Lagrangian

Bruce C. Dieffenbach ()
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Bruce C. Dieffenbach: Independent author

Chapter 8 in Conjugate Duality in Economic Analysis, 2026, pp 65-72 from Springer

Abstract: Abstract Economists commonly introduce a Lagrangian to analyze an optimization problem with a constraint. Generalizing the traditional ad hoc definition employed by economists, we define the Lagrangian as the value of an optimization. Given the perturbation function for a problem, taking the conjugate in the perturbation variable while holding the choice variable fixed obtains the Lagrangian. It is a saddle-value function of the choice variables in the primal and the dual, and its value lies between the values of the primal and dual objective functions. A basic result is the saddle-point theorem that the choice variables form a saddle point of the Lagrangian if and only if they solve the primal and the dual. Let $$F\left ( \boldsymbol {x},\boldsymbol {y}\right )$$ be a proper, closed, convex, perturbation function. For simplicity of exposition, we analyze the special case of perturbation base values zero. The primal $$\inf _{\boldsymbol {x}}F\left ( \boldsymbol {x},\mathbf {0}\right )$$ has the reciprocal dual $$\sup _{\boldsymbol {y}^{\ast }}\left [ -F^{\ast }\left ( \mathbf {0},\boldsymbol {y}^{\ast }\right ) \right ] .$$ By definition, the Lagrangian is $$L\left ( \boldsymbol {x},\boldsymbol {y}^{\ast }\right ) :=-\sup _{\boldsymbol {y}}\left [ \left \langle \boldsymbol {y}^{\ast },\boldsymbol {y}\right \rangle -F\left ( \boldsymbol {x},\boldsymbol {y}\right ) \right ] \!,$$ a saddle-value function—convex in x and concave in y*. One can show that the value of the Lagrangian lies between the value of the primal and the value of the dual: $$F\left ( \boldsymbol {x},\mathbf {0}\right ) \geq L\left ( \boldsymbol {x},\boldsymbol {y}^{\ast }\right ) \geq -F^{\ast }\left ( \mathbf {0},\boldsymbol {y}^{\ast }\right ) \!,$$ for any $$\left ( \boldsymbol {x},\boldsymbol {y}^{\ast }\right )$$ . If there is no duality gap and the optimum value is attained in both the primal and the dual, then all three expressions are equal. The value of the Lagrangian is the optimum value of the problem.

Date: 2026
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DOI: 10.1007/978-3-032-21396-9_8

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