Existence of optimal Lagrange multipliers for certain nonconvex allocation problems

The use of Lagrange multipliers for decentralization of large resource allocation problems is well known. However, these dual techniques may suffer from the drawback ofduality gaps, to guarantee the absence of which various functions are required to be convex. This limits greatly the applicability of the decentralized approach. We show that less restrictive conditions can be formulated for a certain class of allocation problems, which we call resource management problems, which typically occur in large operational systems. We present a theorem for the existence of optimal multipliers, while placing almost no restrictions on the forms of the resource usage functions or the domains of the decision variables. Efficient solution algorithms, with provable convergence properties, have been given in a companion paper. Our results justify the application of dual methods to this class ofreal-world problems.The author is indebted to Mr. G. Karady and Professor Y. C. Ho of Harvard University for their valuable comments, and also to the referees for their helpful suggestions. This research was partially supported by the Office of Naval Research, under the Joint Services Electronic Program, Contract No. N0001475-C-0648, and by the National Science Foundation, Grant No. ENG-78-15231.     

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.     

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

Mathematical Programming - 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...     

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.     

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.     

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 multiplier characterizations of solution sets of constrained pseudolinear optimization problems

In this article, we study the minimization of a pseudolinear (i.e. pseudoconvex and pseudoconcave) function over a closed convex set subject to linear constraints. Various dual characterizations of the solution set of the minimization problem are given. As a consequence, several characterizations of the solution sets of linear fractional programs as well as linear fractional multi-objective constrained problems are given. Numerical examples are also given.     

Existence theorems for multidimensional Lagrange problems

Existence theorems are proved for multidimensional Lagrange problems of the calculus of variations and optimal control. The unknowns are functions of several independent variables in a fixed bounded domain, the cost functional is a multiple integral, and the side conditions are partial differential equations, not necessarily linear, with assigned boundary conditions. Also, unilateral constraints may be prescribed both on the space and the control variables. These constraints are expressed by requiring that space and control variables take their values in certain fixed or variable sets wich are assumed to be closed but not necessarily compact.This research was partially supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-942-65.     

Convergence analysis of a nonlinear Lagrange algorithm for general nonlinear constrained optimization problems

The convergence analysis of a nonlinear Lagrange algorithm for solving nonlinear constrained optimization problems with both inequality and equality constraints is explored in detail. The estimates for the derivatives of the multiplier mapping and the solution mapping of the proposed algorithm are discussed via the technique of the singular value decomposition of matrix. Based on the estimates, the local convergence results and the rate of convergence of the algorithm are presented when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions. Furthermore, the condition number of the Hessian of the nonlinear Lagrange function with respect to the decision variables is analyzed, which is closely related to efficiency of the algorithm. Finally, the preliminary numericM results for several typical test problems are reported.     

Coupled FEM-BEM for elastoplastic contact problems using Lagrange multipliers

The coupling of the elastoplastic finite element and elastic boundary element methods for two-dimensional frictionless contact stress analysis is presented. Interface traction matching (boundary element approach), which involves the force terms in the finite element analysis being transformed to tractions, is chosen for the coupling method. The analysis at the contact region is performed by the finite element method, and the Lagrange multiplier approach is used to apply the contact constraints. Since the analyses of elastoplastic problems are non-linear and involve iterative solution, the reduced size of the final system of equations introduced by combining the two methods is very advantageous, especially for contact problems where the nature of the problem also involves an iterative scheme.     

Further properties of Lagrange multipliers in nonsmooth optimization

We compute the Clarke generalized gradient of the marginal function of a nonconvex optimization problem with respect to usual and non usual parameters and we show how Lagrange multipliers are involved in this formula.     

A Lagrange penalty reformulation method for constrained optimization

In this paper a constrained optimization problem is transformed into an equivalent one in terms of an auxiliary penalty function. A Lagrange function method is then applied to this transformed problem. Zero duality gap and exact penalty results are obtained without any coercivity assumption on either the objective function or constraint functions. The work of the authors was supported by the Australian Research Council (grant DP0343998), the Research Grants Council of Hong Kong (PolyU 5145/02E) and NNSF (10571174) of China, respectively.     

Augmented Lagrangian for cone constrained topology optimization

Algorithmic aspects for the solution of topological shape optimization problems subject to a cone constraint are addressed in this paper. In this framework, an augmented Lagrangian method based on the concept of topological derivative is proposed. It is illustrated by some numerical experiments in structural optimization with compliance and eigenfrequency constraints and multiple loads.     

A nonlinear augmented Lagrangian for constrained minimax problems

A novel smooth nonlinear augmented Lagrangian for solving minimax problems with inequality constraints, is proposed in this paper, which has the positive properties that the classical Lagrangian and the penalty function fail to possess. The corresponding algorithm mainly consists of minimizing the nonlinear augmented Lagrangian function and updating the Lagrange multipliers and controlling parameter. It is demonstrated that the algorithm converges Q-superlinearly when the controlling parameter is less than a threshold under the mild conditions. Furthermore, the condition number of the Hessian of the nonlinear augmented Lagrangian function is studied, which is very important for the efficiency of the algorithm. The theoretical results are validated further by the preliminary numerical experiments for several testing problems reported at last, which show that the nonlinear augmented Lagrangian is promising.     

On lagrange-kuhn-tucker multipliers for pareto optimization problems

We give a simple proof of the existence of Lagrange-Kuhn-Tucker multipliers for Pareto Multiobjective programming problems.     

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.