This section describes the goal attainment method of Gembicki [3]. This method uses a set of design goals, F*={F1*,F2*,...,Fm*}, associated with a set of objectives, F(x) = {F1(x),F2(x),...,Fm(x)}. The problem formulation allows the objectives to be under- or overachieved, enabling the designer to be relatively imprecise about the initial design goals. The relative degree of under- or overachievement of the goals is controlled by a vector of weighting coefficients, w = {w1,w2,...,wm}, and is expressed as a standard optimization problem using the formulation

such that Fi(x)−wiγ≤Fi*,  i=1,...,m.

The term wiγ introduces an element of slackness into the problem, which otherwise imposes that the goals be rigidly met. The weighting vector, w, enables the designer to express a measure of the relative tradeoffs between the objectives. For instance, setting the weighting vector w equal to the initial goals indicates that the same percentage under- or overachievement of the goals, F*, is achieved. You can incorporate hard constraints into the design by setting a particular weighting factor to zero (i.e., wi = 0). The goal attainment method provides a convenient intuitive interpretation of the design problem, which is solvable using standard optimization procedures. Illustrative examples of the use of the goal attainment method in control system design can be found in Fleming ([10] and [11]).

The goal attainment method is represented geometrically in the figure below in two dimensions.

Figure 7-1, Geometrical Representation of the Goal Attainment Method

Specification of the goals, {F1*,F2*}, defines the goal point, P. The weighting vector defines the direction of search from P to the feasible function space, Λ(γ). During the optimization γ is varied, which changes the size of the feasible region. The constraint boundaries converge to the unique solution point F1s, F2s.

The goal attainment method has the advantage that it can be posed as a nonlinear programming problem. Characteristics of the problem can also be exploited in a nonlinear programming algorithm. In sequential quadratic programming (SQP), the choice of merit function for the line search is not easy because, in many cases, it is difficult to “define” the relative importance between improving the objective function and reducing constraint violations. This has resulted in a number of different schemes for constructing the merit function (see, for example, Schittkowski [36]). In goal attainment programming there might be a more appropriate merit function, which you can achieve by posing Equation 2 as the minimax problem

where

Λi=Fi(x)−Fi*wi,  i=1,...,m.

Following the argument of Brayton et al. [1] for minimax optimization using SQP, using the merit function of Equation 30 for the goal attainment problem of Equation 3 gives

When the merit function of Equation 4 is used as the basis of a line search procedure, then, although ψ(x,γ) might decrease for a step in a given search direction, the function max Λi might paradoxically increase. This is accepting a degradation in the worst case objective. Since the worst case objective is responsible for the value of the objective function γ, this is accepting a step that ultimately increases the objective function to be minimized. Conversely, ψ(x,γ) might increase when max Λi decreases, implying a rejection of a step that improves the worst case objective.

Following the lines of Brayton et al. [1], a solution is therefore to set ψ(x) equal to the worst case objective, i.e.,

A problem in the goal attainment method is that it is common to use a weighting coefficient equal to 0 to incorporate hard constraints. The merit function of Equation 5 then becomes infinite for arbitrary violations of the constraints.

To overcome this problem while still retaining the features of Equation 5, the merit function is combined with that of Equation 31, giving the following:

Another feature that can be exploited in SQP is the objective function γ. From the KKT equations it can be shown that the approximation to the Hessian of the Lagrangian, H, should have zeros in the rows and columns associated with the variable γ. However, this property does not appear if H is initialized as the identity matrix. H is therefore initialized and maintained to have zeros in the rows and columns associated with γ.

These changes make the Hessian, H, indefinite. Therefore H is set to have zeros in the rows and columns associated with γ, except for the diagonal element, which is set to a small positive number (e.g., 1e-10). This allows use of the fast converging positive definite QP method described in Quadratic Programming Solution.

The preceding modifications have been implemented in fgoalattain and have been found to make the method more robust. However, because of the rapid convergence of the SQP method, the requirement that the merit function strictly decrease sometimes requires more function evaluations than an implementation of SQP using the merit function of Equation 30.

fminimax uses a goal attainment method. It takes goals of 0, and weights of 1. With this formulation, the goal attainment problem becomes

minimaxx(fi(x)−goaliweighti)=minimaxxfi(x),

which is the minimax problem.

Parenthetically, you might expect fminimax to turn the multiobjective function into a single objective. The function

f(x) = max(F1(x),...Fj(x))