Subdifferentials of multifunctions and Lagrange multipliers for multiobjective optimization

This paper investigates vector optimization problems with objective and the constraints are multifunctions. By using a special scalarization function introduced in optimization by Hiriart-Urruty, we establish optimality conditions in terms of Lagrange-Fritz-John and Lagrange-Kuhn-Tucker multipliers. When all the data of the problem are subconvexlike we derive the results by Li, and hence those of Lin and Corley. We also show how the generalized Moreau-Rockafellar type theorem to multifunctions obtained recently by Lin can be derived from the well-known results in scalar optimization. In the last, vector optimization problem in which objective and the constraints are defined by multifunctions and depends on a parameter u, and the resulting value multifunction M(u) are considered. With the help of the generalized Moreau-Rockafellar type theorem we establish the weak subdifferential of M in terms of the weak subdifferential of objective and constraint multifunctions.     

Lagrange multipliers for set-valued optimization problems associated with coderivatives

In this paper we investigate a vector optimization problem (P) where objective and constraints are given by set-valued maps. We show that by mean of marginal functions and suitable scalarizing functions one can characterize certain solutions of (P) as solutions of a scalar optimization problem (SP) with single-valued objective and constraint functions. Then applying some classical or recent results in optimization theory to (SP) and using estimates of subdifferentials of marginal functions, we obtain optimality conditions for (P) expressed in terms of Lagrange or sequential Lagrange multipliers associated with various coderivatives of the set-valued data.     

Lagrange optimality system for a class of nonsmooth convex optimization

In this paper, we revisit the augmented Lagrangian method for a class of nonsmooth convex optimization. We present the Lagrange optimality system of the augmented Lagrangian associated with the problems, and establish its connections with the standard optimality condition and the saddle point condition of the augmented Lagrangian, which provides a powerful tool for developing numerical algorithms: we derive a Lagrange–Newton algorithm for the nonsmooth convex optimization, and establish the nonsingularity of the Newton system and the local convergence of the algorithm.     

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.     

Bounded sets of Lagrange multipliers for vector optimization problems in infinite dimension

The aim of this paper is to point out some sufficient constraint qualification conditions ensuring the boundedness of a set of Lagrange multipliers for vectorial optimization problems in infinite dimension. In some (smooth) cases these conditions turn out to be necessary for the existence of multipliers as well.     

On the limiting properties of dual trajectories in the Lagrange multipliers method

For the method of Lagrange multipliers (i.e., augmented Lagrangians), possible and typical scenarios for the asymptotic behavior of dual trajectories are examined in the case where the Lagrange multiplier is nonunique. The influence of these scenarios on the convergence rate is also investigated.     

On Lagrange multipliers of trust-region subproblems

Trust-region methods are globally convergent techniques widely used, for example, in connection with the Newton’s method for unconstrained optimization. One of the most commonly-used iterative approaches for solving trust-region subproblems is the Steihaug–Toint method which is based on conjugate gradient iterations and seeks a solution on Krylov subspaces. This paper contains new theoretical results concerning properties of Lagrange multipliers obtained on these subspaces. AMS subject classification (2000)  65F20     

Lagrange multipliers and infinite-dimensional equilibrium problems

We prove the existence of the Lagrange multipliers for a constrained optimization problem, being the constraint set given by the convex set which characterizes the most important equilibrium problems. In order to obtain our result, we’ll make use of the new concept of quasi relative interior.     

Interpretation of Lagrange multipliers in nonlinear pricing problem

We present well-known interpretations of Lagrange multipliers in physical and economic applications, and introduce a new interpretation in nonlinear pricing problem. The multipliers can be interpreted as a network of directed flows between the buyer types. The structure of the digraph and the fact that the multipliers usually have distinctive values can be used in solving the optimization problem more efficiently. We also find that the multipliers satisfy a conservation law for each node in the digraph, and the non-uniqueness of the multipliers are connected to the stability of the solution structure.     

Slopes of shadow prices and Lagrange multipliers

Many economic models and optimization problems generate (endogenous) shadow prices—alias dual variables or Lagrange multipliers. Frequently the “slopes” of resulting price curves—that is, multiplier derivatives—are of great interest. These objects relate to the Jacobian of the optimality conditions. That particular matrix often has block structure. So, we derive explicit formulas for the inverse of such matrices and, as a consequence, for the multiplier derivatives.     

Measures as Lagrange multipliers in multistage stochastic programming

A duality theory is developed for multistage convex stochastic programming problems whose decision (or recourse) functions can be approximated by continuous functions satisfying the same constraints. Necessary and sufficient conditions for optimality are obtained in terms of the existence of multipliers in the class of regular Borel measures on the underlying probability space, these being decomposable, of course, into absolutely continuous and singular components with respect to the given probability measure. This provides an alternative to the approach where the multipliers are elements of the dual of $\text{L}$^∞ with an analogous decomposition. However, besides the existence of strictly feasible solutions, special regularity conditions are required, such as the “laminarity” of the probability measure, a property introduced in an earlier paper. These are crucial in ensuring that the minimum in the optimization problem can indeed be approached by continuous functions.     

A note on the existence of Lagrange multipliers

This note extends a Lagrange multiplier theorem due to J. Zowe and S. Kurcyusz. Under some additional assumptions the required regularity condition can be weakened. The cone corresponding to the explicit constraint is replaced by its closure.     

Convergence of a substructuring method with Lagrange multipliers

Summary. We analyze the convergence of a substructuring iterative method with Lagrange multipliers, proposed recently by Farhat and Roux. The method decomposes finite element discretization of an elliptic boundary value problem into Neumann problems on the subdomains plus a coarse problem for the subdomain nullspace components. For linear conforming elements and preconditioning by the Dirichlet problems on the subdomains, we prove the asymptotic bound on the condition number , or ,where is the characteristic element size and subdomain size. Received January 3, 1995     

Constraint qualifications and Lagrange multipliers in nondifferentiable programming problems

In this paper, we present several constraint qualifications, and we show that these conditions guarantee the nonvacuity and the boundedness of the Lagrange multiplier sets for general nondifferentiable programming problems. The relationships with various constraint qualifications are investigated.The author gratefully acknowledges the comments made by the two referees.