Measure-Based Convexification in Optimization

• Measure-Based Convexification is a suite of methods that integrate measures to convert nonconvex problems into convex formulations, merging analytic and computational strategies.
• It underpins occupation-measure relaxations in variational calculus and optimal control, ensuring no-gap results under joint convexity and linking to convex envelope and Young-measure approaches.
• The framework extends to geometric constructions such as metronoids and volume-based metrics, enhancing relaxation quality in mixed-integer nonlinear optimization scenarios.

Measure-based convexification refers to a broad suite of mathematical and computational methods in which convexified objects, functions, or problems arise naturally via integration (or manipulation) with respect to suitably chosen measures. This class encompasses occupation-measure relaxations in variational and control theory, metronoid constructions in convex geometry, and volume-based quantification of convex relaxations in mixed-integer nonlinear optimization (MINLO). The framework provides a unifying lens for both analytic convexification (e.g., convex envelope, Young measures) and practical convex approximations (e.g., convex hulls, perspective relaxations), with deep implications for tractability, approximation quality, and connections to classical geometric analysis.

1. Occupation Measure Convexification in Calculus of Variations and Optimal Control

In the calculus of variations, occupation-measure convexification recasts function minimization problems as infinite-dimensional linear programs over measures on product spaces. Consider a classical variational problem: $$M := \inf_{y \in W^{1,p}(\Omega;Y)} \int_\Omega L(x, y(x), \nabla y(x)) \, dx$$ subject to pointwise, boundary, and integral constraints involving (possibly uncountable) families of constraints indexed by $i \in I$, with $y: \Omega \to Y$, $\nabla y(x) \in Z \subset \mathbb{R}^{m\times n}$. The occupation-measure relaxation associates to each admissible function $y$ (and its gradient) a pair of nonnegative Radon measures $(\mu, \mu^a)$ on $\Omega \times Y \times Z$ and $\partial \Omega \times Y$, respectively, satisfying marginal and Liouville-type constraints for affine test functions. The convexified problem becomes: $$M_{\text{aff}} := \inf_{(\mu,\mu^a) \in \mathcal{O}_{\text{aff}}} \int_{\Omega \times Y \times Z} L(x, y, z)\, d\mu(x,y,z)$$ with $\text{supp}(\mu)$ restricted by the original constraints, and similar for $i \in I$0 on the boundary (Henrion et al., 2023).

Under joint convexity of the integrand and constraint sets, there is no relaxation gap; i.e., $i \in I$1. The same approach extends to optimal control problems—once the running cost and constraint data are convex in the control variables and their reductions—by lifting to analogous measure spaces over control variables.

2. Equivalence to Convex Envelope and Young-Measure Relaxations

Occupation-measure convexification, when applied to nonconvex problems, is at least as strong as the classical convex-envelope (or relaxation via convexification) approach. For data $i \in I$2 and constraint sets $i \in I$3 that are nonconvex, the framework considers the convex envelope $i \in I$4 over the convexified feasible set $i \in I$5. The convexified variational problem becomes: $i \in I$6 subject to convexified constraints (Henrion et al., 2023). The equivalence $i \in I$7 (when $i \in I$8 is Hausdorff-continuous and $i \in I$9 is continuous) links measure-based occupation relaxations to both Young-measure and convex envelope methods.

Key implications include:

• Occupation-measure relaxations sandwich the original and convexified problem values: $y: \Omega \to Y$0.
• In regimes where the convex envelope method is tight, measure-based relaxations are also tight, extending classical no-gap principles to broader problem classes, e.g., in magnetism or elasticity.

3. Measure-generated Convex Sets and the Metronoid Construction

The metronoid construction defines a convexified set $y: \Omega \to Y$1 for a Borel measure $y: \Omega \to Y$2 on $y: \Omega \to Y$3 as: $y: \Omega \to Y$4 This set is always convex, closed (when $y: \Omega \to Y$5 has finite first moment), and recovers classical convex hulls for discrete measures. For symmetric $y: \Omega \to Y$6, $y: \Omega \to Y$7 precisely describes the zonoid generated by $y: \Omega \to Y$8. In particular, for uniform measures on convex bodies, $y: \Omega \to Y$9 relates to floating bodies via centers of mass of caps of prescribed volume (Huang et al., 2017).

The support function of $\nabla y(x) \in Z \subset \mathbb{R}^{m\times n}$0 is determined via a directional cutoff: $\nabla y(x) \in Z \subset \mathbb{R}^{m\times n}$1 where $\nabla y(x) \in Z \subset \mathbb{R}^{m\times n}$2 is chosen so $\nabla y(x) \in Z \subset \mathbb{R}^{m\times n}$3 and $\nabla y(x) \in Z \subset \mathbb{R}^{m\times n}$4.

4. Quantitative Measures: Volume-based Comparison and Fractional Vertex Index

Measure-based convexification provides explicit quantitative metrics for relaxation quality. In MINLO models, the tightness of a given convex relaxation is measured by the Lebesgue volume of the corresponding region (e.g., epigraph). For the indicator disjunction

$\nabla y(x) \in Z \subset \mathbb{R}^{m\times n}$5

the convex-hull (perspective) relaxation is given by $\nabla y(x) \in Z \subset \mathbb{R}^{m\times n}$6, and weaker relaxations are parameterized as $\nabla y(x) \in Z \subset \mathbb{R}^{m\times n}$7 for $\nabla y(x) \in Z \subset \mathbb{R}^{m\times n}$8 (Lee et al., 2020, Xu et al., 2024).

The ratio of volumes (e.g., $\nabla y(x) \in Z \subset \mathbb{R}^{m\times n}$9) explicitly quantifies the “distance” from a given relaxation to the convex hull: $y$0 For multivariate domains (e.g., a box $y$1), the difference in volumes between naïve and perspective relaxations admits closed-form integrals in terms of the minimal concave upper bound $y$2 (piecewise-linear or constant), and the ratio remains strictly positive for homogeneous convex $y$3 (never vanishing as $y$4), but decays to zero for super-exponential $y$5 (Xu et al., 2024).

In convex geometry, the cost of measure-based convexification is captured by $y$6 and $y$7, representing the minimal total mass or average gauge norm of a measure $y$8 generating $y$9 as a metronoid, linearly invariant analogues of Bezdek–Litvak’s vertex index. As $(\mu, \mu^a)$0, the fractional vertex index $(\mu, \mu^a)$1 provides tight lower and upper bounds for symmetric and general convex bodies, matching up to $(\mu, \mu^a)$2 in dimension (Huang et al., 2017).

5. Algorithmic and Numerical Aspects

Occupation-measure LPs (and boundary-included variants) are amenable to numerical approximation via the Moment–Sum-of-Squares (SOS) hierarchy, converging to the relaxation value (and the true infimum under convexity) through a sequence of semidefinite programs (Henrion et al., 2023). In convexification frameworks for Mean Field Games, strong convexity is established via Carleman estimates, and the unique minimizer is computed by discretizing space and time, then solving a finite-dimensional convex program (e.g., using fmincon) (Chen et al., 9 Dec 2025).

In MINLO contexts, generalized power-cone constraints render measure-oriented relaxations tractable for a range of $(\mu, \mu^a)$3-parameter relaxations, further enabling explicit volume computations and tightness analysis (Lee et al., 2020). For box-shaped domains, triangulation (e.g., classical Kuhn triangulation) provides explicit piecewise-linear descriptions of concave envelopes and integrals thereof (Xu et al., 2024).

6. Applications, Extensions, and Connections

Measure-based convexification has critical consequences in several domains:

• In variational and optimal control problems, occupation-measure relaxations yield no-gap results under convexity and coincide with the classical relaxed (Young-measure) and convex envelope approaches under nonconvexity (Henrion et al., 2023).
• In high-dimensional convex geometry, the metronoid framework extends classical convex hull-based approximation, enables fine-grained control over approximation costs, and leads to sharp bounds on average norms of centroid bodies for probability measures (Huang et al., 2017).
• In combinatorial and mean-variance optimization, measure-based metrics provide closed-form, predictive indicators of continuous relaxation strength and computational performance, as empirically validated on cardinality-constrained portfolio problems (Lee et al., 2020).

Furthermore, the notion of measure-generated convex sets unifies geometric and analytic convexification, offering a general framework for comparing and constructing convex approximations—both in function spaces (via convexifying variational integrals) and finite dimensions (via polytopal or zonotopic relaxation).

7. Summary Table: Measure-Based Convexification Paradigms

Framework	Measure Role	Main Quantitative Metric
Occupation-measure	Enforce PDE/variational constraints	Relaxation gap ($(\mu, \mu^a)$4)
Metronoid (geometry)	Generate convex bodies via integrals	Fractional vertex index ($(\mu, \mu^a)$5)
Volume-based MINLO	Quantify relaxation via epigraph vol	Volume cut-off ratio ($(\mu, \mu^a)$6, $(\mu, \mu^a)$7)

The table summarizes the central paradigms by highlighting the function of measures and the chief quantitative metrics of convexification quality in each setting.

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Measure-based convexification thus connects advanced convex analysis, optimization, and geometric measure theory, providing both rigorous analytic underpinnings (no-gap theorems, explicit convex hull characterizations) and practical computational methodologies for convex relaxations in high-dimensional and infinite-dimensional spaces (Henrion et al., 2023, Huang et al., 2017, Xu et al., 2024, Lee et al., 2020, Chen et al., 9 Dec 2025).