Conditional Flexibility Index (CFI)

• Conditional Flexibility Index (CFI) is a quantitative measure that defines the maximal admissible uncertainty set ensuring system feasibility by conditioning on contextual data.
• CFI extends traditional flexibility indices by leveraging normalizing flows to learn conditional uncertainty sets, thereby enhancing scheduling and decision-making under uncertainty.
• Empirical evaluations, including two-moons and power grid case studies, demonstrate that integrating contextual information significantly improves robustness in complex systems.

The Conditional Flexibility Index (CFI) is a quantitative measure designed to evaluate the robustness and adaptability of scheduling or decision-making procedures subject to uncertainty, with a particular focus on utilizing contextual information and data-driven models to refine the admissible uncertainty sets. By extending traditional flexibility indices, the CFI characterizes the maximal region of admissible uncertainty within which a system remains feasible, leveraging advances in normalizing flow methods to map latent uncertainty sets to observed data distributions while conditioning on available contextual variables. In decision-theoretic settings, a variant of the CFI quantifies the minimum "stretch" in unmodeled uncertainty required for one plan to become preferable to another under constant risk aversion. This article outlines the mathematical foundations, computational formulations, learning strategies, and empirical comparisons underpinning the CFI framework (Wedemeyer et al., 22 Jan 2026, Shachter et al., 2013).

1. Mathematical Definition and Motivation

The classical flexibility index, as introduced by Swaney et al., seeks to maximize the size of an admissible uncertainty set around a nominal value $y_0$ such that a system remains feasible under all realizations within this set. Specifically, letting $x \in X$ denote scheduling or design variables, $y \in Y$ the uncertain parameters, and $z \in Z(x, y)$ the recourse actions, feasibility is defined by the constraint

$$g(x, y, z) \leq 0 \quad \text{for some} \quad z \in Z(x, y).$$

The traditional flexibility index $\delta$ is then

$$\delta^* = \max_{\delta, x \in X} \delta \quad \text{s.t.} \quad \max_{y \in Y(\delta)} \min_{z \in Z(x, y)} g(x, y, z) \leq 0,$$

where $Y(\delta)$ is commonly chosen as a hypercube or hypersphere of radius $\delta$ centered at $y_0$ (Wedemeyer et al., 22 Jan 2026). This construction neglects the empirical distribution and contextual relevance of uncertainty.

The CFI generalizes this by representing the admissible uncertainty set as a region learned from historical data and made conditional on contextual variables $\xi$, reflecting realistic operational conditions. In decision analysis, the CFI is further interpreted as the critical scaling factor of unmodeled uncertainty that causes the certain-equivalent of one plan to surpass another in desirability, assuming constant risk aversion (Shachter et al., 2013).

2. Data-Driven Conditional Uncertainty Sets via Normalizing Flows

The core mechanism of the CFI involves mapping a parametrized latent uncertainty set to data space using a normalizing flow, a bijective transformation $g_\theta: \mathbb{R}^d \to \mathbb{R}^d$ parameterized by $\theta$. Normalizing flows are trained to model the empirical distribution of uncertain parameters. The mapping is described by

$y = g_\theta(l; \xi),$

where $l$ is sampled from a simple latent distribution (e.g., $N(0, I)$), and contextual variables $\xi$ condition the mapping via neural network architectures (e.g., RealNVP coupling layers). The associated data-space conditional density is

$$p_{y|\xi}(y | \xi) = p_l(g_\theta^{-1}(y; \xi)) \cdot |\det J_{g_\theta^{-1}(y; \xi)}|,$$

with $J_{g_\theta^{-1}(y; \xi)}$ the Jacobian of the inverse flow (Wedemeyer et al., 22 Jan 2026).

Within latent space, the admissible uncertainty set is a centered $\ell_2$-ball of radius $\sqrt{\delta}$:

$$L(\delta) = \{ l \in \mathbb{R}^d : \| l \|_2^2 \leq \delta \},$$

which is mapped to the data-space set:

$$U(\xi, \delta) = Y_{\text{cond}}(\xi, \delta) = \{ y = g_\theta(l; \xi) : l \in L(\delta) \}.$$

The coverage probability of $L(\delta)$ in latent space is directly linked to the conditional coverage in data space by the change-of-variables formula.

3. Optimization Formulation of the Conditional Flexibility Index

The CFI is formally defined as the largest latent-ball radius that guarantees existence of feasible recourse for all realizations in the learned, conditional admissible set. Fixing $x \in X$ and context $\xi$, the CFI is

$$\delta_{\text{cond}}^*(\xi) = \max_{\delta \geq 0, x \in X} \delta \quad \text{s.t.} \quad \max_{y \in Y_{\text{cond}}(\xi, \delta)} \min_{z \in Z(x, y)} g(x, y, z) \leq 0,$$

which can be equivalently stated as

$$\max_{\delta, x} \delta \quad \text{s.t.} \quad \max_{\| l \|_2^2 \leq \delta} \min_{z \in Z(x, g_\theta(l; \xi))} g(x, g_\theta(l; \xi), z) \leq 0,$$

a semi-infinite program requiring specialized solution methods. The SIP relaxation introduces a penalty factor $\alpha > 0$:

$$\phi(x, \delta; \xi) = \max_{l, z} \min \left\{ g(x, g_\theta(l; \xi), z), \alpha [\delta - \| l \|_2^2] \right\},$$

with feasibility enforced via adaptive discretization algorithms Blankenship & Falk (1976).

4. Learning Procedures for Conditional Flows

Model parameters $\theta$ are estimated from historical $(y_i, \xi_i)$ pairs by maximizing the conditional log–likelihood:

$$L(\theta) = \sum_{i=1}^N \left[ \log p_l(g_\theta^{-1}(y_i; \xi_i)) + \log | \det J_{g_\theta^{-1}(y_i; \xi_i)} | \right] - \lambda_{\text{WD}} \| \theta \|^2,$$

where $\lambda_{\text{WD}}$ is a possible weight decay regularization. Training is performed via stochastic gradient methods (SGD/Adam) with early stopping on validation data to avoid overfitting. Complexity of the flow architecture is balanced to ensure tractability of the downstream optimization, as nonlinearities (e.g., ReLU, softplus) induce mixed-integer nonlinear programming constraints (Wedemeyer et al., 22 Jan 2026).

5. Decision-Analytic CFI and Risk Aversion

In decision analysis, the CFI is constructed for monetary prospects $X$ and $Y$ under a decision-maker's constant absolute risk aversion (CARA), whose utility satisfies the delta property and is given by the exponential form:

$$u_s(x) = \frac{1 - e^{-r x}}{1 - e^{-r}},~r \neq 0; \quad u_s(x) = x,~r = 0,$$

with $r > 0$ as the risk-aversion coefficient. The certain-equivalent is

$$\mathrm{CE}(X|r) = -\frac{1}{r} \ln \mathbb{E}[e^{-r X}].$$

Unmodeled uncertainty is represented by scaling the prospect by $k \geq 1$, yielding

$$\mathrm{CE}(k X | r) = k \cdot \mathrm{CE}(X | k r).$$

The decision-analytic CFI is then

$$\mathrm{CFI}_r(X; Y) = \inf \{ k \geq 1 : \mathrm{CE}(X | k r) \geq \mathrm{CE}(Y | k r) \},$$

interpreted as the minimum "stretch" in uncertainty required for $X$ to become preferable to $Y$. Robust flexibility corresponds to sensitivity of $\mathrm{CE}(X|k r)$ to $k$; adaptive flexibility is captured by the cross-over behavior in $k$ (Shachter et al., 2013).

6. Empirical Evaluations and Illustrative Examples

Two-Moons Example

An illustrative two-moons dataset in $\mathbb{R}^2$, with binary context $c \in \{0, 1\}$, is employed to compare (a) unconditional hypercube, (b) unconditional flow-based, (c) conditional hypercube, and (d) conditional flow-based uncertainty sets. The feasibility region is specified by $10 - \mathrm{Himmelblau}(y) \leq 0$. Coverage metrics and feasibility under infeasible regions are reported. Flow-based sets more accurately capture data support, and conditional sets exclude irrelevant regions. No universal dominance is evident—conditional sets yield increased coverage (e.g., 79% versus 52%), but performance is context- and data-dependent (Wedemeyer et al., 22 Jan 2026).

Security-Constrained Unit Commitment (SCUC)

A case study on the 3-bus German power grid evaluates the CFI for scheduling generator setpoints under uncertain renewable injections $P_{\text{ren}, n}$. Contextual inputs $\xi_t$ (recent capacity factors, time, seasonal indicators) inform the flow model. Baselines include unconditional hypercube and mean-centered uncertainty sets. Results indicate

• Unconditional hypercube achieves ~71% feasible test realizations.
• Conditional flow model with temporal context achieves ~89–91% feasibility.
• Incorporation of time and season enables improved prediction and hedging against variability. Even centering a hypercube at the flow—predicted mean yields similar benefits, but the CFI automatically adapts via data (Wedemeyer et al., 22 Jan 2026).

7. Limitations, Validity, and Practical Guidance

The CFI requires utility functions satisfying the delta property (CARA form), and prospects admitting appropriate moment-generating functions. The representation of unmodeled uncertainty by scaling may not capture all sources of model error. Practical computation involves evaluating slopes of CE-curves and solving for cross-over points in $k$ space. In optimization, tractability depends on flow architectures and embedding flow parameters into constrained programs, with challenges induced by non-convexities and mixed-integer expressions. There is no general guarantee that data-driven or conditional admissible sets universally outperform simple sets, but both approaches ensure that only relevant regions of the uncertainty space are considered. Choice of additive uncertainty $Z$ and coverage constraints is application-specific. Extensions to non-CARA utilities or alternative uncertainty representations remain open research questions (Wedemeyer et al., 22 Jan 2026, Shachter et al., 2013).