Algorithms and Differential Game Representations for Exploring Nonconvex Pareto Fronts in High Dimensions

TL;DR

This work develops a novel Hamilton-Jacobi and differential-game framework for exploring nonconvex Pareto fronts in high-dimensional MOO by embedding scalarizations through a monotone preference $g$ into a parameterized zero-sum game. The upper value $V^+$ satisfies a first-order HJ equation and admits a Hopf-Lax representation, from which inner minimizers $u_\alpha$ approximate shifted scalarizations; under mild regularity, a dense subset of the weak Pareto front is recovered as $D_R$ vanishes along subsequences. A primal-dual algorithm, with convergence guarantees via the Kurdyka–Łojasiewicz property, solves the resulting optimality systems and scales polynomially in the number of objectives $N$ and decision dimension $d$, enabling exploration of fronts up to $d=100$ in practical runtimes. The method extends to constrained MOO and is validated on semi-algebraic, nonconvex, and high-dimensional problems, demonstrating continuous Pareto curves and effective handling of nonconvex geometry beyond convex envelopes. Overall, the approach links MOO with HJ theory and differential games to provide scalable, interpretable Pareto-front exploration for complex engineering and data-driven tasks.

Abstract

We develop a new Hamiton-Jacobi (HJ) and differential game approach for exploring the Pareto front of (constrained) multi-objective optimization (MOO) problems. Given a preference function, we embed the scalarized MOO problem into the value function of a parameterized zero-sum game, whose upper value solves a first-order HJ equation that admits a Hopf-Lax representation formula. For each parameter value, this representation yields an inner minimizer that can be interpreted as an approximate solution to a shifted scalarization of the original MOO problem. Under mild assumptions, the resulting family of solutions maps to a dense subset of the weak Pareto front. Finally, we propose a primal-dual algorithm based on this approach for solving the corresponding optimality system. Numerical experiments show that our algorithm mitigates the curse of dimensionality (scaling polynomially with the dimension of the decision and objective spaces) and is able to expose continuous curves along nonconvex Pareto fronts in 100D in just $\sim$100 seconds.

Figures (5)

• Figure 1: Overview of our approach. Evolution of the Pareto front is characterized by an associated differential game and HJ equation. A Hopf-Lax representation then yields an efficient primal-dual scheme for the numerical exploration of high-dimensional, potentially nonconvex Pareto fronts. Colors indicate quantities that are equivalent between problem formulations (solid/no lines) and their duals (dotted lines).
• Figure 1: Constrained semi-algebraic MOO problem. Our primal-dual HJ/Hopf-Lax algorithm ($\cdot$) exactly recovers the ground truth Pareto front ($--$) of a 2D constrained convex MOO problem with semi-algebraic feasible set in less than 0.1s. In particular, all discovered Pareto optimal solutions lie on the boundary of the feasible set.
• Figure 2: Nonconvex Pareto fronts with $K= [0,1]^2$. Whereas most conventional MOO algorithms only recover the convex envelopes ($--$) of nonconvex Pareto fronts (--), our primal-dual HJ/Hopf-Lax algorithm (colored dots) is able to perfectly discover highly nonconvex Pareto fronts, including regions that lie strictly above their convex envelopes. In both cases, the computational time of our algorithm is less than 0.1s.
• Figure 3: Nonconvex Pareto fronts with high-dimensional decision spaces. Our HJ/Hopf-Lax solver (colored dots) is able to discover nonconvex Pareto fronts with various high-dimensional decision domains $K = [0,1]^d$, including portions that lie strictly above their convex envelope ($--$). In comparison, we are able to more extensively and continuously explore the Pareto front than the brute force method (greedy Pareto selection with iid Monte Carlo sampling), which, in contrast, only yields an approximate, discrete reference (--) and becomes exponentially more intractable as $d$ increases.
• Figure 4: Projections of a nonconvex Pareto front with high-dimensional decision and objective space into 2D objective space. Our HJ/Hopf-Lax solver is able to recover a nonconvex Pareto front with 20D decision space and 5D objective space in just 13.92s, which highlights its efficiency and tractability even in very high dimensions. While not derived from a particular physical model, the structure of this MOO problem is inspired by those arising in many-body, multi-agent systems, which demonstrates the potential of our algorithm for high-dimensional, real-world problems.

Theorems & Definitions (29)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4: see crandall1983viscositycrandall1984some
• Proposition 2.5: see, for instance, evans1984differential
• Proposition 3.1
• Proof 1
• Proposition 3.2
• Remark 3.3
• Remark 3.4