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Optimization and Lagrange multipliers: non-C1 constraints and “minimal” constraint qualifications

  • Leonid Hurwicz
  • Marcel K. Richter
Chapter
Part of the Advances in Mathematical Economics book series (MATHECON, volume 5)

Abstract

Constrained optimization problems are central to economics, and Lagrange multipliers — when they exist — play a basic role in solving them, in theory and in practice. Examples are well known of optimization problems for which multipliers do not exist. So it is important to know what requirements constraint functions must satisfy to be “Lagrange regular,” i.e. to guarantee existence of multipliers for broad classes of maximand or minimand functions. We relax the requirements in three directions:
  1. 1

    We reduce the smoothness requirements on constraints. This allows weaker and more uniform hypotheses for mixed inequality and equality constraints, permitting, for example, just differentiability at the optimum and continuity in a neighborhood. (We allow much weaker hypotheses, as well.) Beyond smoothness, other requirements have long been imposed on constraint functions, to avoid simple examples lacking multipliers. We examine two types of such “constraint qualifications”.

     
  2. 2

    We provide new, relaxed constraint qualifications of both Jacobian and path types.

     

Keywords

Partial Derivative Lagrange Multiplier Equality Constraint Inequality Constraint Path Criterion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Tokyo 2003

Authors and Affiliations

  • Leonid Hurwicz
    • 1
  • Marcel K. Richter
    • 1
  1. 1.Department of EconomicsUniversity of MinnesotaMinneapolisUSA

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