The Nonlinear Programming Solver
| Constrained Optimization |
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.
The First-Order Necessary Optimality Conditions
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 
, 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.
The Second-Order Necessary Optimality Condition
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 
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.