Modified barrier functions (theory and methods)
Mathematical Programming: Series A and B
Zero duality gap for a class of nonconvex optimization problems
Journal of Optimization Theory and Applications
Nonlinear rescaling and proximal-like methods in convex optimization
Mathematical Programming: Series A and B
Asymptotic analysis for penalty and barrier methods in convex and linear programming
Mathematics of Operations Research
Smoothing Functions for Second-Order-Cone Complementarity Problems
SIAM Journal on Optimization
Penalty/Barrier Multiplier Methods for Convex Programming Problems
SIAM Journal on Optimization
Augmented non-quadratic penalty algorithms
Mathematical Programming: Series A and B
SIAM Journal on Optimization
Perturbation analysis of second-order cone programming problems
Mathematical Programming: Series A and B
Primal-dual nonlinear rescaling method with dynamic scaling parameter update
Mathematical Programming: Series A and B
The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming
Mathematical Programming: Series A and B
Löwner's Operator and Spectral Functions in Euclidean Jordan Algebras
Mathematics of Operations Research
A trust region SQP-filter method for nonlinear second-order cone programming
Computers & Mathematics with Applications
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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.