Mathematics

Asynchronous best-response dynamics in finite exact potential games converge to pure Nash equilibria, but total computational cost depends on player activation schedules. We formulate activation scheduling as a Markov decision process, in which a central regulator learns, with tabular Q-learning, which player to activate in a discrete Shapley facility location game. To handle the combinatorial state space, we present a permutation-invariant observation mapping with four binary flags per player that compresses state representations from $m^n$ to $\left({\displaystyle \genfrac{}{}{0pt}{}{n+15}{n}}\right)$. This reduces per-step decision costs to , compared to for greedy selection. We prove that the regulator asymptotically dominates greedy selection in total cost if and only if their expected iteration ratio is $o\left(n\right)$, and empirically achieve up to a $2.91$-fold cost reduction. A hand-crafted baseline, lowest payoff first, achieves costs 0.34–0.65 times the regulator’s by exploiting an anti-correlation between current payoffs and maximum available gains. The learned Q-tables show the same preference, drawn from the single payoff bit they observe. Finally, restricting the learned choice to unchecked players removes the stalls caused by state aliasing, and a pilot distills the policy into interpretable decision trees that provably reach equilibrium.

Combined heat and power economic dispatch (CHPED) is a challenging nonlinear, nonconvex optimization problem characterized by the simultaneous coordination of electrical and thermal power generation under complex operational constraints. The presence of valve-point loading effects, transmission power losses, practical operating regions, and the strong coupling between heat and power production significantly increases the complexity of obtaining globally optimal dispatch solutions. To address these challenges, this paper proposes an Evolutionary State-Guided Exponential Distribution Optimizer (ESG-EDO), an enhanced version of the recently developed Exponential Distribution Optimizer (EDO). The proposed ESG-EDO incorporates different search enhancement mechanisms to improve both convergence behavior and solution quality. First, a differential searching strategy is integrated to strengthen exploration and improve the capability of escaping local optima. Second, a perturbation module performs selective variable-wise perturbations instead of full-dimensional updates, thereby maintaining useful search directions while preserving population diversity. Third, an evolutionary state factor is introduced to adaptively regulate the activation of the perturbation mechanism according to the evolutionary status of the population, enabling a dynamic balance between exploration and exploitation throughout the optimization process. Finally, a regeneration mechanism periodically rejuvenates a portion of the population to prevent premature convergence and mitigate stagnation in high-dimensional search spaces. The effectiveness of the proposed ESG-EDO is comprehensively evaluated on benchmark CHPED systems of increasing complexity. The first case study considers a small-scale 7-unit CHPED system under two operating scenarios, namely with transmission losses and without transmission losses, to assess the algorithm’s ability to accurately solve conventional nonlinear dispatch problems. The second case study investigates a large-scale 48-unit CHPED system under the same two scenarios to evaluate scalability and robustness under substantially increased dimensionality and constraint interactions. Finally, the proposed algorithm is validated on a very large-scale 192-unit CHPED system comprising one highly complex operating scenario, demonstrating its capability for solving practical large-dimensional economic dispatch problems. The obtained results consistently demonstrate that the proposed ESG-EDO achieves superior optimization accuracy, faster convergence, enhanced population diversity, and improved robustness across all studied systems. For the very large 192-unit system, ESG-EDO reduced the cost, representing an improvement of approximately 15.1% compared with the original EDO. These findings confirm the effectiveness of the proposed ESG-EDO approach in enhancing the search capability of EDO and achieving superior economic costs reduction.

We establish global-in-time, time-weighted Strichartz estimates for the linear inhomogeneous generalized Tricomi equation in spatial dimensions two and higher. At the conformally invariant integrability exponent, we obtain weighted spacetime estimates for an enlarged range of time-weight parameters, including the endpoint. Compared with previously established results, our estimates significantly extend the admissible range of time weights. In particular, in two spatial dimensions, the admissible upper bound reaches twice the degeneracy parameter. We also elucidate the connection between the endpoint estimate in three spatial dimensions and semilinear wave equations on spatially flat Friedmann–Lemaître–Robertson–Walker spacetimes in the decelerating expansion regime. Our results provide a useful analytical tool for investigating the long-time behavior of nonlinear wave equations on such cosmological backgrounds.