Distributionally Robust Optimization for Aerial Multi-Access Edge Computing via Cooperation of UAVs and HAPs

This paper studies how to make reliable task offloading and resource allocation decisions in an aerial multi-access edge computing (MEC) system where unmanned aerial vehicles (UAVs) and a high-altitude platform (HAP) cooperate. The central research question is how to design an optimization framework that remains robust when the wireless environment and service demand are uncertain—specifically, when the distribution of key random variables (e.g., channel quality, workload, or other system states) is not known exactly. This matters because aerial links are highly time-varying and non-stationary, and MEC decisions (which UAV/HAP serves which users, and how much compute/bandwidth to allocate) are sensitive to these uncertainties; naive “average-case” optimization can lead to severe performance degradation under adverse conditions.

A key contribution is the use of distributionally robust optimization (DRO) with a risk-sensitive objective based on conditional value-at-risk (CVaR). CVaR focuses on the tail of the loss distribution, providing a principled way to control worst-case behavior under rare but costly events. The paper begins from the standard variational representation of CVaR for a measurable loss $\varphi (\xi )$\u0009 with safety factor $\alpha \in (0,1)$: $$\mathbb{P}-{\mathrm{CVaR}}_{\alpha }(\varphi (\xi ))=\underset{\beta \in \mathbb{R}}{\inf }\left\{\beta +\frac{1}{1-\alpha }{\mathbb{E}}_{\mathbb{P}}{\left[\varphi (\xi )-\beta \right]}^{+}\right\},$$ where $[\cdot {]}^{+}$ denotes the positive part. The authors then consider a worst-case distribution $\mathbb{P}$ drawn from an ambiguity set $\mathcal{P}$, leading to a maxmin formulation: $$\underset{\mathbb{P}\in \mathcal{P}}{\sup }\underset{\beta \in \mathbb{R}}{\inf }\left\{\beta +\frac{1}{1-\alpha }{\mathbb{E}}_{\mathbb{P}}{\left[\Theta \xi +{\theta }^0-\beta \right]}^{+}\right\}.$$ Here, $\Theta$ and ${\theta }^0$ arise from the system model and decision variables (the paper’s excerpt uses a generic linear form $\Theta \xi +{\theta }^0$ inside the loss). To make the problem tractable, the paper invokes the saddle point theorem to interchange the $\sup$ and $\inf$ operators, turning the problem into an inner maximization over distributions of an expectation of a convex function of the uncertain variable.

Methodologically, the excerpt shows how the inner layer is solved under a moment-based ambiguity set. The ambiguity set is specified as distributions with fixed mean and variance of the uncertainty $\xi$: $$\mathcal{P}=\left\{\mathbb{P}:{\mathbb{E}}_{\mathbb{P}}(\xi )={\mu }_{\xi },{\mathbb{D}}_{\mathbb{P}}(\xi )={\sigma }_{\xi }^2\right\}.$$ By defining $\varrho =\Theta \xi$, the mean and variance of $\varrho$ become $\Theta {\mu }_{\xi }$ and ${\Theta }^2{\sigma }_{\xi }^2$, respectively. The inner maximization of ${\mathbb{E}}_P[(\Theta \xi +{\theta }^0-\beta {)}^{+}]$ is then reformulated as an optimization over nonnegative measures $\phi$ (a standard DRO technique): $$\underset{\phi \in \mathcal{C}}{\sup }{\int }_{\mathbb{R}}(\Theta \xi +{\theta }^0-\beta {)}^{+}\phi d\varrho$$ subject to normalization and moment constraints (constraints on $\int \phi d\varrho$, $\int \varrho \phi d\varrho$, and $\int {\varrho }^2\phi d\varrho$). The authors then apply Lagrangian duality by introducing dual variables ${\chi }_1,{\chi }_2,{\chi }_3$ for these constraints, producing a dual problem with inequality constraints ensuring nonnegativity of a quadratic function over $\varrho$. Strong duality is used to argue equivalence between primal and dual formulations.

A crucial technical step is the reduction of the dual constraints to a finite-dimensional convex program. The excerpt shows that by substituting the optimizer ${\varrho }^{\ast }=-\frac{{\chi }_2}{2{\chi }_3}$ and enforcing strong duality, the worst-case CVaR can be approximated/represented by an optimization problem over auxiliary variables $\beta ,e,q,z,s$. The resulting tractable convex formulation (as shown in the excerpt) is: $$\underset{\beta ,e,q,z,s}{\inf }\beta +\frac{1}{1-\alpha }(e+s)$$ subject to constraints including: - $e-{\theta }^0+\beta +q-\Theta {\mu }_{\xi }-z>0$, - $e\ge 0,z>0$, - $4zs\ge q^2+{\Theta }^2{\sigma }_{\xi }^2$, and an equivalent matrix inequality form: $$\Vert \begin{array}{c}q\\ \Theta {\sigma }_{\xi }\\ z-s\end{array}\Vert \le z+s.$$ This transformation is significant because it converts an intractable distributionally robust CVaR problem into a convex program that can be solved efficiently, enabling practical deployment in MEC decision-making.

Regarding key findings, the excerpt does not provide numerical experiments or system-level performance metrics (e.g., latency reduction, outage probability, or throughput gains) with explicit values. Instead, the “finding” in the provided text is the theoretical result: the worst-case CVaR under moment uncertainty can be reformulated as a tractable convex optimization problem via duality and strong duality, with explicit constraints (the convex program in the excerpt). The paper’s practical contribution therefore lies in making risk-sensitive DRO computationally feasible for aerial MEC.

Limitations are not explicitly stated in the excerpt, but they are implied by the modeling choices: (i) the ambiguity set uses only mean and variance, which may be insufficient if higher-order moments or support constraints matter; (ii) the tractability relies on convexity and duality conditions (e.g., ${\chi }_3>0$, strong duality assumptions); and (iii) the excerpt does not show how $\Theta$ and ${\theta }^0$ map to concrete UAV/HAP scheduling and resource allocation variables, so the end-to-end system impact depends on the full paper’s system model.

Practical implications: if the full paper follows through with an optimization-based controller, then the proposed DRO-CVaR framework can be used by network designers/operators to allocate UAV and HAP resources while explicitly protecting against tail-risk events (e.g., very poor channel realizations or overload scenarios). This is most relevant to engineers building aerial MEC systems for time-sensitive applications (video analytics, AR/VR, emergency response) where worst-case delays or service failures are unacceptable. The approach should be of interest to researchers in robust optimization, wireless communications, and edge computing who need a computationally efficient way to incorporate risk sensitivity under distributional uncertainty.

Overall, based on the provided text, the paper’s core contribution is a rigorous derivation that turns a worst-case CVaR objective under moment-based distributional uncertainty into a convex optimization problem with explicit constraints, enabling robust and risk-aware decision-making in cooperative UAV-HAP MEC networks.