Approximating Rockafellians for Stochastic Optimization

• Approximating Rockafellians are finite-dimensional relaxations of stochastic programs that mitigate instability from distributional perturbations, including discontinuous integrands and chance constraints.
• They utilize auxiliary variable penalties and epigraphical regularization to ensure lower semicontinuity and stability under various probability metrics such as weak, total variation, and Wasserstein.
• The methodology quantifies epi-convergence and near-minimizer convergence, transforming naive plug-in objectives into robust substitute problems with explicit error bounds.

Searching arXiv for the cited paper and closely related Rockafellian literature to ground the article. arXiv search query: (Tian et al., 21 Jul 2025) OR Rockafellian relaxation stochastic optimization perturbations Approximating Rockafellians are finite-dimensional relaxations of stochastic programs designed to mitigate instability caused by distributional perturbations, including for discontinuous integrands and chance constraints. In the formulation developed in "Approximating Rockafellians Mitigate Distributional Perturbations: Discontinuous Integrands and Chance-Constrained Applications" (Tian et al., 21 Jul 2025), the original objective is embedded into a parametric Rockafellian with an auxiliary variable $u$ that is penalized rather than constrained, and approximation operators $G^\nu$ together with penalty scales $\lambda^\nu$ are chosen to counteract inaccuracies in the distribution $\mu^\nu$. This construction extends Rockafellian relaxation beyond special distributions and continuous integrands to general Borel probability measures, weak and setwise modes of convergence, and canonical chance-constrained models.

1. Formal setting and problem classes

The state space is a closed set $\Xi \subseteq \mathbb{R}^d$ equipped with the Borel $\sigma$-algebra, and the set of all Borel probability distributions on $\Xi$ is $\mathcal{M}=\mathcal{P}(\Xi)$. Expectations with respect to $\mu$ are denoted $E_\mu[\cdot]$. The basic stochastic model is a composite expected-value objective

$G^\nu$0

where $G^\nu$1 is proper and lsc, $G^\nu$2 is proper, lsc, and nondecreasing componentwise, and $G^\nu$3 has random-lsc components $G^\nu$4 that are uniformly bounded on $G^\nu$5 locally uniformly in $G^\nu$6. Here “random lsc integrands” means that $G^\nu$7 has component functions $G^\nu$8 that are lower semicontinuous in $G^\nu$9 and measurable in $\lambda^\nu$0 in the sense of measurable epigraphs (Tian et al., 21 Jul 2025).

Chance-constrained programs arise as a special but structurally difficult case. Given osc set-valued mappings $\lambda^\nu$1 and confidence levels $\lambda^\nu$2, the canonical form is

$\lambda^\nu$3

Equivalently, one may write the extended-valued objective

$\lambda^\nu$4

with $\lambda^\nu$5 and $\lambda^\nu$6 (Tian et al., 21 Jul 2025). This representation makes explicit why discontinuity is intrinsic: the indicator structure produces integrands for which naive substitution of $\lambda^\nu$7 for $\lambda^\nu$8 can destabilize feasibility and optimality.

Distributional perturbations are treated in general senses, including weak convergence, setwise convergence, total variation, bounded-Lipschitz, Fortet-Mourier, Wasserstein, and empirical measures. The paper defines, among others, the bounded-Lipschitz metric

$\lambda^\nu$9

the Fortet-Mourier metric of order $\mu^\nu$0,

$\mu^\nu$1

the Wasserstein distance of order $\mu^\nu$2,

$\mu^\nu$3

the minimal information metric

$\mu^\nu$4

and total variation

$\mu^\nu$5

The indicator $\mu^\nu$6 takes value $\mu^\nu$7 on $\mu^\nu$8 and $\mu^\nu$9 otherwise, and $\Xi \subseteq \mathbb{R}^d$0 (Tian et al., 21 Jul 2025).

2. Rockafellian construction and approximation principle

A function $\Xi \subseteq \mathbb{R}^d$1 is a Rockafellian for $\Xi \subseteq \mathbb{R}^d$2 if $\Xi \subseteq \mathbb{R}^d$3 for all $\Xi \subseteq \mathbb{R}^d$4. The Rockafellian used for the stochastic programs above is

$\Xi \subseteq \mathbb{R}^d$5

Minimizers satisfy $\Xi \subseteq \mathbb{R}^d$6 if and only if $\Xi \subseteq \mathbb{R}^d$7 and $\Xi \subseteq \mathbb{R}^d$8 (Tian et al., 21 Jul 2025).

The approximating family replaces the hard anchor $\Xi \subseteq \mathbb{R}^d$9 by a penalty: $\sigma$0 with $\sigma$1, $\sigma$2, and approximants $\sigma$3 chosen to interact favorably with $\sigma$4. The role of $\sigma$5 and $\sigma$6 is explicit. First, $\sigma$7 may be constructed by epigraphical regularization to ensure lsc and continuity in $\sigma$8, to control liminf behavior under weak convergence, and to dominate $\sigma$9. Second, $\Xi$0 penalizes $\Xi$1; by scaling $\Xi$2 appropriately in terms of a metric $\Xi$3, the penalty absorbs distributional discrepancies so that near-minimizers of $\Xi$4 are stable (Tian et al., 21 Jul 2025).

The core approximation theorem is localized epi-convergence. Under the conditions (i) $\Xi$5 for $\Xi$6-a.e. $\Xi$7, (ii) $\Xi$8 for any $\Xi$9, and (iii) $\mathcal{M}=\mathcal{P}(\Xi)$0 and $\mathcal{M}=\mathcal{P}(\Xi)$1, the functions $\mathcal{M}=\mathcal{P}(\Xi)$2 and $\mathcal{M}=\mathcal{P}(\Xi)$3 are lsc, a liminf inequality holds along convergent sequences, and there exist $\mathcal{M}=\mathcal{P}(\Xi)$4 with a matching limsup inequality at $\mathcal{M}=\mathcal{P}(\Xi)$5 (Tian et al., 21 Jul 2025).

These approximation properties propagate to optimization. The convergence theorem states that $\mathcal{M}=\mathcal{P}(\Xi)$6 and $\mathcal{M}=\mathcal{P}(\Xi)$7–argmin $\mathcal{M}=\mathcal{P}(\Xi)$8–argmin $\mathcal{M}=\mathcal{P}(\Xi)$9; moreover, $\mu$0, outer limits of $\mu$1–argmin sets are contained in $\mu$2–argmin $\mu$3, and if $\mu$4 then suitable inner limits recover $\mu$5. When the hypotheses hold for every $\mu$6 with the same $\mu$7, one has full epi-convergence $\mu$8, and set-convergence of near-minimizers follows (Tian et al., 21 Jul 2025). In this framework, the auxiliary variable $\mu$9 is not a modeling nuisance but the device that absorbs the discrepancy $E_\mu[\cdot]$0.

Naive plug-in objectives

$E_\mu[\cdot]$1

can fail catastrophically, especially with discontinuities. The penalty in $E_\mu[\cdot]$2 allows $E_\mu[\cdot]$3 to absorb distributional shifts in expectations. Selecting $E_\mu[\cdot]$4 so that the penalty dominates the discrepancy and designing $E_\mu[\cdot]$5 to be lower than $E_\mu[\cdot]$6 and continuous in $E_\mu[\cdot]$7 yields epi-convergence. Practically, near-minimizers of $E_\mu[\cdot]$8 converge to near-minimizers of $E_\mu[\cdot]$9 (Tian et al., 21 Jul 2025).

3. Epigraphical regularization, probability metrics, and sampling regimes

For weak convergence, the main constructive device is epigraphical regularization. For lsc $G^\nu$00 jointly in $G^\nu$01 and uniformly bounded on $G^\nu$02 locally in $G^\nu$03, the regularized integrand is

$G^\nu$04

with $G^\nu$05 and $G^\nu$06. This $G^\nu$07 is lsc and uniformly bounded locally, the map $G^\nu$08 is continuous and in fact locally Lipschitz, and $G^\nu$09 whenever $G^\nu$10 (Tian et al., 21 Jul 2025).

Two special cases are emphasized. The Pasch-Hausdorff partial envelope corresponds to $G^\nu$11: $G^\nu$12 and the Moreau partial envelope corresponds to $G^\nu$13: $G^\nu$14 Using weak convergence $G^\nu$15, the envelope construction yields the liminf bound on expectations and a subsequence device under which the expectation mismatch at a reference point vanishes sufficiently fast (Tian et al., 21 Jul 2025).

The resulting parameter choices can be summarized compactly.

Perturbation regime	Construction	Sufficient scaling
Weak convergence $G^\nu$16	Envelope $G^\nu$17 with $G^\nu$18	$G^\nu$19
$G^\nu$20	Pasch-Hausdorff envelope	$G^\nu$21, $G^\nu$22, and $G^\nu$23
$G^\nu$24	$G^\nu$25-envelope	$G^\nu$26, $G^\nu$27, and $G^\nu$28
$G^\nu$29 or $G^\nu$30	$G^\nu$31	$G^\nu$32 and $G^\nu$33
Empirical measures	$G^\nu$34	$G^\nu$35

When $G^\nu$36 is used, the inequality $G^\nu$37 implies that the bounded-Lipschitz scaling condition also suffices. Similar statements are given for higher-order Wasserstein metrics through monotonicity in $G^\nu$38. For setwise convergence, one may take $G^\nu$39 and pick $G^\nu$40 sufficiently slowly so that $G^\nu$41. For the minimal information metric or total variation, one again sets $G^\nu$42 and chooses $G^\nu$43 so that $G^\nu$44; the convergence theorem then yields epi-convergence and set-convergence of near-minimizers (Tian et al., 21 Jul 2025).

The same template extends to other divergences: KL, Hellinger, and $G^\nu$45 that upper bound $G^\nu$46 suffice via the same argument. For iid samples $G^\nu$47 and empirical measures $G^\nu$48, almost sure weak convergence combines with the epigraphical law of large numbers and the law of the iterated logarithm to yield almost sure epi-convergence, provided $G^\nu$49 (Tian et al., 21 Jul 2025).

4. Chance constraints, metric subregularity, and quantitative stability

In the single-constraint notation, the feasible set is

$G^\nu$50

For the general multi-constraint form, the set-valued mapping

$G^\nu$51

collects feasible sets under right-hand-side perturbations, and $G^\nu$52 (Tian et al., 21 Jul 2025).

Two constructions are developed. In S1, no $G^\nu$53 is needed. The approximating Rockafellian is

$G^\nu$54

and partial minimization over $G^\nu$55 yields

$G^\nu$56

Under $G^\nu$57 and $G^\nu$58, bounded near-minimizers satisfy $G^\nu$59 and $G^\nu$60 (Tian et al., 21 Jul 2025).

In S2, weak convergence is handled by a Pasch-Hausdorff envelope. The componentwise approximants are

$G^\nu$61

so that $G^\nu$62, and

$G^\nu$63

After partial minimization,

$G^\nu$64

Under $G^\nu$65 and $G^\nu$66, bounded near-minimizers satisfy the same qualitative convergence as in S1 (Tian et al., 21 Jul 2025).

The regularity assumptions are weaker than classical metric regularity assumptions. Metric subregularity (calmness) of $G^\nu$67 at $G^\nu$68 for $G^\nu$69 with modulus $G^\nu$70 means that for every $G^\nu$71 there exists $G^\nu$72 such that

$G^\nu$73

for all $G^\nu$74 near $G^\nu$75. Upper outer-Minkowski content controls the measure of boundary thickening: $G^\nu$76 It is satisfied under tractable conditions, including a decomposition of $G^\nu$77 into an absolutely continuous part with bounded density plus a discrete part with uniformly discrete support, together with geometric assumptions such as convex body, Lipschitz boundary, or compact and $G^\nu$78-rectifiable structure of $G^\nu$79 on a dense set (Tian et al., 21 Jul 2025).

The main stability theorem supplies explicit error quantities. Under boundedness assumptions, $G^\nu$80-Lipschitz continuity of $G^\nu$81 on $G^\nu$82, metric subregularity with modulus $G^\nu$83, and the Minkowski-content condition for S2, if $G^\nu$84–argmin $G^\nu$85 with $G^\nu$86 and $G^\nu$87, then for any $G^\nu$88 there exist $G^\nu$89 and $G^\nu$90 such that, for all $G^\nu$91,

$G^\nu$92

$G^\nu$93

and

$G^\nu$94

The error term is

$G^\nu$95

for S1 and

$G^\nu$96

for S2. If the Minkowski-content assumption is uniform, one can set $G^\nu$97 in S2 (Tian et al., 21 Jul 2025).

Suggested parameter choices sharpen these bounds. For S1,

$G^\nu$98

which yields

$G^\nu$99

For S2,

$\lambda^\nu$00

which yields

$\lambda^\nu$01

If uniform Minkowski content holds, then $\lambda^\nu$02 (Tian et al., 21 Jul 2025).

5. Relation to Rockafellian relaxation, duality, and global approximation theory

The 2025 formulation is part of a broader Rockafellian program. "Rockafellian Relaxation and Stochastic Optimization under Perturbations" introduced an “optimistic” framework in which optimization is conducted jointly over the original decision space and a model perturbation, and developed the notions of exact and limit-exact Rockafellians, including $\lambda^\nu$03-divergence penalization, support perturbations, $\lambda^\nu$04 alternatives, rates of convergence, and first-order optimality conditions (Royset et al., 2022). In that setting, exactness supported by $\lambda^\nu$05 means

$\lambda^\nu$06

strict exactness forces $\lambda^\nu$07 at relaxed minimizers, and strict limit-exactness converts epi-convergence of $\lambda^\nu$08 into convergence of cluster points of $\lambda^\nu$09–argmin sets to actual minimizers. The 2025 paper preserves the optimistic viewpoint but replaces finite-support perturbation models by a construction that works for general Borel $\lambda^\nu$10, discontinuous integrands, and chance constraints (Tian et al., 21 Jul 2025).

The tutorial "Good and Bad Optimization Models: Insights from Rockafellians" framed Rockafellians as perturbation families that diagnose stability and produce Lagrangians and dual functions through

$\lambda^\nu$11

with epi-convergence, level-boundedness, and dual tuning as organizing principles (Royset, 2021). It emphasized that hard right-hand-side perturbations can create “bad” models in which the minimum value jumps to $\lambda^\nu$12, whereas penalty-based constructions can restore stability. This suggests that the penalty term $\lambda^\nu$13 in the approximating Rockafellian of (Tian et al., 21 Jul 2025) is best understood as a concrete realization of that stabilization principle for distributional perturbations.

"Approximations of Rockafellians, Lagrangians, and Dual Functions" extended the theory to a global, epi-convergence-based framework for tilted Rockafellians, Lagrangian relaxations, dual functions, augmentation, and truncated Hausdorff error bounds (Deride et al., 2024). There, exactness yields

$\lambda^\nu$14

while epi-convergence of $\lambda^\nu$15 and tightness produce convergence of substitute problems even when direct approximations of $\lambda^\nu$16 do not epi-converge to $\lambda^\nu$17. The 2025 chance-constrained results fit naturally into that lineage: they supply a class of stochastic programs for which the substitute Rockafellian problems are explicitly finite-dimensional, epi-convergent, and quantitatively stable under probability-metric perturbations (Tian et al., 21 Jul 2025).

Relative to prior Rockafellian relaxation work by Royset et al. 2024, Antil et al. 2024, and De Ridder et al. 2024, the 2025 paper extends the framework to general Borel $\lambda^\nu$18, discontinuous integrands, chance constraints, multiple metrics including BL, FM, Wasserstein, TV, and KL, and weaker assumptions based on metric subregularity and upper outer-Minkowski content. It also contrasts the construction with distributionally robust optimization: DRO is “pessimistic” and searches over ambiguity sets, whereas approximating Rockafellians take an “optimistic” robust-statistics-inspired route by regularizing the decision mapping rather than correcting $\lambda^\nu$19 (Tian et al., 21 Jul 2025).

6. Examples, implementation, and open directions

The 2025 paper reports several instability patterns for naive approximations and the corresponding corrective effect of approximating Rockafellians. In finite distribution examples, one example has $\lambda^\nu$20 infeasible $\lambda^\nu$21 under small $\lambda^\nu$22 perturbations while $\lambda^\nu$23 with penalty recovers stability, and another example shows that the minimum and argmin of $\lambda^\nu$24 shift incorrectly even when $\lambda^\nu$25, whereas $\lambda^\nu$26 avoids erroneous shifts. For discrete countable distributions, weak convergence without total variation can make both $\lambda^\nu$27 and naive $\lambda^\nu$28 fail, but carefully designed $\lambda^\nu$29 based on enlarged indicator sets or envelope constructions restores convergence. For empirical measures, an example shows that $\lambda^\nu$30 is often infeasible almost surely, while $\lambda^\nu$31 with $\lambda^\nu$32 scaled by $\lambda^\nu$33 gives almost sure convergence (Tian et al., 21 Jul 2025).

The implementation pattern is deliberately simple. One first decides the perturbation model. If $\lambda^\nu$34 in $\lambda^\nu$35 or $\lambda^\nu$36, one sets $\lambda^\nu$37 and picks $\lambda^\nu$38 so that $\lambda^\nu$39. If only weak convergence is available, one constructs $\lambda^\nu$40 via epigraphical regularization, such as the Pasch-Hausdorff or Moreau partial envelope with $\lambda^\nu$41, and selects $\lambda^\nu$42 so that

$\lambda^\nu$43

The conceptual algorithmic template then consists of: input $\lambda^\nu$44, $\lambda^\nu$45, $\lambda^\nu$46 or $\lambda^\nu$47, $\lambda^\nu$48, a metric estimate $\lambda^\nu$49, and $\lambda^\nu$50; if chance constraints and weak convergence are present, compute $\lambda^\nu$51 and define $\lambda^\nu$52; define $\lambda^\nu$53; solve either $\lambda^\nu$54 or the partially minimized problem $\lambda^\nu$55; choose $\lambda^\nu$56 to balance fidelity and stability; and return $\lambda^\nu$57 together with the bound $\lambda^\nu$58 on constraint violation and distance to near-minimizers (Tian et al., 21 Jul 2025).

The theory also records its present limits. It focuses on near global minimizers rather than stationary points; extending quantitative results to stationary points likely requires stronger assumptions. General quantitative rates beyond chance constraints remain open, and structural assumptions such as strong growth or convexity may be needed. Computational strategies for envelopes on complex $\lambda^\nu$59 and in high dimensions warrant further research, because evaluating $\lambda^\nu$60 can be expensive (Tian et al., 21 Jul 2025).

A plausible implication is that approximating Rockafellians now occupy a distinct niche within stochastic optimization: they retain finite-dimensional optimization problems, admit epi-convergence and explicit rates under broad distributional perturbations, and remain effective when discontinuity prevents direct expectation-based approximations from being stable. In the terminology of the cited works, they turn “bad” approximations into substitute problems whose near-minimizers, values, and feasibility properties converge in a quantified way (Royset, 2021).