Definition Lagrangian of an optimization problem [lcc-002K]

Let f_0: \mathbb {R}^k \to \mathbb {R}, f_1, \dots f_n, g_1 , \dots g_m be a standard-form convex optimization problem, as in Definition [lcc-001G]. Then the Lagrangian of this problem is the function L: \mathbb {R}^k \times \mathbb {R}^n_+ \times \mathbb {R}^m \to \mathbb {R} defined by L(x;\lambda ,\nu ) = f_0(x) + \sum _i \lambda _i f_i(x) + \sum _i \nu _i g_i(x). Recall that \mathbb {R}_+ denotes the nonnegative reals.