Optimization and Control (math.OC)

Abstract: We provide a duality result linking the value function for a control problem with supremum cost H under an isoperimetric inequality G $\le$ gmax, and the value function for the same controlled dynamics with cost G and state constraint H $\le$ hmax. This duality is proven for initial conditions at which lower semi-continuity of the value functions can be guaranteed, and is completed with optimality considerations. Furthermore, we provide structural assumptions on the dynamics under which such regularity can be established. As a by-product, we illustrate the partial equivalence between recentworks dealing with non-pharmaceutically controlled epidemics under peak or budget restrictions.

Abstract: We extend the concept of well-posedness to the split equilibrium problem and establish Furi-Vignoli-type characterizations for the well-posedness. We prove that the well-posedness of the split equilibrium problem is equivalent to the existence and uniqueness of its solution under certain assumptions on the bifunctions involved. We also characterize the generalized well-posedness of the split equilibrium problem via the Kuratowski measure of noncompactness. We illustrate our theoretical results by several examples.

Abstract: We present a novel method for estimating the circulations and positions of point vortices using trajectory data of passive particles in the presence of Gaussian noise. The method comprises two algorithms: the first one calculates the vortex circulations, while the second one reconstructs the vortex trajectories. This reconstruction is done thanks to a hierarchy of optimization problems, involving the integration of systems of differential equations, over time sub-intervals all with the same amplitude defined by the autocorrelation function for the advected passive particles' trajectories. Our findings indicate that accurately tracking the position of vortices and determining their circulations is achievable, even when passive particle trajectories are affected by noise.

Abstract: In this article, we establish a class of new accelerated modulus-based iteration methods for solving the linear complementarity problem. When the system matrix is an $H_+$-matrix, we present appropriate criteria for the convergence analysis. Also, we demonstrate the effectiveness of our proposed method and reduce the number of iterations and CPU time to accelerate the convergence performance by providing two numerical examples for various parameters. Keywords. Linear complementarity problem, Iteration method, $P$-matrix, $H_{+}$-matrix, Convergence analysis, Matrix splitting.

Abstract: The inverse-square law states that the effect a source has on its surroundings is inversely proportional to the square of the Euclidean distance from that source. Its applicability spans multiple fields including physics, engineering, and computer science. We study the combined effect of multiple point sources surrounding a closed region in multidimensional space. We determine that the maximum effect in D dimensions is on the region's boundary if $D \leq 4$, and the minimum is on the boundary if $D \geq 4$.

Abstract: As climate change evolves, wildfire risk is increasing globally, posing a growing threat to power systems, with grid failures fueling the most destructive wildfires. In day-to-day operations, preemptive de-energization of equipment is an effective tool to mitigate the risk and damage of wildfires. However, such power outages have significant impacts on customers. This paper proposes a novel framework for planning preemptive de-energization of power systems to mitigate wildfire risk and damage. We model wildfire disruptions as stochastic disruptions with random magnitude and timing and formulate a two-stage stochastic program that maximizes the electricity delivered while proactively reducing the risk of wildfires by selectively de-energizing components over multiple time periods. We use cellular automata to sample grid failure and wildfire scenarios based on grid-related risks and environmental factors. We develop a decomposition algorithm that can generate adaptive shutoff plans before a disruption occurs. We test our method on an augmented version of the RTS-GLMC test case in Southern California and compare it with two benchmark cases. We find that our method reduces both wildfire damage costs and load-shedding losses over multiple time periods, and our nominal plan is robust against the uncertainty model perturbation.