Chance constraints

All chapters are equipped with clear figures and tables to help the reader understand the included topics. Furthermore, several appendices are included to explain the general background in the area of stochastic programming, chance constraint programming and robust optimization. 

Schwarm, A., and Nikolaou, M., Chance constraint formulation of model predictive control, AIChE Annual Meeting (1997). 

Thus, this representation of risk is favored and variability, upper partial mean, regret functions, chance constraints, VaR, and the risk premium are disregarded. 

This type of chance constraints is named as joint chance constraints. Where Pr - represents the probability of event established in , a point x is feasible if and only if the probability of event gj x, ) < 0,j = 1,2,...,/ is no less than a, which means the probability of constraint violation is less than (1 — a). 

Under normal circumstances, chance-constrained programming is not necessarily convex, even assuming that aU functions gj x, ),j = 1,2,..., p are always convex. It still cannot guarantee the convexity of the chance constraints set, unless certain conditions are satisfied. These conditions may refer to that gj x, ),j= 1,2,..., p is united convex on (x, and probability Pr is pseudo-concave. In fact, it is... 

Taking into account that decision-makers do not always care about maximizing revenue, but how to achieve the optimal revenue in the sense of probability, we apply stochastic chance-constrained programming theory to translate the model into the stochastic programming model under chance constraints so that the optimal decision objective with a certain confidence level can be expressed. 

Deterministic optimization has been the common approach for batch distillation operation in previous studies. Since uncertainties exist, the results obtained by deterministic approaches may cause a high risk of constraint violations. In this work, we propose to use a stochastic optimization approach under chance constraints to address this problem. A new scheme for computing the probabilities and their gradients applicable to large scale nonlinear dynamic processes has been developed and applied to a semibatch reactive distillation process. The kinetic parameters and the tray efficiency are considered to be uncertain. The product purity specifications are to be ensured with chance constraints. The comparison of the stochastic results with the deterministic results is presented to indicate the robustness of the stochastic optimization. 

A stochastic dynamic optimization approach has been successfully implemented for a reactive semibatch distillation process. The aim is batch time minimization subject to product purity restrictions. A method for computing the probabilities and their gradients is developed to solve the dynamic stochastic optimization problem. The results obtained by the implementation with a higher probability level show that the consideration of uncertainties with chance constraints leads to a trade-off between the objective value and robustness. A comparison of the stochastic results with the deterministic results is made with respect to the objective values and the reliability of satisfying the purity constraints. We thank the Deutsche Forschungsgemeinschaft (DFG) for the financial support under the contract WO 565/12-1. 

Li, T., Li, P., Chen, B., Hu, M., Zhang, X. A Simulation-based Inexact Two-stage Chance Constraint Quadratic Programming for Sustainable Water Quality Management under Dual Uncertainties. J. of Water Resources Planning and Management (2012). doi 10.1061/(ASCE)WR.1943-5452.0000328... 

Most equipment failures occur under abnonnal conditions, especially elevated pressures and temperatures. The design of equipment presents internal and external constraints. External limits may arise from physical laws, while internal limits may depend on tlie process and materials. In any case, if these limits are exceeded, tlie chance of an accident is greatly increased. 

It is not the catalytic activity itself that make zeolites particularly interesting, but the location of the active site within the well-defined geometry of a zeolite. Owing to the geometrical constraints of the zeolite, the selectivity of a chemical reaction can be increased by three mechanisms reactant selectivity, product selectivity, and transition state selectivity. In the case of reactant selectivity, bulky components in the feed do not enter the zeolite and will have no chance to react. When several products are formed within the zeolite, and only some are able to leave the zeolite, or some leave the zeolite more rapidly, we speak about product selectivity. When the geometrical constraints of the active site within the zeolite prohibit the formation of products or transition states leading to certain products, transition state selectivity applies. 

The size of the initial sample is an important consideration in a double-sampling plan. The initial sample should be as large as practicable with the constraint that only a small chance exists that... 

With the number of known exoplanets increasing very fast, current results are giving us the chance to undertake the first statistical studies of the properties of the exoplanets, as well of their host stars [24,21,8]. This is bringing new interesting constraints for the models of planet formation and evolution. 

Despite the linear set of constraints, Yee and Grossmann (1990) proposed an initialization procedure for the solution of the relaxed NLP problem in the OA/ER/AP since this may improve the chances of obtaining the best relaxed solution (note that the relaxed NLP is nonconvex). This initialization procedure is presented in the appendix of their paper. 

Venkat Venkatasubramanian Due to time constraints, I didn t get a chance to talk about that. This is an interactive framework. It is not a one-task deal. You are interacting and guiding the search at any given time, and based on your intuition, you can direct the search and change the weights and so on. 

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