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Assessing the Quality of a Set of Basis Functions for Inverse Optimal Control via Projection onto Global Minimizers

Filip Bečanović, Jared Miller, Vincent Bonnet, Kosta Jovanović, Samer Mohammed

TL;DR

This work tackles inverse optimal control by evaluating how well a convex-basis set $\mathcal{F}$ expresses a given observation $y$ through projection onto the set of global minimizers $\mathcal{G}$ (unconstrained) or $\mathcal{G}^c$ (constrained). It develops a geometric understanding of these optima sets for convex and quadratic bases, and provides practical upper bounds via bilevel gradient methods and lower bounds via LMIs, enabling validation or rejection of candidate bases. The authors demonstrate the method with toy and small-scale examples, showing how LMI-based bounds can robustly detect incompatible bases and offer quantitative distance measures, while also providing a path to extensions such as piecewise costs, local minimizers, and polynomial optimization. The framework thus offers a principled, computationally tractable approach to assessing IOC bases with potential applicability to biomechanics, robotics, and related domains.

Abstract

Inverse optimization (Inverse optimal control) is the task of imputing a cost function such that given test points (trajectories) are (nearly) optimal with respect to the discovered cost. Prior methods in inverse optimization assume that the true cost is a convex combination of a set of convex basis functions and that this basis is consistent with the test points. However, the consistency assumption is not always justified, as in many applications the principles by which the data is generated are not well understood. This work proposes using the distance between a test point and the set of global optima generated by the convex combinations of the convex basis functions as a measurement for the expressive quality of the basis with respect to the test point. A large minimal distance invalidates the set of basis functions. The concept of a set of global optima is introduced and its properties are explored in unconstrained and constrained settings. Upper and lower bounds for the minimum distance in the convex quadratic setting are implemented by bi-level gradient descent and an enriched linear matrix inequality respectively. Extensions to this framework include max-representable basis functions, nonconvex basis functions (local minima), and applying polynomial optimization techniques.

Assessing the Quality of a Set of Basis Functions for Inverse Optimal Control via Projection onto Global Minimizers

TL;DR

This work tackles inverse optimal control by evaluating how well a convex-basis set expresses a given observation through projection onto the set of global minimizers (unconstrained) or (constrained). It develops a geometric understanding of these optima sets for convex and quadratic bases, and provides practical upper bounds via bilevel gradient methods and lower bounds via LMIs, enabling validation or rejection of candidate bases. The authors demonstrate the method with toy and small-scale examples, showing how LMI-based bounds can robustly detect incompatible bases and offer quantitative distance measures, while also providing a path to extensions such as piecewise costs, local minimizers, and polynomial optimization. The framework thus offers a principled, computationally tractable approach to assessing IOC bases with potential applicability to biomechanics, robotics, and related domains.

Abstract

Inverse optimization (Inverse optimal control) is the task of imputing a cost function such that given test points (trajectories) are (nearly) optimal with respect to the discovered cost. Prior methods in inverse optimization assume that the true cost is a convex combination of a set of convex basis functions and that this basis is consistent with the test points. However, the consistency assumption is not always justified, as in many applications the principles by which the data is generated are not well understood. This work proposes using the distance between a test point and the set of global optima generated by the convex combinations of the convex basis functions as a measurement for the expressive quality of the basis with respect to the test point. A large minimal distance invalidates the set of basis functions. The concept of a set of global optima is introduced and its properties are explored in unconstrained and constrained settings. Upper and lower bounds for the minimum distance in the convex quadratic setting are implemented by bi-level gradient descent and an enriched linear matrix inequality respectively. Extensions to this framework include max-representable basis functions, nonconvex basis functions (local minima), and applying polynomial optimization techniques.
Paper Structure (20 sections, 3 theorems, 32 equations, 4 figures, 2 algorithms)

This paper contains 20 sections, 3 theorems, 32 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Problem Statement
  5. Geometry of Unconstrained Optimal Sets
  6. Convex Case
  7. Quadratics Only
  8. Geometry of Constrained Optimal Sets
  9. Convex Case
  10. Quadratic Programming
  11. Numerical Methods
  12. Upper Bounds
  13. Lower Bounds
  14. Unconstrained Lower Bound
  15. Constrained Lower Bounds
  16. ...and 5 more sections

Key Result

Theorem III.1

The map $\kappa(\alpha)$ from eq:opt_x_uncons is a continuous surjection from $\Delta^m$ onto $\mathcal{G}$ under Assumption assum:quad_uncons.

Figures (4)

  • Figure 1: A set of unconstrained global optima in 3D
  • Figure 2: The set $\mathcal{G}^c$ is not necessarily a subset of $\mathcal{G}$
  • Figure 3: Comparison of the projection onto an unconstrained set of optima $\mathcal{G}$.
  • Figure 4: Comparison of the projection onto a constrained set of optima $\mathcal{G}^c$.

Theorems & Definitions (16)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem III.1
  • proof
  • Definition III.1: Compact
  • Theorem III.2
  • proof
  • Definition III.2: Path-Connected
  • Theorem III.3
  • ...and 6 more