(184d) Development of a Deterministic Optimization Approach, the Sdnbi Algorithm for Nonconvex and Combinatorial Bi-Objective Programming and Its Application to Molecular Design

Multi-objective optimization (MOO) techniques have been applied to design problems across a wide range of engineering fields to identify trade-offs between conflicting decision criteria which cannot be easily be placed on the same quantitative footing. Many practical problems often include discrete decision variables and nonlinear model equations in the mathematical formulation. Some of the most widely used approaches to solving MOO problems are based on scalarization methods and include the weighted sum method [1], the normal boundary intersection (NBI) method [2] and the sandwich algorithm [3]. However, these methods suffer from limitations that prevent them from reliably producing optimal solutions along the nonconvex or discrete regions of a Pareto front. As a result, the performance of these methods in the solution of many practical problems is limited when nonconvexities are present in the problem formulation due to the use of discrete decision variables and nonlinear model equations.

In this work, we present a robust bi-objective optimization approach that combines the sandwich algorithm and modified normal boundary intersection method (mNBI) [4]. The main improvements in the development of the algorithm are focused on the effective exploration of the nonconvex regions of the Pareto front and the early identification of regions where no additional Pareto solutions exist. We investigate theoretical properties arising from the interplay between the mNBI and sandwich algorithms: 1) the validity of the inner and outer approximations, 2) the completeness of the decomposition of the objective search space based on the convexity of the Pareto front and 3) the effectiveness of modifications of single-objective subproblems in avoiding unnecessary search steps for the disconnected portion of the Pareto front. The performance of the algorithm is compared to that of the sandwich algorithm and the mNBI method over a set of literature benchmark problems. The efficiency of the proposed algorithm is further investigated through application to solvent design for CO₂ capture [5] and the integrated design of working fluid and Organic Rankine Cycle processes [6] by examining its applicability and reliability to mixed-integer nonlinear problem. The SDNBI is found to provide the most evenly distributed approximation of the Pareto front as well as useful information on regions of the objective space that do not contain a nondominated point.

References

[1] Marler, R.T., and Arora, J.S. (2004). Survey of multi-objective optimization methods for engineering. Structural and multidisciplinary optimization 26, 369–395

[2] Das, I., and Dennis, J.E. (1998). Normal-boundary intersection: A new method for generating the pareto surface in nonlinear multicriteria optimization problems. SIAM Journal on Optimization 8, 631–657

[3] Rennen, G., Van Dam, E.R., Den Hertog, D. (2011). Enhancement of sandwich algorithms forapproximating higher-dimensional convex pareto sets. INFORMS Journal on Computing 23, 493–517

[4] Shukla, P.K. (2007). On the normal boundary intersection method for generation of efficientfront, in: International Conference on Computational Science, Springer. pp. 310–317

[5] Lee, Y.S., Graham, E.J., Galindo, A., Jackson, G., Adjiman, C.S. (2020). A comparative study of multi-objective optimization methodologies for molecular and process design. Computers & Chemical Engineering 136, 106802[6] Bowskill, D. H., Tropp, U. E., Gopinath, S., Jackson, G., Galindo, A., & Adjiman, C. S. (2020). Beyond a heuristic analysis: integration of process and working fluid design for organic rankine cycles. Molecular Systems Design & Engineering, 5, 493–510.