Abstract
We introduce new augmented Lagrangian algorithms for linear programming which provide faster global convergence rates than the augmented algorithm of Polyak and Treti'akov. Our algorithm shares the same properties as the Polyak-Treti'akov algorithm in that it terminates in finitely many iterations and obtains both primal and dual optimal solutions. We present an implementable version of the algorithm which requires only approximate minimization at each iteration. We provide a global convergence rate for this version of the algorithm and show that the primal and dual points generated by the algorithm converge to the primal and dual optimal set, respectively.
Similar content being viewed by others
References
Polyak, B. T., andTreti'akov, N. V.,An Iterative Method for Linear Programming and Its Economic Interpretation, Matekon, Vol. 8, pp. 81–100, 1972.
Güler, O.,New Proximal Point Algorithms for Convex Minimization, SIAM Journal on Optimization (to appear November 1992).
Bertsekas, D. P.,Necessary and Sufficient Conditions for a Penalty Method to Be Exact, Mathematical Programming, Vol. 9, pp. 87–99, 1975.
Rockafellar, R. T.,Monotone Operators and the Proximal Point Algorithm, SIAM Journal on Control and Optimization, Vol. 14, pp. 877–898, 1976.
Lin, Y. Y., andPang, J. S.,Iterative Methods for Large Convex Quadratic Programming: A Survey, SIAM Journal on Control and Optimization, Vol. 25, pp. 383–411, 1987.
Mangasarian, O. L., andDe Leone, R.,Serial and Parallel Solution of Large-Scale Linear Programs by Augmented Lagrangian Successive Overrelaxation, Optimization, Parallel Processing, and Applications, Edited by A. Kurzhanski, K. Neumann, and D. Pallaschke, Springer-Verlag, Berlin, Germany, pp. 103–124, 1988.
Wright, S. J.,Implementing Proximal Point Algorithms for Linear Programming, Journal of Optimization Theory and Applications, Vol. 65, pp. 531–554, 1990.
Polyak, B. T.,The Conjugate Gradient Method in Extremal Problems, USSR Computational Mathematics and Mathematical Physics, Vol. 9, pp. 94–112, 1969.
Gol'shtein, E. G.,An Iterative Linear Programming Algorithm Based on an Augmented Lagrangian, Nonlinear Programming 4, Edited by O. L. Mangasarian and S. M. robinson, Academic Press, New York, New York, pp. 131–146, 1981.
Hestenes, M. R.,Multiplier and Gradient Methods, Journal of Optimization Theory and Applications, Vol. 4, pp. 303–320, 1969.
Powell, M. J. D.,A Method for Nonlinear Constraints in Minimization Problems, Optimization, Edited by R. Fletcher, Academic Press, New York, New York, pp. 283–298, 1969.
Bertsekas, D. P.,Multiplier Methods: A Survey, Automatica, Vol. 12, pp. 133–145, 1976.
Bertsekas, D. P.,Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, New York, 1982.
Rockafellar, R. T.,Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming, Mathematics of Operations Research, Vol. 1, pp. 97–116, 1976.
Rockafellar, R. T.,The Multiplier Method of Hestenes and Powell Applied to Convex Programming, Journal of Optimization Theory and Applications, Vol. 12, pp. 555–562, 1973.
Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.
Güler, O.,On the Convergence of the Proximal Point Algorithm for Convex Minimization, SIAM Journal on Control and Optimization, Vol. 29, pp. 403–419, 1991.
Nesterov, Y. E.,On an Approach to the Construction of Optimal Methods of Minimization of Smooth Convex Functions, Ekonomika i Matematicheskie Metody, Vol. 24, pp. 509–517, 1988 (in Russian).
Nemirovsky, A. S., andYudin, D. B.,Problem Complexity and Method Efficiency in Optimization, Wiley, New York, New York, 1983.
Nesterov, Y. E.,A Method of Solving a Convex Programming Problem with Convergence Rate O 2), Soviet Mathematics Doklady, Vol. 27, pp. 372–376, 1983.
Hoffman, A. J.,On Approximate Solution of Systems of Linear Inequalities, Journal of Research of the National Bureau of Standards, Vol. 49, pp. 263–265, 1952.
Mangasarian, O. L., andMcLinden, L.,Simple Bounds for Solutions of Monotone Complementarity Problems and Convex Programs, Mathematical Programming, Vol. 32, pp. 32–40, 1985.
Cryer, C. W.,The Solution of a Quadratic Problem Using Systematic Overrelaxation, SIAM Journal on Control and Optimization, Vol. 9, pp. 385–392, 1971.
Author information
Authors and Affiliations
Additional information
Communicated by O. L. Mangasarian
Rights and permissions
About this article
Cite this article
Güler, O. Augmented Lagrangian algorithms for linear programming. J Optim Theory Appl 75, 445–470 (1992). https://doi.org/10.1007/BF00940486
Issue Date:
DOI: https://doi.org/10.1007/BF00940486