Lagrangian Optimization for LP
Published Online: 15 FEB 2011
Copyright © 2010 John Wiley & Sons, Inc. All rights reserved.
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. .
- Published Online: 15 FEB 2011
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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.
- linear programming;
- large-scale optimization;
- Lagrangian dual;
- Lagrange multipliers;
- nondifferentiable optimization;
- subgradient algorithm;
- integer programs