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References
Bertsekas, D. (1982). Constrained Optimization and Lagrange Multipliers Methods, Academic Press, New York.
Carroll, C. (1961). “The created response surface technique for optimizing nonlinear restrained systems,” Operations Research, 9, 169–184.
Eggermont, P. (1990). “Multiplicative Iterative Algorithm for Convex Programming,” Linear Algebral and its Applications, 130, 25–42.
Fiacco, A.V. and McCormick, G.P. (1968). Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley, New York.
Frisch K. (1955). “The logarithmic potential method of convex programming,” Technical Memorandum, May 13, University Institute of Economics, Oslo.
Gonzaga, C. (1992). “Path Following Methods for Linear Programming,” SIAM Review, 34, 167–224.
Gill, P., Murray, W., Saunders, M., Tomlin, J., and Wright, M. (1986). “On Projected Barrier Methods for Linear Programming and an Equivalence to Karmarkar's Projective Method,” Mathematical Programming, 36, 183–209.
Golshtein, E.G. and Tretyakov, N.V. (1974). “Modified Lagrangean Functions,” Economics and Math. Methods, 10, 568–591 (Russian).
Hestenes, M. (1969). “Multiplier and gradient methods,” Jl. Optimization Theory & Applications, 4, 303–320.
Huard, P. (1967). “Resolution of Mathematical Programming with Nonlinear Constraints by the Method of Centers,” in Nonlinear Programming, J. Abadie, ed., North-Holland, Amsterdam.
Jensen, D. and Polyak, R. (1994). “The convergence of the modified barrier method for convex programming.” IBM Jl. R&D, 38, 307–321.
Karmarkar, N. (1984). “A New Polynomial-time Algorithm for Linear Programming,” Combinatorica, 4, 373–395.
Lustig, I., Marsten, R., and Shanno, D. (1992). “On implementing Mehrotra's Predictor-Corrector Interior Point Method for Linear Programming,” Siam Jl. Optimization, 2, 435–449.
Mangasarian, O. (1975). “Unconstrained Lagrangians in Nonlinear Programming,” SIAM Jl. Control, 13, 772–791.
Melman, A. and Polyak, R. (1996). “The Newton Modified Barrier Method for QP,” Annals of Operations Research, 62, 465–519.
Mehrotra, S. (1992). “On the Implementation of Primal-Dual Interior Point Method,” SIAM Jl. Optimization, 2, 575–601.
Nesterov, Yu. and Nemirovsky, A. (1994). Interior Point Polynomial Algorithms in Convex Programming. SIAM Studies in Applied Mathematics, Philadelphia.
Polyak, B.T. and Tretyakov, N.V. (1973). “The method of penalty bounds for constrained extremum problems,” Zh. Vych. Mat. iMat. Fiz, 13, 34–46, USSR Computational Methods Math. Phys., 13, 42–58.
Polyak, R. (1986). Controlled processes in extremal and equilibrium problems. VINITY, Moscow (Russian).
Polyak, R. (1992). “Modified Barrier Functions (Theory and Methods),” Mathematical Programming, 54 177–222.
Polyak, R. (1996). “Modified Barrier Functions in Linear Programming,” Research Report, Dept. of Operations Research, George Mason University, 1–56.
Polyak, R. (1997). “Modified Interior Distance Functions,” Contemporary Mathematics, 209, 183–209.
Polyak, R. and Teboulle, M. (1997). “Nonlinear Rescaling and Proximal-like Methods in Convex Optimization,” Mathematical Programming, 76, 265–284.
Powell, M. (1969). “A Method for Nonlinear Constraints in Minimization Problems,” in Optimization, R. Fletcher, ed., Academic Press, New York.
Powell, M. (1995). “Some Convergence Properties of the Modified Log Barrier Method for Linear Programming,” SIAM Jl. Optimization, 5, 695–739.
Renegar, J. (1988). “A polynomial-time algorithm, based on Newton's method for linear programming,” Mathematical Programming, 40, 59–93.
Rockafellar, R.T. (1973). “The Multiplier Method of Hestenes and Powell Applied to Convex Programming,” Jl. Optimization Theory & Applications, 12, 555–562.
Roos, C., Terlaky, T., and Vial, J.-Ph. (1997). Theory and Algorithms for Linear Optimization: An Interior Point Approach, John Wiley, New York.
Smale, S. (1986). “A Newton method estimates from data at one point,” in The Merging of Disciplines in Pure, Applied and Computational Mathematics, R. Ewing, ed., Springer-Verlag, New York/Berlin.
Teboulle, M. (1993). “Entropic proximal mappings with application to nonlinear programming,” Mathematics of Operations Research, 17, 670–690.
Wright, S. (1997). Primal-Dual-Interior Point Methods, SIAM, Philadelphia.
Yinyu, Ye (1997). Interior Point Algorithms: Theory and Analysis, John Wiley, New York.
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Polyak, R.A. (2001). BARRIER FUNCTIONS AND THEIR MODIFICATIONS. In: Gass, S.I., Harris, C.M. (eds) Encyclopedia of Operations Research and Management Science. Springer, New York, NY. https://doi.org/10.1007/1-4020-0611-X_57
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