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The KKT System

  • Robert J. Vanderbei
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 196)

Abstract

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:
$$\displaystyle\begin{array}{rcl} A\Delta x + \Delta w& =& \rho {}\end{array}$$
(19.1)
$$\displaystyle\begin{array}{rcl}{ A}^{T}\Delta y - \Delta z& =& \sigma {}\end{array}$$
(19.2)
$$\displaystyle\begin{array}{rcl} Z\Delta x + X\Delta z& =& \mu e - XZe{}\end{array}$$
(19.3)
$$\displaystyle\begin{array}{rcl} W\Delta y + Y \Delta w& =& \mu e - Y We.{}\end{array}$$
(19.4)

Bibliography

  1. John, F. (1948). Extremum problems with inequalities as subsidiary conditions. In K. Fredrichs, O. Neugebauer, and J. Stoker (Eds.), Studies and essays: Courant anniversary volume (pp. 187–204). New York: Wiley.Google Scholar
  2. Karush, W. (1939). Minima of functions of several variables with inequalities as side conditions (Technical report, M.S. thesis) Department of Mathematics, University of Chicago.Google Scholar
  3. Kuhn, H. (1976). Nonlinear prgramming: A historical view. In R. Cottle and C. Lemke (Eds.), Nonlinear programming, SIAM-AMS proceedings (Vol. 9, pp. 1–26). Providence: American Mathetical Society.Google Scholar
  4. Kuhn, H., and Tucker, A. (1951). Nonlinear prgramming. In J. Neyman (Ed.), Proceedings of the second Berkeley symposium on mathematical statistics and probability (pp. 481–492). Berkeley: University of California Press.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Robert J. Vanderbei
    • 1
  1. 1.Department of Operations Research and Financial EngineeringPrinceton UniversityPrincetonUSA

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