[Paper Review] A Ray Tracing Technique for the Navigation on a Non-convex Pareto Front

Dimitri Nowak, Karl‐Heinz Küfer|arXiv (Cornell University)|Jan 10, 2020

Computational Geometry and Mesh Generation21 references4 citations

TL;DR

This paper presents a real-time ray tracing-based navigation technique for non-convex Pareto fronts using Delaunay triangulation of precomputed Pareto-efficient solutions. By interpolating between these points and enabling interactive exploration in both objective and design spaces, the method supports responsive, high-performance navigation for multi-objective optimization problems with up to 10 objectives and arbitrary parameters.

ABSTRACT

A new interactive approach to navigate on approximations of in general non-convex but connected Pareto fronts is introduced. Given a finite number of precalculated representative Pareto-efficient solutions, an adapted Delaunay triangulation is generated. Based on interpolation and ray tracing techniques, real time navigation in the vicinity of Pareto-optimal solutions is made possible.

Motivation & Objective

• To enable interactive, real-time navigation on non-convex Pareto fronts in multi-objective optimization problems.
• To overcome limitations of existing methods in handling non-convexity and maintaining smooth navigation without abrupt jumps.
• To support decision makers in exploring trade-offs by filtering both objective and design space parameters dynamically.
• To provide a responsive and comprehensive user experience during interactive exploration of Pareto-optimal solutions.
• To extend existing approximation and navigation techniques to work efficiently on non-convex, connected Pareto fronts with up to 10 objectives.

Proposed method

• The method uses a three-stage process: precomputing Pareto-efficient solutions, constructing a Delaunay triangulation of these solutions as a surrogate model, and enabling real-time navigation via ray tracing.
• The triangulation is adapted to preserve Pareto optimality by removing simplices that dominate or are dominated by any solution in the set.
• Ray tracing is employed to project user-defined preferences (e.g., target objectives or parameter ranges) onto the triangulated Pareto front for real-time feedback.
• Interpolation between Pareto points is performed in the objective space, with the triangulation structure enabling smooth navigation even across non-convex regions.
• The approach supports filtering in both objective and design spaces, with implicit updates reflecting changes in restricted parameter ranges.
• The method is optimized for performance, achieving real-time interaction for up to 10 objectives, with triangulation times scaling predictably (e.g., ~184 seconds for 100 points in 10D space).

Experimental results

Research questions

• RQ1 How can interactive navigation on non-convex Pareto fronts be made responsive and intuitive for decision makers?
• RQ2 What triangulation technique preserves Pareto optimality while enabling real-time interpolation and ray tracing?
• RQ3 How does the method handle holes or disconnected components in the Pareto front during navigation?
• RQ4 What is the performance trade-off between accuracy, computational cost, and scalability in high-dimensional objective spaces?
• RQ5 How does the method compare to existing approaches like PAINT, NIMBUS, or Hyperboxing in terms of speed, usability, and robustness?

Key findings

• The proposed method enables real-time navigation on non-convex Pareto fronts with up to 10 objectives, achieving interactive responsiveness for decision makers.
• For 100 Pareto points in 10-dimensional objective space, the triangulation step takes approximately 183.77 seconds, which is comparable to PAINT despite higher dimensionality.
• The method avoids sudden jumps in navigation by handling holes differently than PAINT, maintaining a more predictable user experience.
• The approach outperforms PAINT in computational efficiency for high-dimensional problems, with triangulation times increasing significantly beyond 10 objectives.
• The method supports dynamic filtering in both objective and design spaces, providing implicit feedback on trade-offs during navigation.
• The technique has been validated in industrial applications, particularly in the chemical industry, demonstrating practical usability and success.