Abstract convexity and augmented Lagrangians
- Authors: Burachik, Regina , Rubinov, Alex
- Date: 2007
- Type: Text , Journal article
- Relation: SIAM Journal on Optimization Vol. 18, no. 2 (2007), p. 413-436
- Full Text: false
- Reviewed:
- Description: The ultimate goal of this paper is to demonstrate that abstract convexity provides a natural language and a suitable framework for the examination of zero duality gap properties and exact multipliers of augmented Lagrangians. We study augmented Lagrangians in a very general setting and formulate the main definitions and facts describing the augmented Lagrangian theory in terms of abstract convexity tools. We illustrate our duality scheme with an application to stochastic semiinfinite optimization. © 2007 Society for Industrial and Applied Mathematics.
- Description: C1
- Description: 2003005362
On augmented lagrangians for optimization problems with a single constraint
- Authors: Gasimov, Rafail , Rubinov, Alex
- Date: 2004
- Type: Text , Journal article
- Relation: Journal of Global Optimization Vol. 28, no. 2 (2004), p. 153-173
- Full Text: false
- Reviewed:
- Description: We examine augmented Lagrangians for optimization problems with a single (either inequality or equality) constraint. We establish some links between augmented Lagrangians and Lagrange-type functions and propose a new kind of Lagrange-type functions for a problem with a single inequality constraint. Finally, we discuss a supergradient algorithm for calculating optimal values of dual problems corresponding to some class of augmented Lagrangians.
- Description: C1
- Description: 2003000929
The zero duality gap property and lower semicontinuity of the perturbation function
- Authors: Rubinov, Alex , Huang, X. X. , Yang, Xiao
- Date: 2002
- Type: Text , Journal article
- Relation: Mathematics of Operations Research Vol. 27, no. 4 (2002), p. 775-791
- Full Text: false
- Reviewed:
- Description: We examine the validity of the zero duality gap properties for two important dual schemes: a generalized augmented Lagrangian dual scheme and a nonlinear Lagrange-type dual scheme. The necessary and sufficient conditions for the zero duality gap property to hold are established in terms of the lower semicontinuity of the perturbation functions.
- Description: 2003000117
The nonlinear and augmented Lagrangians for nonconvex optimization problems with a single constraint
- Authors: Rubinov, Alex , Gasimov, Rafail
- Date: 2002
- Type: Text , Journal article
- Relation: Applied and Computational Mathematics Vol. 1, no. 2 (2002), p. 142-157
- Full Text: false
- Reviewed:
- Description: The paper contains the survey of some recent results obtained by the authors and their colleagues. We study zero duality gap properties for optimization problems with a single constraint with respect to a nonlinear penalization. The penalty function is constructed as a convolution of the objective function and the constraint by means of IPH (increasing positively homogeneous) functions. The main results are obtained for penalization by strictly IPH functions. We also examine augmented Lagrangians for optimization problems with a single constraint. We establish some links between augmented Lagrangians and Lagrange-type functions and propose a new kind of Lagrange-type functions for the problems with a single inequality constraint.
- Description: C1
- Description: 2003000115