Abstract
The rediscovery of the affine scaling method in the late 1980s was one of the turning points which led to a new chapter in Modern Optimization—the interior point methods (IPMs). Simultaneously and independently, the theory of exterior point methods for convex optimization arose. The two seemingly unconnected fields turned out to be intrinsically connected. The purpose of this paper is to show the connections between primal exterior and dual IPMs. Our main tool is the Lagrangian transformation (LT), which for inequality constrained has the best features of the classical augmented Lagrangian. We show that the primal exterior LT method is equivalent to the dual interior ellipsoid method (IEM). Using the equivalence we prove convergence, estimate the convergence rate, and establish the complexity bound for the IEM assuming boundedness of both the primal and the dual optimal sets. We show that application of the LT method with modified barrier transformation for linear programming (LP) leads to Dikin’s affine scaling method for the dual LP.
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References
Dikin, I.: Iterative solutions of linear and quadratic programming problems. Sov. Math. Doklady. 8, 674–675 (1967)
Barnes, E.: A variation on Karmarkar’s algorithm for solving linear programming problems. Math. Program. 36, 174–182 (1986)
Vandarbei, R., Meketon, M., Freedman, B.: A modification of Karmarkar’s linear programming algorithm. Algorithmica 1(4), 395–407 (1986)
Karmarkar, N.: A new polynomial time algorithm for linear programing. Combinatorica 4, 373–395 (1984)
Gill, P., Murray, W., Saunders, M., Tomlin, J., Wright, M.: On projected Newton barrier methods for linear programming and an equivalence to Karmarkar’s projected method. Math. Program. 36, 183–209 (1986)
Gonzaga, C.: An algorithm for solving Linear Programming problems in O(\(n^3L\)) operations. In: Megiddo, N. (ed.) Progress in Mathematical Programming, pp. 1–82. Springer, Berlin (1988)
Renegar, J.: A polynomial time algorithm, based on Newton’s method for linear programming. Math. Program. 40, 59–93 (1988)
Potra, F.A., Wright, S.J.: Interior-point methods. J. Comput. Appl. Math. 124, 281–302 (2000)
Nesterov, Yu., Nemirovsky, A.: Interior Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia (1994)
Nesterov, Yu.: Introductory Lectures on Convex Optimization. Kluwer Academic Publishers, Norwell (2004)
Auslender, A., Teboulle, M., Ben-Tiba, S.: Interior primal and multipliers methods based on second-order homogeneous kernels. Math. Oper. Res. 24(3), 645–668 (1999)
Bertsekas, D.: Constrained Optimization and Lagrange Multipliers Methods. Academic Press, New York (1982)
Polyak, R.: Modified barrier functions (theory and methods). Math. Program. 54, 177–222 (1992)
Polyak, R., Teboulle, M.: Nonlinear rescaling and proximal-like methods in convex optimization. Math. Program. 76, 265–284 (1997)
Ben-Tal, A., Yuzefovich, B., Zibulevsky, M.: Penalty-barrier multipliers method for minimax and constrained smooth convex optimization. Technion, Research report, 9–92 (1992).
Ben-Tal, A., Zibulevsky, M.: Penalty—barrier methods for convex programming problems. SIAM J. Optim. 7, 347–366 (1997)
Polyak, R.: Nonlinear rescaling vs. smoothing technique in convex optimization. Math. Program. 92, 197–235 (2002)
Tseng, P., Bertsekas, D.: On the convergence of exponential multiplier method for convex programming. Math. Program. 60, 1–19 (1993)
Teboulle, M.: Entropic proximal mappings with applications to nonlinear programming. Y Math. Oper. 17, 97–116 (1992)
Griva, I., Polyak, R.: Primal–dual nonlinear rescaling method with dynamic scaling parameter update. Math. Program. Ser. A 106, 237–259 (2006)
Kočvara, M., Stingl, M.: On the solution of large-scale SDP problems by the modified barrier method using iterative solvers. Math. Program. Ser. B 109(2–3), 413–444 (2007)
Polyak, R.: Nonlinear rescaling as interior quadratic prox method in convex optimization. Comput. Optim. Appl. 35, 347–373 (2006)
Polyak, R.: Lagrangian Transformation in Convex Optimization, Research Report-072003, pp. 1–23. Department of SEOR and Mathematical Science Department, GMU, Fairfax (2003)
Matioli, L., Gonzaga, C.: A new family of penalties for Augmented Lagrangian methods. Numer. Linear Algebra Appl. 15, 925–944 (2008)
Fiacco, A., Mc Cormick, G.: Nonlinear programming, sequential unconstrained minimization techniques. Y SIAM (1990). doi:10.1137/1.9781611971316
Bregman, L.: The relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7, 200–217 (1967)
Censor, Y., Zenios, S.: The proximal minimization algorithm with d-functions”. J. Optim. Theory Appl. 73, 451–464 (1992)
Chen, G., Teboulle, M.: Convergence analysis of a proximal-like minimization algorithm using Bregman Functions. SIAM J. Optim. 3(4), 538–543 (1993)
Eckstein, J.: Nonlinear proximal point algorithms using Bregman functions with applications to convex programming. Y. Math. Oper. Res. 18(1), 202–226 (1993)
Hestenes, M.R.: Multipliers and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969)
Polyak, B.: Introduction to Optimization. Optimization Software, New York (1987)
Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, (ed.) Optimization, pp. 283–298. Academic Press, London (1969)
Rockafellar, R.T.: A dual approach to solving nonlinear programming problems by unconstrained minimization. Math. Program. 5, 354–373 (1973)
Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Rockafellar, R.T.: Augmented lagrangians and applications of the proximal points algorithms in convex programming. Y Math. Oper. Res. 1, 97–116 (1976)
Guler, O.: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 29, 403–419 (1991)
Polyak, R.: Primal–dual exterior point method for convex optimization. Optim. Methods Softw. 23(1), 141–160 (2008)
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The research was supported by NSF Grant CCF–0836338.
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Communicated by Florian A. Potra.
In memory of Ilya Dikin.
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Polyak, R. Lagrangian Transformation and Interior Ellipsoid Methods in Convex Optimization. J Optim Theory Appl 164, 966–992 (2015). https://doi.org/10.1007/s10957-014-0527-5
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DOI: https://doi.org/10.1007/s10957-014-0527-5
Keywords
- Lagrangian transformation
- Interior point methods
- Duality
- Bregman distance
- Augmented lagrangian
- Interior ellipsoid method