A Safe First-Order Method for Pricing-Based Resource Allocation in Safety-Critical Networks

TL;DR

This paper addresses safe, pricing-based resource allocation under safety-critical constraints by solving NUM with an arbitrary convex, compact feasible set ${\cal X}$. It introduces SPNUM, which posts prices and uses observed user profit-maximization responses to realize feasible primal iterates at each step; safety is achieved by operating on a shrunk feasible set ${\cal X}_{\Delta^t}$ and progressively reducing shrinkage and step sizes, while a Jacobian-based price update aligns induced demand with the desired target. The authors prove a sublinear static regret $R(T)=O(\log(T))$ and a convergence rate $\|x^{T/2}-x^*\|^2=O(\log(T)/T)$, and validate the approach through numerical studies against existing first-order methods. The work has practical implications for pricing-based control in safety-critical networks, such as electricity demand response, where no back-and-forth with users is possible.

Abstract

We introduce a novel algorithm for solving network utility maximization (NUM) problems that arise in resource allocation schemes over networks with known safety-critical constraints, where the constraints form an arbitrary convex and compact feasible set. Inspired by applications where customers' demand can only be affected through posted prices and real-time two-way communication with customers is not available, we require an algorithm to generate ``safe prices''. This means that at no iteration should the realized demand in response to the posted prices violate the safety constraints of the network. Thus, in contrast to existing distributed first-order methods, our algorithm, called safe pricing for NUM (SPNUM), is guaranteed to produce feasible primal iterates at all iterations. At the heart of the algorithm lie two key steps that must go hand in hand to guarantee safety and convergence: 1) applying a projected gradient method on a shrunk feasible set to get the desired demand, and 2) estimating the price response function of the users and determining the price so that the induced demand is close to the desired demand. We ensure safety by adjusting the shrinkage to account for the error between the induced demand and the desired demand. In addition, by gradually reducing the amount of shrinkage and the step size of the gradient method, we prove that the primal iterates produced by the SPNUM achieve a sublinear static regret of ${\cal O}(\log{(T)})$ after $T$ time steps.

Figures (3)

• Figure 1: Results for the benchmarking study. In all plots, SPNUM is shown in blue, SDGM in green, and DG in red. The shaded areas correspond to one standard deviation. In (a), we plot the convergence of the primal variables measured by $\|x^t-x^\star\|^2$ (left) and the infeasibility amount measured by $\|[Ax^t-c]_+\|/\|c\|$ (right) for all three algorithms when $A\in\{0,1\}^{m\times n}$ is a binary matrix. In (b), we plot the convergence of the primal variables measured by $\|x^t-x^\star\|^2$ (left) and the infeasibility amount measured by $\|[Ax^t-c]_+\|/\|c\|$ (right) for all three algorithms when $A\in{\mathbb R}^{m\times n}$ is a real matrix.
• Figure 2: Results for the numerical study on the impact of sharpness on regret. The figures on each row share the same y-axis. The shaded areas correspond to one standard deviation. The title of each subfigure denotes the $(\beta,\Gamma_{\cal X})$ configuration, and the regret incurred by different values of $n$ are plotted for each configuration. We observe that in the top row of figures, i.e., when $\beta$ is small, both $\Gamma_{\cal X}$ and $n$ have little effect on the regret (e.g., increasing ${\Gamma}_{\cal X}$ by 8 times only doubles the regret for all $n$). On the other hand, the bottom row of figures demonstrates that when $\beta$ is larger, then $\Gamma_{\cal X}$ and $n$ have a significant impact.
• Figure 3: Results for the numerical study on SPNUM on non-linear feasible set. In the left figure, the regret divided by ${\log(1+t)}$ is plotted in green, and constraint violation is plotted in blue, where constraint violation is $0$ if $x^t\in {\cal X}$ and $1$ otherwise. In the right figure, we plot the primal convergence measured as $\|x^t-x^\star\|^2$. Shaded areas correspond to one standard deviation.

Theorems & Definitions (23)

• Definition 1
• Definition 2
• Definition 3
• Example 1: Utility function
• Definition 4
• Example 2
• Remark 1
• Definition 5
• Remark 2
• Remark 3