The paper proposes a novel adaptive search space decomposition method and a novel gradient-free optimization-based formulation for the pre- and post-buckling analyses of space truss structures. Space trusses are often employed in structural engineering to build large steel constructions, such as bridges and domes, whose structural response is characterized by large displacements. Therefore, these structures are vulnerable to progressive collapses due to local or global buckling effects, leading to sudden failures. The method proposed in this paper allows the analysis of the load-equilibrium path of truss structures to permanent and variable loading, including stable and unstable equilibrium stages and explicitly considering geometric nonlinearities. The goal of this work is to determine these equilibrium stages via optimization of the Lagrangian kinematic parameters of the system, determining the global equilibrium. However, this optimization problem is non-trivial due to the undefined parameter domain and the sensitivity and interaction among the Lagrangian parameters. Therefore, we propose formulating this problem as a nonlinear, multimodal, unconstrained, continuous optimization problem and develop a novel adaptive search space decomposition method, which progressively and adaptively re-defines the search domain (hypersphere) to evaluate the equilibrium of the system using a gradient-free optimization algorithm. We tackle three benchmark problems and evaluate a medium-sized test representing a real structural problem in this paper. The results are compared to those available in the literature regarding displacement-load curves and deformed configurations. The accuracy and robustness of the adopted methodology show a high potential of gradient-free algorithms in analyzing space truss structures.

We focus on adversarial patrolling games on arbitrary graphs, where the Defender can control a mobile resource, the targets are alarmed by an alarm system, and the Attacker can observe the actions of the mobile resource of the Defender and perform different attacks exploiting multiple resources. This scenario can be modeled as a zero-sum extensive-form game in which each player can play multiple times. The game tree is exponentially large both in the size of the graph and in the number of attacking resources. We show that when the number of the Attacker's resources is free, the problem of computing the equilibrium path is NP-hard, while when the number of resources is fixed, the equilibrium path can be computed in poly-time. We provide a dynamic-programming algorithm that, given the number of the Attacker's resources, computes the equilibrium path requiring poly-time in the size of the graph and exponential time in the number of the resources. Furthermore, since in real-world scenarios it is implausible that the Defender knows the number of attacking resources, we study the robustness of the Defender's strategy when she makes a wrong guess about that number. We show that even the error of just a single resource can lead to an arbitrary inefficiency, when the inefficiency is defined as the ratio of the Defender's utilities obtained with a wrong guess and a correct guess. However, a more suitable definition of inefficiency is given by the difference of the Defender's utilities: this way, we observe that the higher the error in the estimation, the higher the loss for the Defender. Then, we investigate the performance of online algorithms when no information about the Attacker's resources is available. Finally, we resort to randomized online algorithms showing that we can obtain a competitive factor that is twice better than the one that can be achieved by any deterministic online algorithm.

The most common solution concept for a strategic interaction situation is the Nash equilibrium, in which no agent can do better by deviating unilaterally. However, the Nash equilibrium underlays on the assumption of common information that is hardly verified in many practical situations. When information is not common, rational agents are assumed to learn from their observations to derive beliefs over their opponents' play and payoffs. In these situations, there are steady states composed of beliefs and strategies in which the strategies do not constitute a Nash equilibrium. These stable states are called in the game theory literature self-confirming equilibria. They are such that every agent plays the best response to her beliefs and these are correct on the equilibrium path, while off the equilibrium path they may be incorrect. We present some mathematical programming formulations for computing self-confirming equilibria and its refinements in two-player extensive-form games and we study their properties.