This paper tackles the Sugeno Integral Optimisation (SIO) problem, which involves optimising objectives defined by the Sugeno integral, which generally is neither convex nor concave. Existing non-smooth optimisation methods yield only roughly accurate solutions, while mixed-integer linear programming (MIP) formulations with auxiliary binary variables become computationally prohibitive for n>12. We address large-scale cases ( n>12 ) by developing efficient bisection algorithms which involve linear constraint satisfaction problems, treated by linear and quadratic programming, and iterative methods of Alternating Direction Method of Multipliers (ADMM), Projection Onto Convex Sets (POCS), the Dykstra method, Douglas–Rachford methods and Farkas’ Lemma, emphasizing their scalability. We further consider optimisation with k-maxitive or k-minitive capacities to reduce constraint complexity, other improvements to bisection, and also treat the Shilkret integral in the same framework.
Malaria in the Greater Mekong Subregion now persists in a small number of residual foci in which transmission is low, strongly seasonal, and sustained in part by importation. We analyse a climate-sensitive transmission model for Tak Province, on the Thailand–Myanmar border, whose formulation and calibration were introduced in our companion study. The model extends a human–mosquito compartmental system with imperfect, waning vaccination and infected immigration by replacing its constant vector parameters with climate-forced functions of lagged rainfall, temperature and vegetation. We establish well-posedness, obtain the basic reproduction number ℛ_0 in closed form, and prove that the slope at the origin of the endemic consistency map equals ℛ_0^2 , so that existence of an endemic equilibrium is governed by ℛ_0 and is decidable by a finite check. An explicit vaccination threshold shows that a leaky vaccine cannot reduce ℛ_0^2 below its efficacy-limited floor ε _vℛ_0(0)^2 , and we introduce a climate reproduction index ℛ_c(t) admitting a closed form in three composite climate elasticities. Calibration to weekly 2025 surveillance and climate data reproduced the epidemic peak in week 24, although it underestimated the peak magnitude by about 30% . At the reference calibration, ℛ_0=0.96 and no endemic equilibrium exists, but because the reporting scale is not identifiable, ℛ_0 ranges from 0.94 to 1.14, placing the province close to threshold. Transmission is driven by seasonal excursions of ℛ_c(t) above unity; importation is examined by scenario. The net rainfall elasticity is negative, consistent with larval flushing. This threshold structure explains why a 30% vector-directed reduction averts 95– 98% of predicted cases, why vaccination alone averts little, and why deployment timed to the rise in transmission potential–which peaks about nine weeks before cases–averts far more than peak-time deployment.
The main purpose of this paper is to develop and analyze a proximal-type method for the local minimization of weakly convex functions. Our motivation for introducing this algorithm is to increase its versatility and applicability in a broader range of scenarios in optimization and introduce a more adaptable version of the method for typical generalizations. Our study relies heavily on the Moreau envelope, a well-known mathematical tool used to analyze the behavior of the proximal operator. By leveraging the properties of the Moreau envelope, we are able to prove that the proximal algorithm converges in local contexts. Moreover, we present a complexity result to assess the practical feasibility of the proximal algorithm and provide a numerical example to illustrate the practical behavior of the algorithm.
This paper proposes a multi-interval Legendre spectral collocation method based on a modified geometric mesh for nonlinear third-kind Volterra integral equations with pantograph delay, which deals with the singularity of equations and the sophisticated effect of pantograph delays. The rigorous convergence analysis of the hp-version is derived under the L^2 -norm. Numerical experiments are presented to demonstrate the theoretical expectations.
We investigate a discrete-time predator–prey ecosystem that features a generalist predator and a prey species with defense mechanisms, with a focus on the function of herding behavior. First, we provide sufficient one-step positivity conditions, determine the fixed points, and derive conditions for the local stability of the interior equilibrium. Then, by combining bifurcation theory with center manifold techniques, we obtain analytical criteria for the occurrence of flip (period-doubling) and Neimark–Sacker bifurcations with respect to the discretization step size. The analysis reveals that the discrete model exhibits rich dynamical behaviors, including period-doubling cascades, quasi-periodic oscillations, and chaos, highlighting a more complex bifurcation structure than that of the continuous formulation. To suppress chaotic dynamics, we propose an OGY-based control strategy and demonstrate that the unstable interior fixed point can be effectively stabilized. The robustness of the control scheme is further validated through basin of attraction and parameter sensitivity analyses. Numerical simulations, including bifurcation diagrams, phase portraits, and maximal Lyapunov exponents, support the theoretical results and emphasize the crucial role of discretization in shaping the qualitative dynamics of the system. These results highlight the role of prey herd behavior in shaping the qualitative dynamics of predator–prey interactions and have potential implications for ecological modeling and population dynamics.
