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Structure Preserving Approximation of Semiconcave Functions

Karl Kunisch, Donato Vásquez-Varas

TL;DR

The paper addresses the smooth, structure-preserving approximation of $C$-semiconcave functions on bounded convex domains, motivated by optimal control and Hamilton–Jacobi–Bellman theory. It introduces a finite-dimensional parametrization based on a smooth minimum operator $\psi_{n,\varepsilon}$ applied to a family of $C^2$ functions $\phi_i$, yielding approximants $v_{n,\varepsilon}$ that preserve semiconcavity. The authors prove universality and convergence results in $C(\overline{\Omega})$ and $W^{1,p}(\Omega)$, with gradient convergence on subsets away from gradient discontinuities and a probabilistic interpretation of gradient contributions through active sets. They illustrate the method on an Exponential Distance Function, showing that the MoreauRegMin-based approximation outperforms Log-Sum-Exp in capturing gradient discontinuities and approximately solving HJB equations, supported by numerical experiments. These results have practical implications for control synthesis and PDE-based learning approaches, enabling stable, semiconcavity-respecting approximations in high dimensions and suggesting pathways for ML-assisted HJB solvers.

Abstract

This article addresses structure-preserving smooth approximation of semiconcave functions. semiconcave functions are of particular interest because they naturally arise in a variety of variational problems, including {optimal feedback control, game theory, and optimal transport}. We leverage the fact that any semiconcave function can be represented as the {infimum of a countable family of \(C^2\) functions}. This infimum is expressed in a form that allows {approximation by finitely many functions}, combined with {smoothing operations}, such that each element of the approximating sequence remains semiconcave. The {active sets of indices} contributing to the representation of the semiconcave function and its approximations are analyzed in detail. Moreover, we show that the {gradients of the elements in the expansion of the approximating functions form a probability distribution}, a property of particular interest for the {value function in optimal control}. Approximation results are established in \(C(\bar Ω)\) and in \(W^{1,p}(Ω)\) for \(p \in [1,\infty)\) and \(p = \infty\). Finally, {numerical results} are presented to illustrate the approach on a test example.

Structure Preserving Approximation of Semiconcave Functions

TL;DR

The paper addresses the smooth, structure-preserving approximation of -semiconcave functions on bounded convex domains, motivated by optimal control and Hamilton–Jacobi–Bellman theory. It introduces a finite-dimensional parametrization based on a smooth minimum operator applied to a family of functions , yielding approximants that preserve semiconcavity. The authors prove universality and convergence results in and , with gradient convergence on subsets away from gradient discontinuities and a probabilistic interpretation of gradient contributions through active sets. They illustrate the method on an Exponential Distance Function, showing that the MoreauRegMin-based approximation outperforms Log-Sum-Exp in capturing gradient discontinuities and approximately solving HJB equations, supported by numerical experiments. These results have practical implications for control synthesis and PDE-based learning approaches, enabling stable, semiconcavity-respecting approximations in high dimensions and suggesting pathways for ML-assisted HJB solvers.

Abstract

This article addresses structure-preserving smooth approximation of semiconcave functions. semiconcave functions are of particular interest because they naturally arise in a variety of variational problems, including {optimal feedback control, game theory, and optimal transport}. We leverage the fact that any semiconcave function can be represented as the {infimum of a countable family of functions}. This infimum is expressed in a form that allows {approximation by finitely many functions}, combined with {smoothing operations}, such that each element of the approximating sequence remains semiconcave. The {active sets of indices} contributing to the representation of the semiconcave function and its approximations are analyzed in detail. Moreover, we show that the {gradients of the elements in the expansion of the approximating functions form a probability distribution}, a property of particular interest for the {value function in optimal control}. Approximation results are established in \(C(\bar Ω)\) and in \(W^{1,p}(Ω)\) for \(p \in [1,\infty)\) and . Finally, {numerical results} are presented to illustrate the approach on a test example.
Paper Structure (9 sections, 13 theorems, 200 equations, 6 figures, 1 table)

This paper contains 9 sections, 13 theorems, 200 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Semiconcavity preserving parametrization
  3. Probabilistic interpretation and active sets
  4. Approximation
  5. Example
  6. Exponential Distance Function
  7. Setting and approximation
  8. Numerical experiments
  9. Conclusions

Key Result

Theorem 2.1

Let $v\in C(\overline{\Omega})$. Then $v$ is Lipschitz on $\overline{\Omega}$ and semiconcave with constant $C>0$ if and only if there exists a family of functions $\{\phi_{i}\}_{i\in \mathcal{I}}\subset C^2(\overline{\Omega})$ uniformly bounded in $C^2(\overline{\Omega})$ such that and where $\mathcal{I}$ is an uncountable index set.

Figures (6)

  • Figure 1: Illustrative example for \ref{['rem:prob:HJB:eq3']}.
  • Figure 2: Active sets
  • Figure 3: $\Omega_{\delta}$ sets of $v_d$.
  • Figure 4: Error for $\delta=0$.
  • Figure 5: Error for $\delta=10^{-2}$
  • ...and 1 more figures

Theorems & Definitions (37)

  • Definition 2.1
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['theo:SemiConcaveRepresentation']}
  • Proposition 2.1
  • proof : Proof of Proposition \ref{['prop:discreteRep']}
  • Remark 2.2
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 2.3
  • ...and 27 more