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Lagrangian Optimization for LP

  1. Jitamitra Desai

Published Online: 15 FEB 2011

DOI: 10.1002/9780470400531.eorms0447

Wiley Encyclopedia of Operations Research and Management Science

Wiley Encyclopedia of Operations Research and Management Science

How to Cite

Desai, J. 2011. Lagrangian Optimization for LP. Wiley Encyclopedia of Operations Research and Management Science. .

Author Information

  1. Nanyang Technological University, Division of Systems and Engineering Management, School of Mechanical and Aerospace Engineering, Singapore

Publication History

  1. Published Online: 15 FEB 2011

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Abstract

In this article, we present a methodological framework for solving large-scale linear programming (LP) problems via a Lagrangian optimization approach. Beginning with a formulation of the Lagrangian dual problem, we relate this approach to standard LP theory in two steps: first, we demonstrate that the optimal objective function value obtained by solving the Lagrangian dual is optimal to the original linear program; and second, we include theorems and propositions that characterize the mathematical properties of the Lagrangian dual and establish the equivalence of this approach to classical LP duality. Recognizing that the Lagrangian dual formulation is a nondifferentiable optimization problem, a subgradient algorithm is prescribed for solving this problem, and computational results for an illustrative example are presented. A brief extension of this approach for solving integer programs can also be found in this article.

Keywords:

  • linear programming;
  • large-scale optimization;
  • Lagrangian dual;
  • Lagrange multipliers;
  • nondifferentiable optimization;
  • subgradient algorithm;
  • integer programs