Lagrange Multipliers with Optimal Sensitivity Properties in Constrained Optimization
نویسنده
چکیده
We consider optimization problems with inequality and abstract set constraints, and we derive sensitivity properties of Lagrange multipliers under very weak conditions. In particular, we do not assume uniqueness of a Lagrange multiplier or continuity of the perturbation function. We show that the Lagrange multiplier of minimum norm defines the optimal rate of improvement of the cost per unit constraint violation. 1 Research supported by NSF Grant ECS-0218328. 2 Dept. of Electrical Engineering and Computer Science, M.I.T., Cambridge, Mass., 02139.
منابع مشابه
Some Properties of the Augmented Lagrangian in Cone Constrained Optimization
A large class of optimization problems can be modeled as minimization of an objective function subject to constraints given in a form of set inclusions. We discuss in this paper augmented Lagrangian duality for such optimization problems. We formulate the augmented Lagrangian dual problems and study conditions ensuring existence of the corresponding augmented Lagrange multipliers. We also discu...
متن کاملEnhanced Fritz John Conditions for Convex Programming
We consider convex constrained optimization problems, and we enhance the classical Fritz John optimality conditions to assert the existence of multipliers with special sensitivity properties. In particular, we prove the existence of Fritz John multipliers that are informative in the sense that they identify constraints whose relaxation, at rates proportional to the multipliers, strictly improve...
متن کاملState Constrained Optimal Control Problems with States of Low Regularity
We consider first order optimality conditions for state constrained optimal control problems. In particular we study the case where the state equation has not enough regularity to admit existence of a Slater point in function space. We overcome this difficulty by a special transformation. Under a density condition we show existence of Lagrange multipliers, which have a representation via measur...
متن کاملInvited Review Paper OPTIMAL DESIGN OF NONLINEAR MAGNETIC SYSTEMS USING FINITE ELEMENTS
An inverse finite element method was developed to find optimal geometric parameters of a magnetic device to approximate a desired magnetic flux density distribution at certain test points and directions selected in the device. The augmented Lagrange multipliers method was utilized to transform the constrained problem consisting of a least-square objective function and a set of constraint equati...
متن کاملCritical solutions of nonlinear equations: stability issues
It is known that when the set of Lagrange multipliers associated with a stationary point of a constrained optimization problem is not a singleton, this set may contain so-called critical multipliers. This special subset of Lagrange multipliers defines, to a great extent, stability pattern of the solution in question subject to parametric perturbations. Criticality of a Lagrange multiplier can b...
متن کامل