A CMOP with m objectives and (q+k) constraints, can be defined as:
\(\minF(x)=(f_1(x),\dots,f_m(x))^T\)
\(st\ g_i(x)\ge0,\ i=1,\dots,q\)
\(h_j(x)=0,\ j=1,\dots,k\)
\(x∈R^n\)
where F(x) defines the multi-dimensional objectives to be optimised, and gi(x) and hj(x) define the inequality and equality constraints, respectively. A solution is an n-dimensional vector of decision variables, x. To determine the objective value of a solution, a Pareto-optimal solution x1 dominates another solution x2 if F(x1) ≤ F(x2) where they are feasible. A total set of all feasible and Pareto-optimal solutions can then be defined as the Pareto Set, or Pareto Front (PF) when mapped onto the objective space. This PF represents all solutions with the optimal trade-off between objectives.
A commonly defined materials discovery problem is usually of combinatorial nature with unexplored regions of objective space, given some mixture of chemicals, precursors, and other process parameters. This problem can be formulated as a CMOP with an unknown PF to be extrapolated to, with minimal evaluation budget. \cite{Yong_2022,Lim_2021,Sabharwal_2016,klein2015towards} This is achieved through selection and evaluation of available solutions , where each solution represents the set of experimental input parameters (chemicals, temperature settings etc.) used in the screening. The number of data points is typically low, with most works generally limited to around 102-103 data points due to practical bottlenecks such as time taken to synthesize and characterize, or simply due to a limited time/cost budget.
In addition, the PF can be discontinuous with multiple infeasible regions due to underlying property limitations such as phase boundaries/solubility limits, or engineering rules, for example summing mixtures to 100%. \cite{Gopakumar_2018} Such constraints can also be knowledge-based, where a domain expert with prior knowledge sets them to pre-emptively ‘avoid’ poor results and converge faster. \cite{niculescu2006bayesian,Asvatourian_2020,Liu_2022} Figure 1 illustrates an example of such a problem.