Compromise solution

Given the problem 4.3, it is possible to define $y^\star$ as the ideal outcome of the cost function $f(\mathbf{x})$ without any constraints, so that $y^\star=\displaystyle\inf_{\mathbf{x}\in X} f(\mathbf{x})$; the compromise solution is defined as the minimum of regret:

$$r(\mathbf{y})=\Vert \mathbf{y} - \mathbf{y^\star}\Vert;$$

typically, the $\mathcal{L}_p$-norm (the distance between the actual solution and the ideal point) ) it is used:

$$r(\mathbf{y})=r(\mathbf{y};p)= \left[ \sum_{i=1}^n \vert y_i - y^\star_i\vert^p\right]^\frac{1}{p}.$$

Again, a weight can be associated for each term of the sum:

$$r(\mathbf{y};p,\mathbf{w})= \left[ \sum_{i=1}^n w_i^p\vert y_i - y^\star_i\vert^p\right]^\frac{1}{p}.$$

Definition 5.26 (Compromise solution)   The compromise solution with respect to $\mathcal{L}_p$-norm is $\mathbf{y}^p\in Y$ that minimizes $r(\mathbf{y};p,\mathbf{w})$ over $Y$.

The compromise solution enjoys several properties, the most important is:

Property 5.27 (Pareto optimality)   The compromise solution $\mathbf{y}^p\in Y$ is an N-point, for $1 \leq p < \infty$ with respect to Pareto preference (definition 4.20). If $\mathbf{y}^\infty$ is unique, then it is also an N-point.

When the ideal point is not known, one can use an approximation, or, even, a constraint; in the latter case the more appropriate term is satisfying level. To point out the differences between constraints and satisfying level, one must observe:

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The constraints are, typically, a disequality constraints: the solution must be as lesser as possible than the specified constraints. In term of a $\mathcal{L}_p$-norm the solution must be as farther as possible from the constraints, that is the $\mathcal{L}_p$-norm must not to be minimized. So the method of the penalty function is the only suitable for this kind of problem.

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The satisfying levels are, typically, equality constraints: the solution must be as closer as possible to the levels indicated, that is the $\mathcal{L}_p$-norm must be minimized. So the method of the compromise solution can be devised.