We consider the problem of selecting and scheduling TV advertisements to fill a commercial break as effectively as possible. The gain generated by selecting and airing an ad depends on the ad’s profit and on the position in the break where the ad is scheduled, the latter being represented by an impact function. The resulting problem turns out to be a generalization of the Knapsack Problem, for which we revise and extend results from the literature and show strong NP-hardness in its general form. Then, we focus on the relevant case where the impact function is negative unimodal. For this case, we introduce two new relaxations based on splitting ads into time slices, determine their worst-case optimality gaps, and introduce constructive algorithms for their computation. In addition, a dynamic programming algorithm is introduced for computing an exact solution of the problem. Finally, we present three greedy-type algorithms and analyze their worst-case performance guarantee.
This paper proposes a robust fractional optimal control framework governed by higher-order variational structure and higher-order dynamics. The formulation combines Riemann–Liouville weighted curvilinear action functionals with uncertainty in both the objective and constraints, extending deterministic higher-order variational frameworks to a robust fractional setting. Robust necessary optimality conditions of Euler-Lagrange type are derived, together with a higher-order variational Kuhn-Tucker concept and sufficient conditions for robust weak optimality under convexity assumptions. A numerical example with third-order dynamics illustrates the applicability and accuracy of the proposed approach.
This paper examines structural properties of the solution set of absolute value equations (AVE) of the form Ax + |x| = b . Motivated by recent developments and related open problems in the literature of absolute value equations, and as a continuation of our earlier work [Optimization Letters 20 (2026) 611–623] on the alternative form Ax - |x| = b , we revisit these questions in their original setting. In particular, we analyze solvability of AVE, uniqueness of its solution, computational aspects, as well as its dependence on the properties of the matrix A , and construct counterexamples to show that certain proposed implications of solvability do not hold in general. We then identify sufficient conditions on the matrix A under which these implications become valid, notably establishing that one of them holds when A is symmetric.
The widespread use of UAVs in various application areas has led to an increasing demand for their route planning. However, complex 3D environments and diverse mission requirements make traditional route planning methods challenging. Meta-heuristic algorithms have become a powerful tool to solve this problem due to their good performance in search and optimization problems. The Honey Badger method (HBA), a recently developed meta-heuristic method for 2021, is prone to falling into a local optimal state and has low convergence accuracy. First, this paper proposes a novel improved honey badger algorithm (CTHBA), in which the efficient convergence ability of center-of-mass adversarial computation and the perturbation ability of adaptive t-distribution variability are utilized in the HBA algorithm to overcome these two problems of the HBA algorithm. Secondly, the flight path, threat cost, and flight altitude of the UAV are designed to establish a three-dimensional complex simulation environment. Finally, the convergence performance and engineering application capability of the proposed CTHBA algorithm are experimentally simulated. According to the testing results, the CTHBA method in the CEC2021 function outperformed the many meta-heuristic algorithms that are already in use, and the impact of UAV 3D path planning is clearly superior.
The aim of the paper mentioned in the title is to obtain several results related to Pareto efficiency in spaces without topological requirements. In this short note we show that the conclusions of the main results of that paper are not true.
