Advances and applications of chance-constrained approaches to systems optimisation under uncertainty

This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. A chance-constrained optimisation (CCOPT) model has a dual goal: guaranteeing performance as well as system reliability under uncertainty. The beginning of CCOPT methods dates back in the 1950s. Recently, CCOPT approaches are gaining momentum as modern engineering and finance applications are forced to consider reliability and risk metrics at the design and planning stages. Although theoretical development and practical applications have been made, many open problems remain to be addressed in this area. This article attempts to provide a brief survey of major application areas, structure properties, challenges and solution approaches to CCOPT. In particular, we present our research results achieved in the past few years. 1. Introduction Engineering and financial applications have to deal with uncertainties. Frequently, it is required to guarantee an optimal and reliable performance in the presence of uncertainties. The optimisation of some performance function under uncertainty is most preferably accomplished through stochastic optimisation The conventional way of solving optimisation problems with uncertainties is by using expected (nominal) values for the uncertain parameters. Nevertheless, the solutions obtained through such simplifications may not be robust and reliable. Therefore, stochastic optimisation methods are essential to consider the full effect of stochastic parameters. In practical applications, constraint violations are unavoidable due to unexpected and extreme events, measurement errors, etc. In addition, a requirement for deterministic satisfaction of constraints may incur high operating costs. In practice, to cope with inevitable uncertainties, some degree of violations of constraints are tolerated. Therefore, it is reasonable to demand constraint satisfaction with some level of probability. This calls for the use of chance constraints. Chance constraints can be used to specify a guaranteed level of fulfilment of product specifications, a guaranteed level of availability of products or outputs, safety