Hierarchical multilevel optimization with multiple-leaders multiple-followers setting and nonseparable objectives

Hierarchical multilevel multi-leader multi-follower problems are non-cooperative decision problems in which multiple decision-makers of equal status in the upper-level and multiple decision-makers of equal status are involved at each of the lower-levels of the hierarchy. Much of solution methods proposed so far on the topic are either model specific which may work only for a particular sub-class of problems or are based on some strong assumptions and only for two level cases. In this paper, we have considered hierarchical multilevel multi-leader multi-follower problems in which the objective functions contain separable and non-separable terms (but the non-separable terms can be written as a factor of two functions, a function which depends on other level decision variables and a function which is common to all objectives across the same level) and shared constraint. We have proposed a solution algorithm to such problems by equivalent reformulation as a hierarchical multilevel problem involving single decision maker at all levels of the hierarchy. Then, we applied a multi-parametric algorithm to solve the resulting single leader single followers problem.

Complexity of near-optimal robust versions of multilevel optimization problems

Near-optimality robustness extends multilevel optimization with a limited deviation of a lower level from its optimal solution, anticipated by higher levels. We analyze the complexity of near-optimal robust multilevel problems, where near-optimal robustness is modelled through additional adversarial decision-makers. Near-optimal robust versions of multilevel problems are shown to remain in the same complexity class as the problem without near-optimality robustness under general conditions.

Sequential sampling framework for metamodeling uncertainty reduction in multilevel optimization of hierarchical systems

Metamodels instead of computer simulations are often adopted to reduce the computational cost in the uncertainty based multilevel optimization. However, metamodel techniques may bring prediction discrepancy, which is defined as metamodeling uncertainty, due to the limited training data. An unreliable solution will be obtained when the metamodeling uncertainty is ignored, while an overly conservative solution, which contradicts the original intension of the design, may be got when both parametric and metamodeling uncertainty are treated concurrently. Hence, an adaptive sequential sampling framework is developed for the metamodeling uncertainty reduction of multilevel systems to obtain a solution that approximates the true solution. Based on the Kriging model for the probabilistic analytical target cascading, the proposed framework establishes a revised objective-oriented sampling criterion and sub-model selection criterion, which can realize the location of additional samples and the selection of subsystem requiring sequential sampling. Within the sampling criterion, the metamodeling uncertainty is decomposed by the Karhunen-Loeve expansion into a set of stochastic variables, and then polynomial chaos expansion is employed for uncertainty quantification. The polynomial coefficients are encoded and integrated in the selection criterion to obtain subset sensitivity indices for the sub-model selection. The effectiveness of the developed framework for metamodeling uncertainty reduction is demonstrated on a mathematical example and an application.

Assessing the Vulnerability of Power Systems Using Multilevel Programming: A Literature Review

Vulnerability studies can identify critical elements in electric power systems in order to take protective measures against possible scenarios that may result in load shedding, which can be caused by natural events or deliberate attacks. This article is a literature review on the latter kind, i.e., the interdiction problem, which assumes there is a disruptive agent whose objective is to maximize the damage to the system, while the network operator acts as a defensive agent. The non-simultaneous interaction of these two agents creates a multilevel optimization problem, and the literature has reported several interdiction models and solution methods to address it. The main contribution of this paper is presenting the considerations that should be taken into account to analyze, model, and solve the interdiction problem, including the most common solution techniques, applied methodologies, and future studies. This literature review found that most research in this area is focused on the analysis of transmission systems considering linear approximations of the network, and a few interdiction studies use an AC model of the network or directly treat distribution networks from a multilevel standpoint. Future challenges in this field include modeling and incorporating new defense options for the network operator, such as distributed generation, demand response, and the topological reconfiguration of the system.f the system.