Lagrange Hessian

Lagrange multiplier General perturbation strength Hessian shift parameter Angular momentum quantum number Lagrange function... 

The Hessian of the Lagrange function is defined by the following relation ... 

This example also demonstrates that the Hessian of the Lagrange function, which coincides with the original function in this case, is not necessarily positive definite. It is positive definite only when we move away from the solution along a direction that satisfies the constraints. 

The BzzMatrixSparseSyimnetricLocked class manages the structure of the original objective function and the Lagrange function. It is exploited in different situations in the calculation of gradient and Hessian (see Chapter 4) of the objective function and in the solution of the appropriate KKT system. 

The BzzMatrixSparseLocked class manages the structure of the linear and nonlinear constraints. The objects in this class allow the calculation of the Jacobian of nonlinear constraints and, when necessary, their Hessians, useful to building the Hessian of the Lagrange function they are indispensable in efficiently solving the appropriate KKT system. 

An estimation of the Hessians of each nonlinear constraint is useful in calculating the Hessian of the Lagrange function when a SQP method is adopted... 

The updating of the Hessian of the Lagrange function is often computationally ( ) infeasible because of matrix filling due to the BFGS formula. Conversely, the updating of each Hessian with Schubert s formula minimizes the overall matrix filling. 

Some authors force the Hessian of the Lagrange function to be positive definite to ensure that the QP subproblem is feasible. As demonstrated in Section 12.2.3, this is not strictly necessary, although usually no calculation problems crop up in the overall procedure. 

In nonlinear programming (NLP) problems, either the objective function, the constraints, or both the objective and the constraints are nonlinear. Unlike LP, NLP solution does not always lie at the vertex of the feasible region. NLP optimum lies where the Jacobean of the function obtained by combining constraints with the objective function (using Lagrange multiphers as follows) is zero. The solution is local minimum if the Jacobian J is zero and the Hessian H is positive definite, and it is a local maximum if J is zero and H is negative definite. 

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