Multi-Objective Optimization identify non-dominated individuals (individuals, for which in the multi-objective, is not dominated); this forms the “pareto frontier” create all combinations of input parameters, and create a pareto frontier for them identify a weighting between the variations you desire, and identify the elements which align with the Pareto frontier Pareto Optimiality see Pareto Optimality Pareto Frontier A Pareto frontier is the entire set of pareto optimal points—i. the set that’s not dominated. Solving for Pareto Frontier Constraint Method We can convert this into a single-objective optimization problem; first, sort the constraints by order of importance:

we can set c_{j} to calibrate which point we want on our Pareto Frontier. By setting c_{j} large, we identify that we don’t care about that constraint as much; as we track c_{j} small, we start tracing out the Frontier along the j th direction. At some point, as you lower c_{j}, we will run out of points because we would have left the criterion space. Lexicographic Method Iteratively perform optimization; again sort constraints in order of importance, then: Weight Method see weighted sum method you can use a linear weight to obtain some Pareto optimal answers:

this fits a line against the Pareto region, which will miss some points but will give some Pareto optimal answers—there will not be any weighting which preserves points in the zone. Goal Programming

just minimize some distance (L1, L2, L-inf norms are all good) to a “good” point, usually the Utopia Point. L1 norms have the same problem as Weight Method as it is a line. Utopia Point The Utopia Point is the most optimal point, component wise. Multi-Objective Population Method Classic population method create m subpopulations optimize each population for one objective shuffle them together after each cohort’s optimization, create crossovers and mutations This method is good to get the pareto frontier, but often we get clumping at the extremas of each objective. Often, we try to encourage dispersion. Non-Dominating Ranking You can rank all points (including those not on the Pareto Frontier), with pareto-frontier non-dominated except for 1) non-dominated except 1) or 2) … Pareto filter Identify points on the Pareto Frontier, and keep top k, perhaps with dispersion Niche Technique a niche disperses the design along the Pareto Frontier fitness sharing: penalize if neighbors are too close) equivalence class sharing: if two individuals are compared, their non-dominating ranks are compared first, then fitness sharing is used as a tie breaker