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This intuitive notion of global robustness is not used often in practice because the robust optimization problems that it induces are usually (not always) very difficult to solve.
=== Example ===
Consider the robust optimization problem
:<math>z(U):= \max_{x\in X}\ \{f(x): g(x,u)\le b, \forall u\in U\}</math>
where <math>g</math> is a real-valued function on <math>X\times U</math>, and assume that there is no feasible solution to this problem because the robustness constraint <math>g(x,u)\le b, \forall u\in U</math> is too demanding.
To overcome this difficult, let <math>\mathcal{N}</math> be a relatively small subset of <math>U</math> representing "normal" values of <math>u</math> and consider the following robust optimization problem:
:<math>z(\mathcal{N}):= \max_{x\in X}\ \{f(x): g(x,u)\le b, \forall u\in \mathcal{N}\}</math>
Since <math>\mathcal{N}</math> is much smaller than <math>U</math>, its optimal solution may not perform well on a large portion of <math>U</math> and therefore may not be robust against the variability of <math>u</math> over <math>U</math>.
One way to fix this difficulty is to relax the constraint <math>g(x,u)\le b</math> for values of <math>u</math> outside the set <math>\mathcal{N}</math> in a controlled manner so that larger violations are allowed as the distance of <math>u</math> from <math>\mathcal{N}</math> increases. For instance, consider the relaxed robustness constraint
: <math>g(x,u) \le b + \beta \cdot dist(u,\mathcal{N}) \ , \ \forall u\in U</math>
where <math>\beta \ge 0</math> is a control parameter and <math>dist(u,\mathcal{N})</math> denotes the distance of <math>u</math> from <math>\mathcal{N}</math>. Thus, for <math>\beta =0</math> the relaxed robustness constraint reduces back to the original robustness constraint.
This yields the following (relaxed) robust optimization problem:
:<math>z(\mathcal{N},U):= \max_{x\in X}\ \{f(x): g(x,u)\le b + \beta \cdot dist(u,\mathcal{N}) \ , \ \forall u\in U\}</math>
The function <math>dist</math> is defined in such a manner that
:<math>dist(u,\mathcal{N})\ge 0,\forall u\in U</math>
and
: <math>dist(u,\mathcal{N})= 0,\forall u\in \mathcal{N}</math>
and therefore the optimal solution to the relaxed problem satisfies the original constraint <math>g(x,u)\le b</math> for all values of <math>u</math> in <math>\mathcal{N}</math>. In addition, it also satisfies the relaxed constraint
: <math>g(x,u)\le b + \beta \cdot dist(u,\mathcal{N})</math>
outside <math>\mathcal{N}</math>.
== Non-probabilistic robust optimization models ==
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