multioobjective

Multiobjective optimization aims to optimize multiple objectives simultaneously, defined as F(x) = (f1(x), ..., fm(x)) to be minimized (or maximized) subject to x in X. Most often there is no single x that optimizes all objectives; instead, Pareto optimality concept used. The standard term is multiobjective optimization; multioobjective is a less common spelling.

Pareto dominance: x dominates y if fi(x) ≤ fi(y) for all i and fi(x) < fi(y) for at least one i, for minimization. A solution is Pareto optimal if no feasible x dominates it. The set of all Pareto optimal solutions forms the Pareto frontier.

Solution approaches: scalarization converts to single-objective problems, using weighted sums, epsilon-constraints, or goal programming. Evolutionary algorithms like NSGA-II and SPEA2 are popular for approximating Pareto fronts, as are decomposition methods such as MOEA/D. Gradient-based methods exist for differentiable objectives; surrogate modeling aids expensive evaluations. Handling constraints and uncertainty are common extensions.

Applications and challenges: used in engineering design, finance, energy systems, logistics, environmental planning. Challenges include non-convexity, many-objective scenarios with more than three objectives, computational cost, and decision-maker preference incorporation. Performance metrics include hypervolume, inverted generational distance, and spread to assess Pareto front quality.

Variants and related terms: many-objective optimization, constrained optimization, robust optimization, and stochastic optimization are related. The field emphasizes trade-off visualization and interactive decision support.