Reduced gradient method, generalized

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... 

In many cases the equality constraints may be used to eliminate some of the variables, leaving a problem with only inequality constraints and fewer variables. Even if the equalities are difficult to solve analytically, it may still be worthwhile solving them numerically. This is the approach taken by the generalized reduced gradient method, which is described in Section 8.7. 

The constraint in the original problem has now been eliminated, and fix2) is an unconstrained function with 1 degree of freedom (one independent variable). Using constraints to eliminate variables is the main idea of the generalized reduced gradient method, as discussed in Section 8.7. 

Abadie, J. and J. Carpentier. Generalization of the Wolfe Reduced Gradient Method to the Case of Nonlinear Constraints. In Optimization, R. Fletcher, ed. Academic Press, New York (1969), pp. 37-47. 

Solve the following problems by the generalized reduced-gradient method. Also, count the number of function evaluations, gradient evaluations, constraint evaluations, and evaluations of the gradient of the constraints. 

The nonlinear programming problem based on objective function (/), model equations (b)-(g), and inequality constraints (was solved using the generalized reduced gradient method presented in Chapter 8. See Setalvad and coworkers (1989) for details on the parameter values used in the optimization calculations, the results of which are presented here. 

The solution listed in Table E15.2B was obtained from several nonfeasible starting points, one of which is shown in Table E15.2C, by the generalized reduced gradient method. 

In this section, the important aspects of the mathematical basis for optimization methods are described. This will provide the necessary background to understand the most widely used method, LP. Then descriptions of two more effective NLP methods are outlined the generalized reduced gradient method and the successive LP method. Then methods for mixed-integer and multicriteria optimization problems are summarized. 

There are essentially six types of procedures to solve constrained nonlinear optimization problems. The three methods considered more successful are the successive LP, the successive quadratic programming, and the generalized reduced gradient method. These methods use different strategies but the same information to move from a starting point to the optimum, the first partial derivatives of the economic model, and constraints evaluated at the current point. Successive LP is used in a number of solvers including MINOS. Successive quadratic programming is the method of... 

In the generalized reduced gradient method, the independent variables are separated into basic and nonbasic ones. There are m basic variables Xb, and (n - m) nonbasic variables x b from Eqs. (2) and (3) with the inequalities converted to equalities using slack and surplus variables. In theory, the m constraint equations could be solved for the m basic variables in terms of the n - m) nonbasic variables, i.e.. 

In the proposed framework, the objective functions are formulated in Excel for the modelled process in HYSYS. The multi-objective optimisation technique, e-constraint, is formulated with the Premium Solver Platform (by Frontline Systems), which is an upgrade of the standard Excel solver, that uses the standard non-linear GRG (Generalized Reduced Gradient) method. However, any other optimisation method, such as that mentioned earlier, can be easily formulated in Excel and evaluated accordingly. 

This method of optimization is known as the generalized reduced-gradient (GRG) method. The objective function and the constraints are linearized ia a piecewise fashioa so that a series of straight-line segments are used to approximate them. Many computer codes are available for these methods. Two widely used ones are GRGA code (49) and GRG2 code (50). 

A generalization of this method, known as the generalized reduced gradient (GRG) method, is treated by Himmelblau (H4) and discussed in Section IV,B,3. 

The minimal cost design problem formulated above was solved by Bickel et al. (B7) using the generalized reduced gradient (GRG) method of Abadie and Guigou (H4). If x and u are vectors of state and decision (independent) variables and u) is the objective function in a minimization subject to constraints [Eq. (90)], then the reduced gradient d/du is given by... 

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 following strategies are all examples of Generalized Reduced Gradient (GRG) methods. 

---