(373i) An Objective Reduction Algorithm for Nonlinear Many-Objective Optimization Problems

In recent times, sustainable investing has gained significant traction in the mainstream [1]. It involves evaluating industrial performance not only based on traditional profit metrics but also placing emphasis on environmental, social, and governance (ESG) factors. An approach to sustainable decision making is to solve a multi-objective optimization problem. It gives results as Pareto frontiers, which depict a manifold of solutions representing the best one objective can do without making another one worse. However, a challenge with multi-objective approaches in sustainable decision making arises when dealing with problem of four or more objectives (many-objective problems, or MaOPs), where visualization of objective trade-offs becomes less intuitive and generating a complete set of solution points becomes computationally prohibitive.

In our previous work, we devised an algorithm capable of systematically reducing objective dimensionality for (mixed-integer) linear MaOPs [2]. Derivatives of objectives are projected onto the obtained constraint surfaces obtained. Strength of interaction is defined as the inner product of the projected vectors and a weighted sum of the constraint interaction strengths is used to determine the total objective correlation strength. This information is embedded into an objective correlation graph and community detection is applied to identify groups of objectives that are strongly correlated within the group and competing with objectives in other groups. In this work, we will extend the algorithm to reduce the dimensionality of nonlinear MaOPs. An outer approximation-like method is used to systematically replace nonlinear objectives and constraints with a set of linear approximations that, when the nonlinear problem is convex, provides a relaxation of the original problem. [3]. We demonstrate that identifying correlation strengths along the linearly relaxed constraint space can be sufficient for developing correlation strength weights for objective grouping. To generate new fixed linear appoximation points, we adopt a random step direction within the cone defined by all objective gradient vectors. We document how adding different linear relaxation constraints impacts objective groupings in order to refine our approach on systematically determining fixed points and better understand the tradeoff between number of fixed points analyzed (computational effort) and the accuracy of obtained objective groupings.

The nonlinear objective reduction algorithm is validated through its application to various systems. A system with three-dimensional elliptical constraints and nonlinear objectives will be employed for easy analysis of results. Results from this example clearly demonstrate that objectives pointing to nearby points on the ellipsoid are grouped together, while those pointing towards faraway points are kept separate. Additionally, we will test the algorithm on commonly used test problem sets for many objective optimization such as the DTLZ and ZDT sets [4,5]. Finally, we demonstrate the applicability of our approach to the practical case study of optimal design of a hydrogen production system to demonstrate its versatility.

[1] Pástor, Ľ., Stambaugh, R.F. and Taylor, L.A., 2021. Sustainable investing in equilibrium. Journal of financial economics, 142(2), pp.550-571.

[2] Russell, J.M., Allman, A., 2023. Sustainable decision making for chemical process systems via dimensionality reduction of many objective problems. AIChE Journal, 69(2), e17692

[3] Viswanathan, J. and Grossmann, I.E., 1990. A combined penalty function and outer-approximation method for MINLP optimization. Computers & Chemical Engineering, 14(7), pp.769-782.

[4] Deb, K., Thiele, L., Laumanns, M. and Zitzler, E., 2005. Scalable test problems for evolutionary multiobjective optimization. In Evolutionary multiobjective optimization: theoretical advances and applications (pp. 105-145). London: Springer London.

[5] Zitzler, E., Deb, K. and Thiele, L., 2000. Comparison of multiobjective evolutionary algorithms: Empirical results. Evolutionary computation, 8(2), pp.173-195.