Decision-Dependent Uncertainty

• Decision-dependent uncertainty is an optimization framework in which the uncertainty model—such as support, ambiguity sets, or probability laws—changes in response to decision-maker actions.
• It introduces structural nonlinearities and computational challenges, often requiring tailored reformulations and decomposition methods to address NP-completeness and nonstandard gradients.
• DDU is critical in applications like energy systems and pricing models, where decisions directly influence the uncertainty environment, leading to improved risk management and performance.

Searching arXiv for recent and foundational papers on decision-dependent uncertainty to ground the article. Decision-dependent uncertainty (DDU), also called endogenous uncertainty in several strands of the literature, denotes optimization settings in which the feasible realizations, support, ambiguity set, or probability law of uncertain parameters is influenced by the decision-maker’s own actions. The defining contrast with conventional exogenous uncertainty is therefore structural: instead of a fixed uncertainty set $\mathcal U$ or fixed distribution $D$, one faces objects such as $\mathcal U(x)$, $D(x)$, or beliefs supported on a moving set $S(x)$. Across robust optimization, stochastic programming, distributionally robust optimization, and chance-constrained models, this shift alters both the meaning of uncertainty and the validity of classical decomposition, duality, and statistical estimation procedures (Nohadani et al., 2016, Hikima et al., 2023, Cotrina et al., 10 Sep 2025).

1. Formalizations and terminology

A common formal distinction runs through the literature. In robust optimization, DDU appears when constraints must hold for all $\xi\in\mathcal U_i(x)$ rather than for all $\xi\in\mathcal U_i$. In stochastic optimization, it appears when the objective is $$\min_{x\in\mathcal C}\mathbb E_{\xi\sim D(x)}[f(x,\xi)]$$, so that both the integrand and the sampling law depend on the decision. In support-based models, the measure $\beta_x$ is required to satisfy $\beta_x(S(x))=1$, meaning that the support itself moves with the decision. In contextual chance-constrained models, the relevant feasibility probability is $D$0, so the decision conditions the outcome law directly (Nohadani et al., 2016, Hikima et al., 2023, Cotrina et al., 10 Sep 2025, Liu et al., 7 Feb 2026).

A second distinction, emphasized in robust optimization, is that DDU is not the same as adjustable robustness. In adjustable models, decisions adapt to realized uncertainty; in DDU models, decisions may remain static while changing the uncertainty environment itself. The paper on robust optimization under DDU makes this explicit by writing the uncertainty set as $D$1 and noting that “the decisions do not adapt to realizations; rather, the decisions affect the uncertainty set itself” (Nohadani et al., 2016).

The main formal patterns can be summarized as follows.

Paradigm	Representative formulation	Decision-dependent object
Robust optimization	$D$2 or $D$3	Uncertainty set
Stochastic optimization	$D$4	Probability law
Moving-support expectation	$D$5, $D$6	Support set
Contextual chance constraints	$D$7	Conditional feasibility law
Distributionally robust optimization	$D$8	Ambiguity set

This taxonomy also clarifies why DDU is broader than price-responsive demand or state-dependent dynamics alone. The hallmark is not merely that outcomes vary with decisions through a known formula, but that the admissible uncertainty description itself changes with the decision.

2. Structural forms of decision dependence

The literature contains several recurrent structural realizations of DDU. One is the decision-dependent uncertainty set. In robust optimal operation of virtual power plants, the uncertain price elasticity $D$9 enters the demand model

$\mathcal U(x)$0

while the relevant elasticity interval depends on the chosen time-of-use price ratio. The first-stage tariff $\mathcal U(x)$1 therefore determines which interval of admissible elasticity values is active, so the uncertainty set is $\mathcal U(x)$2 rather than a fixed box (Tan et al., 25 Mar 2026).

A second form is the decision-dependent ambiguity set. In renewable-powered fast-charging planning, the support of EV adoption scenarios is fixed, but the admissible probability vectors over that support depend on prior charging-station decisions through decision-dependent moment bounds. The mean and second moment of pathwise EV adoption are affine functions of prior siting decisions, so the ambiguity set $\mathcal U(x)$3 changes with infrastructure deployment (Li et al., 2023). In wind planning, both the center and the radius of a Wasserstein ambiguity set depend on the capacity allocation vector because aggregate wind generation is $\mathcal U(x)$4, and the induced variance $\mathcal U(x)$5 changes with siting; the smoothing effect is therefore itself a DDU mechanism (Chen et al., 14 Aug 2025).

A third form is the decision-dependent nominal distribution in data-driven DRO. In residuals-based contextual DRO, the nominal empirical law is

$\mathcal U(x)$6

so both the center of the ambiguity set and, potentially, the ambiguity radius depend on the candidate decision $\mathcal U(x)$7 and covariate realization $\mathcal U(x)$8 (Zhu et al., 2024). In contextual chance-constrained programming, the same issue appears nonparametrically: historical samples were generated under past $\mathcal U(x)$9, not under the candidate $D(x)$0, so the feasible region depends on a decision-conditioned estimator of the law $D(x)$1 (Liu et al., 7 Feb 2026).

These examples also show that DDU commonly produces secondary nonlinearities. Decision-dependent sets often induce bilinear or mixed-integer couplings, such as the product of an uncertain elasticity with a tariff deviation in the VPP model, or the interaction between hardening decisions, power-flow prescriptions, and line-failure ambiguity sets in wildfire-aware distribution planning (Tan et al., 25 Mar 2026, Piancó et al., 2024).

3. Computational consequences and algorithmic frameworks

Once uncertainty depends on the decision, standard tractability results weaken sharply. In robust optimization, the general affine decision-dependent polyhedral case is NP-complete, even though classical robust linear optimization with exogenous polyhedral uncertainty is tractable by duality. That hardness result motivates the search for structured subclasses, such as uncertainty sets with reducible upper bounds under binary influence decisions, where exact reformulations become possible (Nohadani et al., 2016).

For multistage robust optimization, the main additional difficulty is that future uncertainty sets may depend on prior recourse decisions. The multistage robust mixed-integer framework under endogenous uncertainty addresses this by lifting the uncertainty process and combining it with nonlinear decision rules, including discontinuous piecewise linear decision rules for continuous recourse, producing a tractable MILP reformulation (Feng et al., 2020).

Decomposition methods also require modification. In piecewise-constant stochastic DDU, the first-stage region is partitioned into subsets $D(x)$2, with each subset inducing a different distribution $D(x)$3. Classical L-shaped cuts are then invalid globally, because a cut derived under $D(x)$4 need not apply under $D(x)$5. The extended L-shaped method resolves this with distribution-specific feasibility and optimality cuts that are activated only when the current solution belongs to the corresponding subset, and the paper proves finite convergence for both linear and mixed-integer recourse (Pantuso et al., 15 Jun 2025). In the VPP setting, standard column-and-constraint generation fails for the same reason: a worst-case scenario $D(x)$6 generated under one tariff may not belong to the uncertainty set induced by another tariff. The remedy is to return a worst-case vertex pattern $D(x)$7, not the raw $D(x)$8, so that the scenario is remapped to the currently active interval as prices change (Tan et al., 25 Mar 2026).

When the law $D(x)$9 is differentiable, gradient-based optimization becomes possible, but only after differentiating through the distribution map. The key identity is

$S(x)$0

which adds a score-function term absent in exogenous stochastic optimization. The same paper constructs an unbiased stochastic gradient estimator with a baseline parameter $S(x)$1 and establishes convergence to a stationary point via projected stochastic approximation (Hikima et al., 2023).

For contextual chance constraints, the main obstacle is the combination of endogeneity and discontinuity. The Contextual Cluster Weights method addresses this by assigning equal weights to a local cluster $S(x)$2 of similar historical $S(x)$3 pairs, turning the estimated feasibility probability into a cluster average and enabling mixed-integer reformulations with pre-calculated clusters. Under a nestedness condition on scenario-wise feasible sets, the resulting approximated feasible region becomes convex (Liu et al., 7 Feb 2026).

4. Statistical and geometric theory

Theoretical analysis of DDU has developed along at least three lines: smooth stochastic gradients, support-based regularity, and data-driven consistency. The support-based line begins from the observation that many endogenous laws are naturally specified by a moving support $S(x)$4 together with a density on that support. The expected value function

$S(x)$5

is then analyzed via the geometry of $S(x)$6, not by assuming regularity of the full measure-valued map $S(x)$7 directly. Sufficient conditions for Lipschitz continuity include a Lipschitz support map with convex compact full-dimensional images, a constant affine dimension with a Lipschitz parametrization, or a fully linear parametric lower-level solution set. The same work also proves that support Lipschitzness alone is insufficient: a family of trapezoids can move in a $S(x)$8-Lipschitz Hausdorff way while the induced expectation is not Lipschitz because normalization over the moving support changes too sharply (Cotrina et al., 10 Sep 2025).

A second line concerns nonparametric estimation under endogenous laws. In contextual chance constraints, consistency must be uniform in the decision, because the optimizer searches over $S(x)$9. The CCW framework therefore proves strong or weak uniform-in-decision consistency, together with explicit rates for $\xi\in\mathcal U_i(x)$0-nearest-neighbor and local-set-average constructions, for the estimated objective and chance-feasibility functions (Liu et al., 7 Feb 2026).

A third line extends residuals-based DRO to decision-dependent settings. ER-$\xi\in\mathcal U_i(x)$1 learns a regression $\xi\in\mathcal U_i(x)$2, transports empirical residuals onto the new pair $\xi\in\mathcal U_i(x)$3, and centers a Wasserstein, sample-robust, or same-support ambiguity set at the induced nominal law. The paper provides finite-sample certificate guarantees, asymptotic optimality, and rates of convergence, and it also discusses decision- and covariate-dependent radius calibration by cross-validation (Zhu et al., 2024).

Taken together, these results show that DDU theory is not limited to one modeling idiom. Smooth parametric models support score-function differentiation; moving-support models require volumetric and geometric control; nonparametric and residual-based methods require uniform-in-decision guarantees because the data-generating law varies with the optimized action.

5. Representative application domains

Energy systems provide the densest concentration of DDU models. In virtual power plants, pricing decisions select the relevant elasticity interval and thereby alter the support of uncertain demand response; the proposed DDU-aware algorithm converges in $\xi\in\mathcal U_i(x)$4 iterations and $\xi\in\mathcal U_i(x)$5 seconds, whereas a traditional C&CG method that returns $\xi\in\mathcal U_i(x)$6 directly fails to converge (Tan et al., 25 Mar 2026). In generalized energy storage dispatch, the uncertain available SoC bounds depend on incentive signals and accumulated discomfort, so chance-constrained dispatch under DDU yields conservative but credible strategies and reduced penalty cost relative to DIU-only models (Qi et al., 2022). In fast-charging planning, DDU is attached to EV diffusion through decision-dependent ambiguity sets, and incorporating DDU yields expected cost savings of $\xi\in\mathcal U_i(x)$7 for IF $\xi\in\mathcal U_i(x)$8 and $\xi\in\mathcal U_i(x)$9 for IF $\xi\in\mathcal U_i$0, peaking at $\xi\in\mathcal U_i$1 when traffic flow is $\xi\in\mathcal U_i$2 and incentive factor is $\xi\in\mathcal U_i$3 (Li et al., 2023). In wind planning, the smoothing effect makes the aggregate wind distribution and Wasserstein radius depend on turbine allocation, and the decision-dependent DRO model outperforms both empirical optimization and a normalized decision-independent DRO baseline (Chen et al., 14 Aug 2025). In wildfire-aware distribution planning, high power-flow prescriptions in high-threat areas raise line-failure likelihood, so the ambiguity set over outage patterns depends on flows and hardening; in the case study, modeling DDU changes the optimal investment portfolio and reduces average annual active load loss from $\xi\in\mathcal U_i$4 to $\xi\in\mathcal U_i$5 (Piancó et al., 2024). In capacity credit evaluation, the capacity credit of generalized energy storage increases with more strategic capacity withholding but decreases with more DDU levels (Qi et al., 2024).

Pricing and revenue-management applications furnish a different but equally canonical form of DDU: prices change the demand law. The general stochastic pricing model with decision-dependent uncertainty develops gradient estimators for $\xi\in\mathcal U_i$6 and shows that retraining methods that ignore the derivative of the distribution map can fail in pricing (Hikima et al., 2023). In carsharing, the operator’s price vector determines the Bernoulli adoption probabilities of potential customers, hence the demand distribution $\xi\in\mathcal U_i$7; explicitly modeling that dependence improves expected profits by $\xi\in\mathcal U_i$8 compared to a deterministic price-elastic benchmark and by $\xi\in\mathcal U_i$9 compared to an exogenous random-demand benchmark, on average (Deng et al., 10 Jun 2026).

Strategic and human-in-the-loop systems produce another family of endogenous laws. In strategic classification, the current classifier changes the future uncertainty over manipulation costs, producing a two-stage robust model with a decision-dependent uncertainty set over cost matrices. In the reported semi-synthetic experiment, the DD-aware model yields total loss $$\min_{x\in\mathcal C}\mathbb E_{\xi\sim D(x)}[f(x,\xi)]$$0 versus $$\min_{x\in\mathcal C}\mathbb E_{\xi\sim D(x)}[f(x,\xi)]$$1 for the decision-independent baseline, with unqualified manipulations $$\min_{x\in\mathcal C}\mathbb E_{\xi\sim D(x)}[f(x,\xi)]$$2 versus $$\min_{x\in\mathcal C}\mathbb E_{\xi\sim D(x)}[f(x,\xi)]$$3 (Alhanouti et al., 29 Jun 2026). In Bayesian bilevel formulations, the uncertainty may be a follower response drawn from a decision-dependent optimal-solution set, which motivates the moving-support framework described above (Cotrina et al., 10 Sep 2025).

6. Conceptual pitfalls and open issues

Several recurring misconceptions are corrected by the current literature. First, DDU is not exhausted by writing $$\min_{x\in\mathcal C}\mathbb E_{\xi\sim D(x)}[f(x,\xi)]$$4 with a fixed $$\min_{x\in\mathcal C}\mathbb E_{\xi\sim D(x)}[f(x,\xi)]$$5-law. The distinctive feature is that the uncertainty description itself changes with the decision, whether as $$\min_{x\in\mathcal C}\mathbb E_{\xi\sim D(x)}[f(x,\xi)]$$6, $$\min_{x\in\mathcal C}\mathbb E_{\xi\sim D(x)}[f(x,\xi)]$$7, $$\min_{x\in\mathcal C}\mathbb E_{\xi\sim D(x)}[f(x,\xi)]$$8, or $$\min_{x\in\mathcal C}\mathbb E_{\xi\sim D(x)}[f(x,\xi)]$$9; this is precisely why treating DDU as ordinary exogenous uncertainty with a more complicated objective is generally incorrect (Nohadani et al., 2016, Hikima et al., 2023).

Second, standard decomposition and estimation routines can become algorithmically inconsistent when they ignore endogeneity. In the VPP case, using traditional C&CG with raw worst-case scenarios leads to nearly unchanged upper and lower bounds and even $\beta_x$0, because the returned scenario is meaningful only relative to the uncertainty set generated by the current tariff (Tan et al., 25 Mar 2026). In stochastic price optimization, retraining-style baselines perform badly because they optimize against a frozen distribution and ignore the derivative of the decision-to-distribution map (Hikima et al., 2023).

Third, support regularity is subtler than it appears. The moving-support theory shows that Hausdorff-Lipschitz motion of the support alone does not control the induced normalized-volume measure strongly enough; additional volumetric conditions or structural decompositions are necessary (Cotrina et al., 10 Sep 2025).

Finally, most tractable DDU models still rely on restrictive structure. Existing approaches frequently assume binary influence decisions or piecewise-constant distribution partitions, differentiable strictly positive densities with bounded score functions, convex compact supports, linearized network physics, or big-$\beta_x$1 linearizations. Mixed-integer first-stage restrictions appear in the L-shaped method for stochastic DDU and in contextual chance-constrained reformulations with pre-calculated clusters, while energy applications often rely on LinDistFlow, polygonal approximations, or coarse interval bins for endogenous response (Pantuso et al., 15 Jun 2025, Zhu et al., 2024, Qi et al., 2022). This suggests that the main open frontier is not whether DDU matters—it clearly does—but how to model and solve richer endogenous uncertainty mechanisms without sacrificing either statistical credibility or tractability.

In that sense, DDU marks a shift in optimization theory from hedging against an external environment to reasoning about environments that are partly shaped by the optimizer itself. The technical consequences are visible in every paradigm: uncertainty sets become mappings, distributions become decision-conditioned laws, ambiguity sets acquire endogenous centers or radii, and feasibility probabilities cease to be fixed attributes of the world.