Singular Continuation: Generating Piecewise Linear Approximations to Pareto Sets via Global Analysis

Abstract

Abstract. We propose a strategy for approximating Pareto optimal sets based on the global analysis framework proposed by Smale [Global analysis and economics. I. Pareto optimum and a generalization of Morse theory, in Dynamical Systems, Academic Press, New York, 1973, pp. 531-544]. The method highlights and exploits the underlying manifold structure of the Pareto sets, approximating Pareto optima by means of simplicial complexes. The method distinguishes the hierarchy between singular set, Pareto critical set, and stable Pareto critical set, and it can handle the problem of superposition of local Pareto fronts, occurring in the general nonconvex case. Furthermore, a quadratic convergence result in a suitable setwise sense is proven and tested in a number of numerical examples.

Active Learning of Pareto Fronts

“…Reasonable approximations obtained within few function evaluations are in many cases preferred to more accurate solutions which require a heavier computational effort. In fact, as noted by Lovison in work [11], "even a roughly sketched global picture of the whole situation can give crucial information on the problem at hand, suggesting correctly the location of paramount zones".…”

“…Under mild smoothness conditions, the Pareto front of continuous MOPs is an (m − 1)-dimensional piecewise-continuous manifold [1], [11], [12], with m being the number of objectives. The dominance relation defined in the objective space enables a further characterization of the PF by expressing an arbitrary objective z d (dependent function variable) as a function of the remaining objectives z I (independent function variables): z d = g(z I ).…”

PAINT–SiCon: constructing consistent parametric representations of Pareto sets in nonconvex multiobjective optimization

“…Singular Continuation [19,20], referred to as SiCon, is a method setting forth from the characterization of Pareto optimality based on first and second order derivatives of the objective functions given by S. Smale in [24] (see also [25,26]) and further using piece-wise linear approximations of implicitly defined manifolds (see, e.g., [27]). According to the SiCon method, linearized algebraic equations of the PS are determined for all the simplexes of a Delaunay tessellation of the feasible set based on a given sample of points.…”

“…These ideas are reminiscent of "connecting the dots" children's games. Alternatively, a global continuation could start from a geometrical decomposition of the feasible set (e.g., a Delaunay tessellation, a cubic tessellation) and next produce a linear approximation of the portion of the Pareto set passing through each tessellation cell [12,19,20]. The strengths and weaknesses of these methods emerge when dealing with highly nonlinear and nonconvex problems, where the Pareto front can be disconnected, or composed by different local branches intersecting one another, as illustrated in Figure 1.…”

“…Thus, the method facilitates decision making by embedding the use of the approximation in any interactive multiobjective optimization method. The algorithm proposed in this paper draws ideas from two Pareto set approximation algorithms, i.e., PAINT [13,21] and SiCon [19]. The PAINT method differs from the new PAINT-SiCon in the fact that the former works only in the objective space, producing an interpolation of the Pareto front consistent with the dominance structure (see Section 2.2 for more details).…”