University of Washington
Decisions often need to be made before all the facts are in. A portfolio must be purchased in the face of only statistical knowledge, at best, about how markets will perform. A structure must be built to withstand storms, floods, or earthquakes of magnitudes that can only be guessed at. In optimization, this implies that constraints often have to be envisioned in terms of safety margins instead of exact requirements. But what does that really mean in problem formulation, and what are the consequences for optimization structure and computation?
The idea of a "chance constraint" has long had a seemingly irresistible appeal in this setting. Roughly speaking, rather than requiring a barrier along the side of a river to hold back the water with absolute certainty, say, one can require it to do so with a probability of 99.9%, based on the past history of flooding. Analogous approaches in finance utilize so-called value-at-risk in a similar way. Unfortunately, such strategies, even when applied to what otherwise would be ordinary linear programming, can lead to problem formulations in which functions not only lack convexity but even can be badly discontinuous. Nowadays, however, substitutes for chance constraints are available which avoid these troubles and arguably do a much better job anyway at capturing the essence of the situation, while greatly enhancing numerical capabilities.
University of Washington