Nonlinear programming problem definition

This form is convenient in that the active inequality constraints can now be replaced in the QP by all of the inequalities, with the result that Sa is determined directly from the QP solution. Finally, since second derivatives may often be hard to calculate and a unique solution is desired for the QP problem, the Hessian matrix, is approximated by a positive definite matrix, B, which is constructed by a quasi-Newton formula and requires only first-derivative information. Thus, the Newton-type derivation for (2) leads to a nonlinear programming algorithm based on the successive solution of the following QP subproblem ... 

This approach enables the incorporation of input, output, and final-time constraints and is flexible with respect to the crystallizer configuration, the objective function definition, and the choice of manipulated variables. Examples of the use of nonlinear programming to solve this problem are given subsequently. 

An important aspect in LP is the requirement that all mathematical expressimis must be Unear. Obviously, in practice, not all factors are linear, so nonlinear expressions are sometimes necessary to model specific situations, for example, in the case of economies of scale. Fortunately, it has been shown that many factors (work hours, use of machinery, benefits) are reasonably linear or can be approximated by expressions of this type. However, if there is found to be a factor that definitely cannot be expressed linearly, then the problem caimot be solved by LP. This is certainly is the case with nonlinear programming (NLP), which is beyond the scope of this textbook. 

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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