= Optimization Lagrangian
{title2=$L(u,\lambda,\mu)$}
= Lagrangian in optimization
{c}
{synonym}
For a <minimization problem> with $g_i(u)\leq0$ and $h_j(u)=0$, the optimization Lagrangian is $L=f+\sum_i\lambda_i g_i+\sum_j\mu_jh_j$ with $\lambda_i\geq0$ and unrestricted equality <Lagrange multipliers>. Its infimum over the original variable domain gives a dual lower bound. For a <maximization problem>, use $L=f-\sum_i\lambda_i g_i-\sum_j\mu_jh_j$ to obtain upper bounds. The <Lagrangian sufficiency theorem> combines a global extremum of this function with feasibility and <complementary slackness>. This is distinct from a <Lagrangian> density in variational physics.
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