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Soft open points (SOPs) are power electronic devices that replace the normal open points in active distribution systems. They provide resiliency in terms of transferring electrical power between adjacent feeders and delivering the benefits of meshed networks. In this work, a multi-objective bilevel optimization problem is formulated to maximize the hosting capacity (HC) of a real 59-node distribution system in Egypt and an 83-node distribution system in Taiwan, using distribution system reconfiguration (DSR) and SOP placement. Furthermore, the uncertainty in the load is considered to step on the real benefits of allocating SOPs along with DSR. The obtained results validate the effectiveness of DSR and SOP allocation in maximizing the HC of the studied distribution systems with low cost.

In a multi-objective optimization problem, a decision maker has more than one objective to optimize. In a bilevel optimization problem, there are the following two decision-makers in a hierarchy: a leader who makes the first decision and a follower who reacts, each aiming to optimize their own objective. Many real-world decision-making processes have various objectives to optimize at the same time while considering how the decision-makers affect each other. When both features are combined, we have a multi-objective bilevel optimization problem, which arises in manufacturing, logistics, environmental economics, defence applications and many other areas. Many exact and approximation-based techniques have been proposed, but because of the intrinsic nonconvexity and conflicting multiple objectives, their computational cost is high. We propose a hybrid algorithm based on batch Bayesian optimization to approximate the upper-level Pareto-optimal solution set. We also extend our approach to handle uncertainty in the leader’s objectives via a hypervolume improvement-based acquisition function. Experiments show that our algorithm is more efficient than other current methods while successfully approximating Pareto-fronts.

Nowadays, the bilevel multi-objective optimization problem has been received extensive attention from scholars. As two mainstream algorithms to solve it, intelligent algorithm and interactive algorithm based on satisfaction have disadvantages such as slow operation speed and too much subjectivity in weight selection, respectively. This paper proposes a multi-attribute decision-making method based on fuzzy weight, which is a natural generalization of deterministic weight, and it can cope with the decision-making risks brought by uncertain environment, which has strong robustness. Meanwhile the new deviation function index proposed could reflect that the upper and lower decision-makers execute the final decision with the same enthusiasm, which is convenient to make exploratory decisions in unknown environment. Finally, compared with the traditional existing methods, the effectiveness of the proposed method and indicator is verified by the garbage classification game in the main urban district of Chongqing.

Renewable energy integration has been recently promoted by many countries as a cleaner alternative to fossil fuels. In many research works, the optimal allocation of distributed generations (DGs) has been modeled mathematically as a DG injecting power without considering its intermittent nature. In this work, a novel probabilistic bilevel multi-objective nonlinear programming optimization problem is formulated to maximize the penetration of renewable distributed generations via distribution network reconfiguration while ensuring the thermal line and voltage limits. Moreover, solar, wind, and load uncertainties are considered in this paper to provide a more realistic mathematical programming model for the optimization problem under study. Case studies are conducted on the 16-, 59-, 69-, 83-, 415-, and 880-node distribution networks, where the 59- and 83-node distribution networks are real distribution networks in Cairo and Taiwan, respectively. The obtained results validate the effectiveness of the proposed optimization approach in maximizing the hosting capacity of DGs and power loss reduction by greater than 17% and 74%, respectively, for the studied distribution networks.

Bilevel programming is characterized by the existence of two optimization problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimization problem. This hierarchical design of optimization is suitable to model a large number of real-life applications. However, when dealing with a non linear multi-objective optimization context, new complexities arise due to conflicting objectives. In this paper, an exact method is described to solve an integer indefinite quadratic bilevel maximization problem with multiple objectives at the upper level, where the objective functions at both levels are the product of two linear functions. The algorithm suggested aims to produce a set of efficient solutions by employing a branch and cut approach. It optimizes the indefinite quadratic problem of the upper level within the feasible region of the original problem in an iterative manner. Then, it introduces the Dantzig cut technique to identify the optimal solution for the integer indefinite quadratic bilevel programming problem. Additionally, the algorithm utilizes an efficient cut that reduces the search process for obtaining the set of efficient solutions of the main problem, along with a branching constraint for the integer decision variables. The algorithm was implemented and tested on instances generated randomly, yielding positive outcomes.

The paper deals with the optimal parameter tuning for the elastic net problem. This process is formulated as an optimization problem over a Pareto set. The Pareto set is associated with a convex multi-objective optimization problem, and, based on the scalarization theorem, we give a parametrical representation of it. Thus, the problem becomes a bilevel optimization with a unique response of the follower (strong Stackelberg game). Then, we apply this strategy to the parameter tuning for the elastic net problem. We propose a new algorithm called Ensalg to compute the optimal regularization path of the elastic net w.r.t. the sparsity-inducing term in the objective. In contrast to existing algorithms, our method can also deal with the so-called “many-at-a-time” case, where more than one variable becomes zero at the same time and/or changes from zero. In examples involving real-world data, we demonstrate the effectiveness of the algorithm.

A bilevel programming is a two-level optimization problem, namely, the upper level (leaders) and the lower level (followers). The two level’s decision variables are entwined with each other which increases the complexity to obtain the global solution for both the optimization problems. Each level aims to optimize their own objective function under the given constraints at both the levels. To reduce the complexity partial cooperation between the two levels has been exploited in obtaining the Pareto solution. A novel solution procedure is proposed for a multi-objective fuzzy stochastic bilevel programming (MOFSBLP) problem is studied and solved using genetic algorithm. In this paper, previous information of the lower level is used as a fuzzy stochastic constraints in the upper level along with its constraints. Then with the solution of the combine constraints, the lower level solution is evaluated. The proposed solution procedure is illustrated by a numerical example taken from Zheng et al., and results are compared. A simpler version is solved using GAMs software to analyze the result of the numerical example. The proposed method highlights the importance of partial cooperation in solving bilevel programming problem. The advantage of the proposed solution method is that it creates common constraint space which helps in convergence of the algorithm.

In practical problems, which can be stated in the form of multi objective programming problem, we have usually a large set of efficient solutions. So, which solution from this set should be taken, appears as a natural question here. In the paper we propose some sustainable principles and simple numerical method for such choice. The method respects aspirations and priorities of decision makers and enables iterations for possible improvement of the solution. In this way decision makers can clearly understand why a particular solution is obtained and when and how it can be improved. The features of the method are explained by a practical example. An applications to bilevel programming problems are also presented, where the both side mapping, between the set of efficient solutions and the set of all possible priorities, is shown. It is illustrated in detail through several linear and nonlinear examples.

Classic bilevel programming deals with two level hierarchical optimization problems in which the leader attempts to optimize his/her objective, subject to a set of constraints and his/her follower’s solution. In modelling a real-world bilevel decision problem, some uncertain coefficients often appear in the objective functions and/or constraints of the leader and/or the follower. Also, the leader and the follower may have multiple conflicting objectives that should be optimized simultaneously. Furthermore, multiple followers may be involved in a decision problem and work cooperatively according to each of the possible decisions made by the leader, but with different objectives and/or constraints. Following our previous work, this study proposes a set of models to describe such fuzzy multi-objective, multi-follower (cooperative) bilevel programming problems. We then develop an approximation K th-best algorithm to solve the problems.

In this paper, we address a class of semivectorial bilevel programming problem in which the upper level is a scalar optimization problem and the lower level is a linear multi-objective optimization problem. Then, we present a new penalty function method, which includes two different penalty parameters, for solving such a problem. Furthermore, we give a simple algorithm. Numerical examples show that the proposed algorithm is feasible.