Robust optimization
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.[1][2] 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,[3] electrical engineering,[4][5][6] control theory,[7] finance,[8] portfolio management[9] logistics,[10] manufacturing engineering,[11] chemical engineering,[12] medicine,[13] and computer science.In engineering problems, these formulations often take the name of "Robust Design Optimization", RDO or "Reliability Based Design Optimization", RBDO.is finite (consisting of finitely many elements), then this robust optimization problem itself is a linear programming problem: for eachis 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.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.There are cases where robustness is sought against small perturbations in a nominal value of a parameter.In words, the robustness (radius of stability) of decisionall of whose elements satisfy the stability requirements imposed onConsider the simple abstract robust optimization problem whereexists, the constraint can be too "conservative" in that it yields a solutionthat only slightly violates the robustness constraint but yields a very large payoffIn such cases it might be necessary to relax a bit the robustness constraint and/or modify the statement of the problem.In words, the robustness of decision is the size of the largest subset of, and assume that there is no feasible solution to this problem because the robustness constraint, its optimal solution may not perform well on a large portion ofin a controlled manner so that larger violations are allowed as the distance ofThis yields the following (relaxed) robust optimization problem: The functionis defined in such a manner that and and therefore the optimal solution to the relaxed problem satisfies the original constraintThe dominating paradigm in this area of robust optimization is Wald's maximin model, namely where theThis is the classic format of the generic model, and is often referred to as minimax or maximin optimization problem.The non-probabilistic (deterministic) model has been and is being extensively used for robust optimization especially in the field of signal processing.[14][15][16] The equivalent mathematical programming (MP) of the classic format above is Constraints can be incorporated explicitly in these models.The generic constrained classic format is The equivalent constrained MP format is defined as: These models quantify the uncertainty in the "true" value of the parameter of interest by probability distribution functions.Recently, probabilistically robust optimization has gained popularity by the introduction of rigorous theories such as scenario optimization able to quantify the robustness level of solutions obtained by randomization.The solution method to many robust program involves creating a deterministic equivalent, called the robust counterpart.