(373aj) Global Optimization for Chance-Constrained Nonlinear Programs and Its Application on the Gasoline Blending Problem
In this work, we develop a chance-constrained nonlinear program (CCNP) to address the normally distributed parametric uncertainties and obtain a global optimum in the energy production problem. The key idea is to use the McCormick relaxation to construct a second-order cone program (SOCP). Solving the SOCP will generate a lower bound solution. Then, substituting that solution into the CCNP and solving the resulting nonlinear program will yield an upper bound solution. By continuously refining the variable bounds, characterizing the cumulative distribution function (CDF) of the normal distribution, and adding more reformulation-linearization technique (RLT) cuts, the resulting upper and lower bounds will finally converge to the global optimum within a short time. We compare our method with the state-of-the-art solver, such as BARON , to show its efficiency.
An important application of the CCNP in the energy production is the gasoline blending problem. The objective is to produce regular and premium gasoline to meet market demands, satisfy quality specification for sulfur and octane number, and maximize the profits. One of the challenges is that properties of feedstock tend to significantly fluctuate, such that a blending recipe developed by an optimization method based on the nominal parameters may violate the quality specification. Under the linear mixing law, this problem has been addressed by the author . However, for the practical blending, a nonlinear mixing law should be used to calculate the octane number. Based on our best knowledge, the optimal blending recipe for such a problem under uncertainties has not been obtained so far. We test the proposed global optimization method and compare it with our recently developed sampling-based method  in the gasoline blending to show the effectiveness of the global optimization on the CCNP.
 N. V. Sahinidis, Optimization under uncertainty: state-of-the-art and opportunities, Computers & Chemical Engineering, 28, 971-983, 2004.
 N. V. Sahinidis, BARON: A general purpose global optimization software package. Journal of Global Optimization, 8, 201-205, 1996.
 Y. Yang, P. Vayanos and P. I. Barton, Chance-constrained optimization for refinery blend planning under uncertainty, Industrial & Engineering Chemistry Research, 56, 12139-12150, 2017.
 Y. Yang and C. Sutanto, Chance-constrained optimization for nonconvex programs using scenario-based methods, ISA Transactions, in press.