Existence Of Augmented Lagrange Multipliers Research Articles

Second-Order Conditions for the Existence of Augmented Lagrange Multipliers for Sparse Optimization

Second-Order Conditions for the Existence of Augmented Lagrange Multipliers for Sparse Optimization

A Unified Approach to the Global Exactness of Penalty and Augmented Lagrangian Functions I: Parametric Exactness

In this two-part study we develop a unified approach to the analysis of the global exactness of various penalty and augmented Lagrangian functions for finite-dimensional constrained optimization problems. This approach allows one to verify in a simple and straightforward manner whether a given penalty/augmented Lagrangian function is exact, i.e. whether the problem of unconstrained minimization of this function is equivalent (in some sense) to the original constrained problem, provided the penalty parameter is sufficiently large. Our approach is based on the so-called localization principle that reduces the study of global exactness to a local analysis of a chosen merit function near globally optimal solutions. In turn, such local analysis can usually be performed with the use of sufficient optimality conditions and constraint qualifications. In the first paper we introduce the concept of global parametric exactness and derive the localization principle in the parametric form. With the use of this version of the localization principle we recover existing simple necessary and sufficient conditions for the global exactness of linear penalty functions, and for the existence of augmented Lagrange multipliers of Rockafellar-Wets' augmented Lagrangian. Also, we obtain completely new necessary and sufficient conditions for the global exactness of general nonlinear penalty functions, and for the global exactness of a continuously differentiable penalty function for nonlinear second-order cone programming problems. We briefly discuss how one can construct a continuously differentiable exact penalty function for nonlinear semidefinite programming problems, as well.

Existence of Augmented Lagrange Multipliers for Semi-infinite Programming Problems

Using an augmented Lagrangian approach, we study the existence of augmented Lagrange multipliers of a semi-infinite programming problem and discuss their characterizations in terms of saddle points. In the case of a sharp Lagrangian, we obtain a first-order necessary condition for the existence of an augmented Lagrange multiplier for the semi-infinite programming problem and some first-order sufficient conditions by assuming inf-compactness of the data functions and the extended Mangasarian---Fromovitz constraint qualification. Using a valley at 0 augmenting function and assuming suitable second-order sufficient conditions, we obtain the existence of an augmented Lagrange multiplier for the semi-infinite programming problem.

Existence of augmented Lagrange multipliers: reduction to exact penalty functions and localization principle

In this article, we present new general results on existence of augmented Lagrange multipliers. We define a penalty function associated with an augmented Lagrangian, and prove that, under a certain growth assumption on the augmenting function, an augmented Lagrange multiplier exists if and only if this penalty function is exact. We also develop a new general approach to the study of augmented Lagrange multipliers called the localization principle. The localization principle allows one to study the local behaviour of the augmented Lagrangian near globally optimal solutions of the initial optimization problem in order to prove the existence of augmented Lagrange multipliers.

Second-order conditions for existence of augmented Lagrange multipliers for eigenvalue composite optimization problems

In this paper, we mainly consider the augmented Lagrangian duality theory and explore second-order conditions for the existence of augmented Lagrange multipliers for eigenvalue composite optimizati...

Second-order conditions for existence of augmented Lagrange multipliers for eigenvalue composite optimization problems

In this paper, we mainly consider the augmented Lagrangian duality theory and explore second-order conditions for the existence of augmented Lagrange multipliers for eigenvalue composite optimization problems. In the approach, we reformulate the augmented Lagrangian introduced by Rockafellar into a new form in terms of the Moreau envelope function and characterize second-order conditions via the epi-derivatives of the augmented Lagrangian.

Augmented Lagrangian Duality for Composite Optimization Problems

In this paper, augmented Lagrangian duality is considered for composite optimization problems, and first- and second-order conditions for the existence of augmented Lagrange multipliers are presented. The analysis is based on the reformulation of the augmented Lagrangian in terms of the Moreau envelope functions and the technique of epi-convergence via the calculation of second-order epi-derivatives of the augmented Lagrangian. It is also proved that the second-order conditions for optimization problems with abstract constraints given in a form of set inclusions, obtained by Shapiro and Sun, can be derived directly from our general results.

Existence of augmented Lagrange multipliers for cone constrained optimization problems

In this paper, by using an augmented Lagrangian approach, we obtain several sufficient conditions for the existence of augmented Lagrange multipliers of a cone constrained optimization problem in Banach spaces, where the corresponding augmenting function is assumed to have a valley at zero. Furthermore, we deal with the relationship of saddle points, augmented Lagrange multipliers, and zero duality gap property between the cone constrained optimization problem and its augmented Lagrangian dual problem.

On a characterization of convergence for the Hestenes method of multipliers

The connection between the convergence of the Hestenes method of multipliers and the existence of augmented Lagrange multipliers for the constrained minimum problem (P): minimizef(x), subject tog(x)=0, is investigated under very general assumptions onX,f, andg. In the first part, we use the existence of augmented Lagrange multipliers as a sufficient condition for the convergence of the algorithm. In the second part, we prove that this is also a necessary condition for the convergence of the method and the boundedness of the sequence of the multiplier estimates. Further, we give very simple examples to show that the existence of augmented Lagrange multipliers is independent of smoothness condition onf andg. Finally, an application to the linear-convex problem is given.