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Wikipedia / 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. 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

must be satisfied by the worst.

is finite (consisting of finitely many elements), then this robust optimization problem itself is a linear programming.

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:

. The picture is this:

Global robustness

Consider the simple abstract robust optimization problem

denotes the set of all possible under consideration.

This is a global represents all the possible.

. In such cases it might be necessary to relax a bit the robustness constraint and/or modify the statement of the problem.

Example 2

, then the following intuitive measure of robustness suggests itself:

could be defined as the cardinality.

in this set. An optimal decision is then a decision whose robustness is the largest.

This yields the following robust optimization problem:

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 3

Consider the robust optimization problem

is a real-valued function is too demanding.

and consider the following robust optimization problem:

increases. For instance, consider the relaxed robustness constraint

the relaxed robustness constraint reduces back to the original robustness constraint. This yields the following (relaxed) robust optimization problem:

is defined in such a manner that

and

. It also satisfies the relaxed constraint

Non-probabilistic robust optimization models

The dominating paradigm in this area of robust optimization is Wald's maximin model, namely

represents Nature, namely uncertainty. This 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.

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:

Probabilistically robust optimization models

These models quantify the uncertainty in the "true" value of the parameter of interest by probability distribution functions. They have been traditionally classified as stochastic programming and stochastic optimization models. 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. These methods are also relevant to data-driven optimization methods.

Robust counterpart

The solution method to many robust program involves creating a deterministic equivalent, called the robust counterpart. The practical difficulty of a robust program depends on if its robust counterpart is computationally tractable.