Abstract
This paper focuses on the study of a class of nonlinear Lagrangians for solving nonconvex second order cone programming problems. The nonlinear Lagrangians are generated by Löwner operators associated with convex real-valued functions. A set of conditions on the convex real-valued functions are proposed to guarantee the convergence of nonlinear Lagrangian algorithms. These conditions are satisfied by well-known nonlinear Lagrangians appeared in the literature. The convergence properties for the nonlinear Lagrange method are discussed when subproblems are assumed to be solved exactly and inexactly, respectively. The convergence theorems show that, under the second order sufficient conditions with sigma-term and the strict constraint nondegeneracy condition, the algorithm based on any of nonlinear Lagrangians in the class is locally convergent when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter. Compared to the analysis in nonlinear Lagrangian methods for nonlinear programming, we have to deal with the sigma term in the convergence analysis. Finally, we report numerical results by using modified Frisch’s function, modified Carroll’s function and the Log-Sigmoid function.
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References
Auslender, A., Cominetti, R., Haddou, M.: Asymptotic analysis of penalty and barrier methods in convex and linear programming. Math. Oper. Res. 22, 43–62 (1997)
Ben-Tel, A., Zibulevsky, M.: Penalty/barrier multiplier methods for convex programming problems. SIAM J. Optim. 7(2), 347–366 (1997)
Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Academic Press, New York (1982)
Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)
Bonnans, J.F., Ramírez, C.H.: Perturbation analysis of second order cone programming problems. Math. Program., Ser. B 104, 205–227 (2005)
Breitfeld, M., Shanno, D.: A globally convergent penalty-barrier algorithm for nonlinear programming and its computational performance. Technical Report, Rutcor Research Report, Rutgers University, New Jersey (1994)
Charalambous, C.: Nonlinear least pth optimization and nonlinear programming. Math. Program. 12, 195–225 (1977)
Dussault, J.P.: Augmented non-quadratic penalty algorithms. Math. Program. 99, 467–486 (2004)
Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Clarendon, Oxford (1994)
Fukushima, M., Luo, Z.-Q., Tseng, P.: Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim. 12, 436–460 (2001)
Griva, I., Polyak, R.A.: Primal-dual nonlinear rescaling method with dynamic scaling parameter update. Math. Program. 106, 237–259 (2006)
Hayashi, S., Yamashita, N., Fukushima, M.: A combined smoothing and regularization method for monotone second-order cone complementarity problems. SIAM J. Optim. 15, 593–615 (2005)
Hestenes, M.R.: Multiplier and gradient method. J. Optim. Theory Appl. 4, 303–320 (1969)
Kato, H., Fukushima, M.: An SQP-type algorithm for nonlinear second-order cone programs. Optim. Lett. 1, 129–144 (2007)
Li, D.: Zero duality gap for a class of nonconvex optimization problems. J. Optim. Theory Appl. 85, 309–323 (1995)
Liu, Y.J., Zhang, L.W.: Convergence analysis of the augmented Lagrangian method for nonlinear second-order cone optimization problems. Nonlinear Anal. 67, 1359–1373 (2007)
Liu, Y.J., Zhang, L.W.: On the approximate augmented Lagrangian for nonlinear second order cone programming. Nonlinear Anal. Theory Methods Appl. (2008). doi:10.1016/j.na.2006.12.016
Mangasarian, O.L.: Unconstrained Lagrangians in nonlinear programming. SIAM J. Control 13, 772–791 (1975)
Polyak, R.A.: Modified barrier function: theory and methods. Math. Program. 54, 177–222 (1992)
Polyak, R.A.: Log-Sigmoid multipliers method in constrained optimization. Ann. Oper. Res. 101, 427–460 (2001)
Polyak, R.A.: Nonlinear rescaling vs. smoothing technique in convex optimization. Math. Program. 92, 197–235 (2002)
Polyak, R.A., Griva, I.: Primal-Dual nonlinear rescaling method for convex optimization. J. Optim. Theory Appl. 122, 111–156 (2004)
Polyak, R.A., Teboulle, M.: Nonlinear rescaling and proximal-like methods in convex optimization. Math. Program. 76, 265–284 (1997)
Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283–298. Academic Press, New York (1969)
Rockafellar, R.T.: A dual approach to solving nonlinear programming problems by unconstrained optimization. Math. Program. 5, 354–373 (1973)
Rockafellar, R.T.: The multiplier method of Hestenes and Powell applied to convex programming. J. Optim. Theory Appl. 12, 555–562 (1973)
Rockafellar, R.T.: Augmented Lagrange multiplier functions and duality in nonconvex programming. SIAM J. Control 12, 268–285 (1974)
Sun, D.F., Sun, J.: Löwner’s operator and spectral functions in Euclidean Jordan algebras. Math. Oper. Res. 33 421–445 (2008)
Sun, J., Zhang, L.W., Wu, Y.: Properties of the augmented Lagrangian in nonlinear semidefinite optimization. J. Optim. Theory Appl. 129, 437–456 (2006)
Sun, D.F., Sun, J., Zhang, L.W.: The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Math. Program. (2008) doi:10.1007/s10107-007-0105-9
Stingl, M.: On the Solution of Nonlinear Semidefinite Programs by Augmented Lagrangian Methods. Shaker, Aachen (2006). Dissertation
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The research is supported by the National Natural Science Foundation of China under project No. 10771026.
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Zhang, L., Gu, J. & Xiao, X. A class of nonlinear Lagrangians for nonconvex second order cone programming. Comput Optim Appl 49, 61–99 (2011). https://doi.org/10.1007/s10589-009-9279-9
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DOI: https://doi.org/10.1007/s10589-009-9279-9