Convergence of an augmented Lagrange algorithm for nonlinear optimizations with second-order cone constraints

Abstract

An augmented Lagrange algorithm for nonlinear optimizations with second-order cone constraints is proposed based on a Löwner operator associated with a potential function for the optimization problems with inequality constraints. The favorable properties of both the Löwner operator and the corresponding augmented Lagrangian are discussed. And under some mild assumptions, the rate of convergence of the augmented Lagrange algorithm is studied in detail.

Key words： Potential function; Löwner operator; augmented Lagrange algorithm; nonlinear second-order cone optimizations

References

1 Alizadeh F , Goldfarb D . Second-order cone programming. Mathematical Programming, 2003, 95 (1): 3- 51

2 Bonnans J F , Ramírez C H . Perturbation analysis of second-order cone programming problems. Mathematical Programming, 2005, 104 (2): 205- 227

3 Bonnans J F , Shapiro A . Perturbation Analysis of Optimization Problems. New York: Springer-Verlag, 2000

4 Debreu G . Definite and semidefinite quadratic forms. Econometrica, 1952, 20: 295- 300

5 Faraut J , Korányi A . Analysis on Symmetric Cones. Oxford: Clarendon Press, 1994

6 Gu J , Zhang L W , Xiao X T . Log-Sigmoid nonlinear Lagrange method for nonlinear optimization problems over second-order cones. Journal of Computational and Applied Mathematics, 2009, 229 (1): 129- 144

7 He S X , Zhang L W , Li X S . A Potential Function for Solving Inequality Constrained Optimiztion Problems. Advances in Mathematics (China), 2004, 33 (3): 343- 351

8 Hestenes M R . Multiplier and gradient methods. Journal of Optimization Theory and Applications, 1969, 4 (5): 303- 320

9 Liu Y J , Zhang L W . Convergence of the Augmented Lagrangian Method for Nonlinear Optimization Problems over Second-Order Cones. Journal of Optimization Theory and Applications, 2008, 139 (3): 557- 575

10 Lobo M S , Vandenberghe L , Boyd S , Lebret H . Applications of second-order cone programming. Linear Algebra and its Applications, 1998, 284 (1): 193- 228

11 Lou J M , Zhong Q B . Cooperative motion control method for humanoid robot. Journal of Beijing University of Posts and Telecommunications, 2017, 40 (3): 123- 126 (in Chinese)

12 Polyak R A . Log-Sigmoid multipliers method in constrained optimization. Annals of Operations Research, 2001, 101 (1): 427- 460

13 Powell M J D . A Method for Nonlinear Constraints in Minimization Problems. Optimization, 1969: 283- 298

14 Rockafellar R T . A dual approach to solving nonlinear programming problems by unconstrained optimization. Mathematical Programming, 1973, 5 (1): 354- 373

15 Rockafellar R T . Augmented Lagrange Multiplier Functions and Duality in Nonconvex Programming. SIAM Journal on Control, 1974, 12 (2): 268- 285

16 Zhang L W , Gu J , Xiao X T . A class of nonlinear Lagrangians for nonconvex second order cone programming. Computational Optimization and Applications, 2011, 49 (1): 61- 99

17 Zhang L W , Ren Y H , Wu Y , Xiao X T . A class of nonlinear Lagrangians: theory and algorithm. Asia-Pacific Journal of Operational Research, 2008, 25 (3): 327- 371