A function that plays a pivotal role in establishing conditions that characterize a local minimum of an NLP problem is the Lagrangian function , which is defined as
Note that the Lagrangian function can be seen as a linear combination of the objective and constraint functions. The coefficients of the constraints, , and , are called the Lagrange multipliers or dual variables. At a feasible point , an inequality constraint is called active if it is satisfied as an equality—that is, . The set of active constraints at a feasible point is then defined as the union of the index set of the equality constraints, , and the indices of those inequality constraints that are active at ; that is,
An important condition that is assumed to hold in the majority of optimization algorithms is the so-called linear independence constraint qualification (LICQ). The LICQ states that at any feasible point , the gradients of all the active constraints are linearly independent. The main purpose of the LICQ is to ensure that the set of constraints is well-defined in a way that there are no redundant constraints or in a way that there are no constraints defined such that their gradients are always equal to zero.
If is a local minimum of the NLP problem and the LICQ holds at , then there are vectors of Lagrange multipliers and , with components , and , respectively, such that the following conditions are satisfied:
where is the gradient of the Lagrangian function with respect to x, defined as
The preceding conditions are often called the Karush-Kuhn-Tucker (KKT) conditions. The last group of equations () is called the complementarity condition. Its main aim is to try to force the Lagrange multipliers, , of the inactive inequalities (that is, those inequalities with ) to zero.
The KKT conditions describe the way the first derivatives of the objective and constraints are related at a local minimum . However, they are not enough to fully characterize a local minimum. The second-order optimality conditions attempt to fulfill this aim by examining the curvature of the Hessian matrix of the Lagrangian function at a point that satisfies the KKT conditions.
Let be a local minimum of the NLP problem, and let and be the corresponding Lagrange multipliers that satisfy the first-order optimality conditions. Then for all nonzero vectors d that satisfy the following conditions:
,
, , such that
, , such that
The second-order necessary optimality condition states that, at a local minimum, the curvature of the Lagrangian function along the directions that satisfy the preceding conditions must be nonnegative.