Robust optimization is a field of mathematical optimization theory that deals with optimization problems in which a certain measure of robustness is sought against uncertainty that can be represented as deterministic variability in the value of the parameters of the problem itself and/or its solution. It is related to, but often distinguished from, probabilistic optimization methods such as chance-constrained optimization.
History The origins of robust optimization date back to the establishment of modern decision theory in the 1950s and the use of worst case analysis and Wald's maximin model as a tool for the treatment of severe uncertainty. It became a discipline of its own in the 1970s with parallel developments in several scientific and technological fields. Over the years, it has been applied in statistics, but also in operations research, electrical engineering, control theory, finance, portfolio management logistics, manufacturing engineering, chemical engineering, medicine, and computer science. In engineering problems, these formulations often take the name of "Robust Design Optimization", RDO or "Reliability Based Design Optimization", RBDO.
Example 1 Consider the following linear programming problem
max x , y { 3 x + 2 y } s u b j e c t t o x , y ≥ 0 ; c x + d y ≤ 10 , ∀ ( c , d ) ∈ P {\displaystyle \max _{x,y}\ \{3x+2y\}\ \ \mathrm {subject\ to} \ \ x,y\geq 0;cx+dy\leq 10,\forall (c,d)\in P}
where P {\displaystyle P} is a given subset of R 2 {\displaystyle \mathbb {R} ^{2}} . What makes this a 'robust optimization' problem is the ∀ ( c , d ) ∈ P {\displaystyle \forall (c,d)\in P} clause in the constraints. Its implication is that for a pair ( x , y ) {\displaystyle (x,y)} to be admissible, the constraint c x + d y ≤ 10 {\displaystyle cx+dy\leq 10} must be satisfied by the worst ( c , d ) ∈ P {\displaystyle (c,d)\in P} pertaining to ( x , y ) {\displaystyle (x,y)} , namely the pair ( c , d ) ∈ P {\displaystyle (c,d)\in P} that maximizes the value of c x + d y {\displaystyle cx+dy} for the given value of ( x , y ) {\displaystyle (x,y)} . If the parameter space P {\displaystyle P} is finite (consisting of finitely many elements), then this robust optimization problem itself is a linear programming problem: for each ( c , d ) ∈ P {\displaystyle (c,d)\in P} there is a linear constraint c x + d y ≤ 10 {\displaystyle cx+dy\leq 10} . If P {\displaystyle P} is not a finite set, then this problem is a linear semi-infinite programming problem, namely a linear programming problem with finitely many (2) decision variables and infinitely many constraints.
Classification There are a number of classification criteria for robust optimization problems/models. In particular, one can distinguish between problems dealing with local and global models of robustness; and between probabilistic and non-probabilistic models of robustness. Modern robust optimization deals primarily with non-probabilistic models of robustness that are worst case oriented and as such usually deploy Wald's maximin models.
Local robustness There are cases where robustness is sought against small perturbations in a nominal value of a parameter. A very popular model of local robustness is the radius of stability model:
ρ ^ ( x , u ^ ) := max ρ ≥ 0 { ρ : u ∈ S ( x ) , ∀ u ∈ B ( ρ , u ^ ) } {\displaystyle {\hat {\rho }}(x,{\hat {u}}):=\max _{\rho \geq 0}\ \{\rho :u\in S(x),\forall u\in B(\rho ,{\hat {u}})\}}
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