Bézier Simplex Fitting: Describing Pareto Fronts of´ Simplicial Problems with Small Samples in Multi-Objective Optimization

Abstract

Multi-objective optimization problems require simultaneously optimizing two or more objective functions. Many studies have reported that the solution set of an M -objective optimization problem often forms an (M − 1)-dimensional topological simplex (a curved line for M = 2, a curved triangle for M = 3, a curved tetrahedron for M = 4, etc.). Since the dimensionality of the solution set increases as the number of objectives grows, an exponentially large sample size is needed to cover the solution set. To reduce the required sample size, this paper proposes a Bézier simplex model and its fitting algorithm. These techniques can exploit the simplex structure of the solution set and decompose a high-dimensional surface fitting task into a sequence of low-dimensional ones. An approximation theorem of Bézier simplices is proven. Numerical experiments with synthetic and real-world optimization problems demonstrate that the proposed method achieves an accurate approximation of high-dimensional solution sets with small samples. In practice, such an approximation will be conducted in the postoptimization process and enable a better trade-off analysis. PreliminariesLet us introduce notations for defining simplicial problems and review an existing method of Bézier curve fitting. Simplicial ProblemA multi-objective optimization problem is denoted by its objective map f = (f 1 , . . . , f M ) : X → R M . Let I := { 1, . . . , M } be the index set of objective functions and ∆ M −1 := (t 1 , . . . , t M ) ∈ R M 0 ≤ t m , m∈M t m = 1 be the standard simplex in R M . For each non-empty subset J ⊆ I, we call∆ J := (t 1 , . . . , t M ) ∈ ∆ M −1 t m = 0 (m ∈ J) the J-face of ∆ M −1 andThe problem class we are interested in is as follows:We call such φ and f • φ a triangulation of the Pareto set X * (f ) and the Pareto front f X * (f ), respectively. For each non-empty subset J ⊆ I, we call X * (f J ) the J-face of X * (f ) and f X * (f J ) the J-face of f X * (f ). For each 0 ≤ m ≤ M − 1, we callthe m-skeleton of X * (f ) and f X * (f ), respectively.By definition, any subproblem of a simplicial problem is again simplicial. As shown in Figure 1b, the Pareto sets forms a simplex. The second condition asserts that f | X * (f ) : X * (f ) → R M is a C 0 -embedding. This means that the Pareto front of each subproblem is homeomorphic to its Pareto set as shown in Figure 1c. Therefore, the Pareto set/front of an M -objective simplicial problem can be identified with a curved (M − 1)-simplex. We can find its J-face by solving the J-subproblem.

Ensuring Smoothly Navigable Approximation Sets by Bézier Curve Parameterizations in Evolutionary Bi-objective Optimization

“…The Pareto front of nondegenerate problems with m objectives is an m−1-dimensional manifold. Instead of a one-dimensional Bézier curve, the Pareto set can then be modeled by an (m−1)-dimensional Bézier simplex [15]. For the navigation of higher-dimensional manifolds, a one-dimensional path through all obtained solutions could still be used.…”

“…Optimizing only the control points of the Bézier curve, that define its curvature, enforces the decision variables of solutions in the approximation set to vary in a smooth, continuous fashion, thereby likely improving intuitive navigability of the approximation set. Previous work on parameterizations of the approximation set has been applied mainly in a post-processing step after optimization, or was performed in the objective space [3,15,19], but this does not aid in the navigability of the approximation set in decision space. Moreover, fitting a smooth curve through an already optimized set of solutions might result in a bad fit, resulting in a lower-quality approximation set.…”

Asymptotic Risk of Bézier Simplex Fitting

“…is a multinomial coefficient, and (Kobayashi et al 2019) proposed two Bézier simplex fitting algorithms: the all-at-once fitting and the inductive skeleton fitting. They are different in not an only fitting algorithm but also sampling strategy.…”

“…There are a lot of practical problems being simplicial: location problems (Kuhn 1967) and a phenotypic divergence model in evolutionary biology (Shoval et al 2012) are shown to be simplicial, and an airplane design (Mastroddi and Gemma 2013) and a hydrologic modeling (Vrugt et al 2003) have numerical solutions which imply those problems are simplicial. The Pareto set and front of any simplicial problem can be approximated with arbitrary accuracy by a Bézier simplex of an appropriate degree (Kobayashi et al 2019). There are two fitting algorithms for Bézier simplices: the all-at-once fitting is a naïve extension of Borges-Pastva algorithm for Bézier curves (Borges and Pastva 2002), and the inductive skeleton fitting (Kobayashi et al 2019) exploits the skeleton structure of simplicial problems discussed above.…”

“…The Pareto set and front of any simplicial problem can be approximated with arbitrary accuracy by a Bézier simplex of an appropriate degree (Kobayashi et al 2019). There are two fitting algorithms for Bézier simplices: the all-at-once fitting is a naïve extension of Borges-Pastva algorithm for Bézier curves (Borges and Pastva 2002), and the inductive skeleton fitting (Kobayashi et al 2019) exploits the skeleton structure of simplicial problems discussed above.…”