inequality-constrained optimization. Kuhn’s survey paper (Kuhn 1976) gives a historical account of the development of the subject.Google Scholar

Copyright information

© Robert J. Vanderbei 2001

Authors and Affiliations

  • Robert J. Vanderbei
    • 1
  1. 1.Dept. of Operations Research & Financial EngineeringPrinceton UniversityUSA

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