HCR - Lagrangian Theorem

Lagrangian Theorem

Given a minimization problem, with constraints such that every constraint is an equation (not an inequality):

$$\begin{array}{l}\min z=a_o+a_1{x}_1+a_2{x}_2+\dots \\ x_{i1}+x_{k1}+\dots =h_1\\ ⋮\\ x_{im}+x_{km}+\dots =h_m\end{array}$$

We can construct a new minimization problem without constraints, such that the solution of the first problem is the same as that of the second problem:

$$\begin{array}{lcc}\min & \bar{z}=& a_o+a_1{x}_1+a_2{x}_2+\dots +\\ & & +{\lambda }_1(x_{i1}+x_{k1}+\dots -h_1)\\ & & ⋮\\ & & +{\lambda }_m(x_{im}+x_{km}+\dots -h_m)\end{array}$$

→ So to find the minimum of the $\bar{z}$ function we can simply resolve the following system:

$$(\bar{x},\bar{\lambda }):\{\begin{array}{l}\frac{\partial Q(\bar{x},\bar{\lambda })}{\partial x_1}=0\\ \frac{\partial Q(\bar{x},\bar{\lambda })}{\partial x_2}=0\\ ⋮\\ \frac{\partial Q(\bar{x},\bar{\lambda })}{\partial x_m}=0\\ \frac{\partial Q(\bar{x},\bar{\lambda })}{\partial {\lambda }_1}=0\\ \frac{\partial Q(\bar{x},\bar{\lambda })}{\partial {\lambda }_2}=0\\ ⋮\\ \frac{\partial Q(\bar{x},\bar{\lambda })}{\partial {\lambda }_m}=0\end{array}$$