Feasible criteria space

Each point in this plot relates to a pair of those criteria values. The space occupying these points is called the feasible criteria space. 

Figure 4.17 Plot of the feasible criteria space of the crushing strength and the disintegration time = Pareto-optimal point o = inferior point...

A point in the feasible criteria space is a Pareto Optimal point if there exists no other point in that space which yields an improvement in one criterion without causing a degradation in the other. 

In contrast to single-objective problems where optimization methods explore the feasible search space to find the single best solution, in multi-objective settings, no best solution can be found that outperforms all others in every criterion (3). Instead, multiple best solutions exist representing the range of possible compromises of the objectives (11). These solutions, known as non-dominated, have no other solutions that are better than them in all of the objectives considered. The set of non-dominated solutions is also known as the Pareto-front or the trade-off surface. Figure 3.1 illustrates the concept of non-dominated solutions and the Pareto-front in a bi-objective minimization problem. 

The geometrical interpretation of MLR is given in Fig. 29.10. The n rows (objects) of X form a pattern P" of points (represented by x,) which is projected upon an (unknown) axis b. This causes the axis b in S to be imaged by X in the dual space S" at the point y. The vector of observed measurements y has dimension n and, hence, is also represented as a point in 5". Is it possible then to define an axis b in S " such that the predicted y coincides with the observed y Usually this will not be feasible. One may propose finding the best possible b such that y comes as close to y as possible. A criterion for closeness is to ask for the distance between y and y, which is equal to the normlly - yll, to be as small as possible. 

---