Generalized reduced gradient algorithm

The MINLP-problems were implemented in GAMS [7, 8] and solved by the outer approximation/equality relaxation/augmented penalty-method [9] as implemented in DICOPT. The algorithm generates a series of NLP and MILP subproblems, which were solved by the generalized reduced gradient method [10] as implemented in CONOPT and the integrality relaxation based branch and cut method as... 

If dof(x) = n — act(x) = d > 0, then there are more problem variables than active constraints at x, so the (n-d) active constraints can be solved for n — d dependent or basic variables, each of which depends on the remaining d independent or nonbasic variables. Generalized reduced gradient (GRG) algorithms use the active constraints at a point to solve for an equal number of dependent or basic variables in terms of the remaining independent ones, as does the simplex method for LPs. 

The generalized reduced gradient (GRG) algorithm was first developed in the late 1960s by Jean Abadie (Abadie and Carpentier, 1969) and has since been refined by several other researchers. In this section we discuss the fundamental concepts of GRG and describe the version of GRG that is implemented in GRG2, the most widely available nonlinear optimizer [Lasdon et al., 1978 Lasdon and Waren, 1978 Smith and Lasdon, 1992]. 

First, we define the objective function (see Figure 8.22b) such as energy consumption, the number of trays of a column, and the conversion of a reactor. We have up to three optimization algorithms for this case SQR, general reduced gradient, and simultaneous modular SQP. The particularities of the methods can be found elsewhere [44,45]. We define the independent variables and the constraints over variables from the streams or the equipment just by selecting the unit or the stream and the variable and the range of values of operation and an initial value. 

Nonlinear programming (NLP), as the name implies, is similar to LP, but the objective function or constraints can be nonlinear functions. There are no algorithms (like the simplex method) that guarantee a solution for NLP problems. Many methods have been developed, and Solver has one of these built in (called Generalized Reduced Gradient). The subject of NLP is quite complex and far beyond what can be covered ho-e. NLP is introduced by way of a simple example. Even the simplest of chemical and biomolecu-lar engineering NLP problems can be too complex to warrant coverage here. 

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