Budgeted Multi-Objective Optimization with a Focus on the Central Part of the Pareto Front -- Extended Version

Abstract

Optimizing nonlinear systems involving expensive computer experiments with regard to conflicting objectives is a common challenge. When the number of experiments is severely restricted and/or when the number of objectives increases, uncovering the whole set of Pareto optimal solutions is out of reach, even for surrogate-based approaches: the proposed solutions are sub-optimal or do not cover the front well. As non-compromising optimal solutions have usually little point in applications, this work restricts the search to solutions that are close to the Pareto front center. The article starts by characterizing this center, which is defined for any type of front. Next, a Bayesian multi-objective optimization method for directing the search towards it is proposed. Targeting a subset of the Pareto front allows an improved optimality of the solutions and a better coverage of this zone, which is our main concern. A criterion for detecting convergence to the center is described. If the criterion is triggered, a widened central part of the Pareto front is targeted such that sufficiently accurate convergence to it is forecasted within the remaining budget. Numerical experiments show how the resulting algorithm, C-EHI, better locates the central part of the Pareto front when compared to state-of-the-art Bayesian algorithms.

Modeling and optimization with Gaussian processes in reduced eigenbases

“…m(•) and s(•) are the conditional mean and standard deviation of Y (•) (Equation ( 10)), respectively, while φ N and ϕ N stand for the normal cumulative distribution function and probability density function. The threshold a is usually set as the current minimum, f min := min i=1,...,n y i , while other values have also been investigated [23,21].…”

Targeting solutions in Bayesian multi-objective optimization: sequential and batch versions

“…When the objectives are modeled by independent GPs and the reference point is not dominated by the empirical front, P Y R (in the sense that no vector in P Y dominates R), one has EHI(•; R) = mEI(•; R). The proof is straightforward and given in [20]. mEI is particularly appealing from a computational point of view.…”

“…We assume local convergence and stop the algorithm when the line-uncertainty, L p(y)(1 − p(y))dy, is small enough, where L is the broken line going from I to R and N, a line which crosses the empirical front. The convergence detection is described more thoroughly in [20]. A flow chart of this Bayesian targeting search is given in Algorithm 1.…”

“…Other properties concerning the center of the Pareto front, and further details regarding the estimations of I and N's are given in [20].…”

“…In the current article, in addition to the GPs, we make use of an aspiration point [48]. This aspiration point can be implicitly defined as the center of the Pareto front [20] or a neutral point [48] or an extension of it [52,9]. Alternatively, it can be given by the DM as a level to attain, and if possible, to improve.…”