Levenberg–Marquardt method and partial exact penalty parameter selection in bilevel optimization

Bilevel optimization problems require every feasible upper-level solution to satisfy optimality of a lower-level optimization problem. These problems commonly appear in many practical problem solving tasks including optimal control, process optimization, game-playing strategy development, transportation problems, and others. In the context of a bilevel single objective problem, there exists a number of theoretical, numerical, and evolutionary optimization results. However, there does not exist too many studies in the context of having multiple objectives in each level of a bilevel optimization problem. In this paper, we address bilevel multi-objective optimization issues and propose a viable algorithm based on evolutionary multi-objective optimization (EMO) principles. Proof-of-principle simulation results bring out the challenges in solving such problems and demonstrate the viability of the proposed EMO technique for solving such problems. This paper scratches the surface of EMO-based solution methodologies for bilevel multi-objective optimization problems and should motivate other EMO researchers to engage more into this important optimization task of practical importance.

Combinatorial bi-level optimization remains a challenging topic, especially when the lower-level is an NP-hard problem. In this paper, we tackle large-scale and combinatorial bi-level problems using GP hyper-heuristics, i.e., an approach that permits to train heuristics like a machine learning model. Our contribution aims at targeting the intensive and complex lower-level optimizations that occur when solving a large-scale and combinatorial bi-level problem. For this purpose, we consider hyper-heuristics through heuristic generation. Using a GP hyper-heuristic approach, we train greedy heuristics in order to make them more reliable when encountering unseen lower-level instances that could be generated during bi-level optimization. To validate our approach referred to as GA+AGH, we tackle instances from the bi-level cloud pricing optimization problem (BCPOP) that model the trading interactions between a cloud service provider and cloud service customers. Numerical results demonstrate the abilities of the trained heuristics to cope with the inherent nested structure that makes bi-level optimization problems so hard. Furthermore, it has been shown that training heuristics for lower-level optimization permits to outperform human-based heuristics and metaheuristics which constitute an excellent outcome for bi-level optimization.

A distributionally robust optimization model for batch nonlinear switched time-delay system considering uncertain output measurements

Bi-level optimisation problems (BOPs) are a class of challenging problems with two levels of optimisation tasks. These problems allow to model a large number of real-life situations in which a first decision maker, hereafter the leader, optimises his objective by taking the follower's response to his decisions explicitly into account. In this context, a new proposed algorithm called CODBA-II was suggested to solve combinatorial BOPs. The latter was able to improve the quality of generated bi-level solutions regarding to recently proposed methods. In fact, a wide range of applications fit the bi-level programming framework and real-life implementations still scarce. For this reason, we propose in this paper a co-evolutionary decomposition-based bi-level algorithm for the bi-level knapsack optimisation problem. The computational algorithm turned out to be quite efficient on both computation time and solution quality regarding to other competitive EAs.

Uncertainties in variables and parameters cause optimization problems to move away from globally-optimal and uncertain solutions. Practitioners resort to finding robust and reliable solutions in such situations. Bilevel optimization problems involving a hierarchy of two nested optimization problems have received a growing attention in the recent past due to their relevance in practice. While a number of studies on bilevel solution methodologies and applications are available for a deterministic setup, but studies on uncertainties in bilevel optimization are rare. In this paper, we suggest methodologies for handling uncertainty in both lower and upper level variables that may occur from different practicalities. For the first time, we perform a systematic study demonstrating the effect of uncertainties in each level along with the definition of robustness and reliability in the context of bilevel optimization. The issues and complexities introduced due to such uncertainties are then studied through a number of test cases, for brevity, we only show results on three test cases. Finally, two real-world bilevel problems involving uncertainties in their variables are solved. The study provides foundations and demon- strates viable directions for further research in uncertainty-based bilevel optimization problems.

In this paper, we consider the problem of scheduling N robots interacting with a moving target. Both, the sequence of the robots and their trajectories are unknown and subject to optimization. Such type of problems appear in highly automated production plants and in the simulation of virtual factories. The purpose of the paper is to provide a mathematical model and to suggest a numerical solution approach. Our approach is based on the formulation of the problem as a bilevel optimization problem, where the lower level problem is an optimal control problem, while the upper level problem is a finite dimensional mixed-integer optimization problem. We approach the problem by exploitation of necessary optimality conditions for the lower level problem and by application of a Branch & Bound method for the resulting single level optimization problem. Two settings are taken into account. Firstly, no state constraints are assumed on the lower level problem, thus the local minimum principle applies directly. Secondly, the problem setting is augmented by pure state constraints, which are being handled by virtual controls in order to regularize the problem.

MotivationWithin Flux Balance Analysis, the investigation of complex subtasks, such as finding the optimal perturbation of the network or finding an optimal combination of drugs, often requires to set up a bilevel optimization problem. In order to keep the linearity and convexity of these nested optimization problems, an ON/OFF description of the effect of the perturbation (i.e. Boolean variable) is normally used. This restriction may not be realistic when one wants, for instance, to describe the partial inhibition of a reaction induced by a drug.ResultsIn this paper we present a formulation of the bilevel optimization which overcomes the oversimplified ON/OFF modeling while preserving the linear nature of the problem. A case study is considered: the search of the best multi-drug treatment which modulates an objective reaction and has the minimal perturbation on the whole network. The drug inhibition is described and modulated through a convex combination of a fixed number of Boolean variables. The results obtained from the application of the algorithm to the core metabolism of E.coli highlight the possibility of finding a broader spectrum of drug combinations compared to a simple ON/OFF modeling.ConclusionsThe method we have presented is capable of treating partial inhibition inside a bilevel optimization, without loosing the linearity property, and with reasonable computational performances also on large metabolic networks. The more fine-graded representation of the perturbation allows to enlarge the repertoire of synergistic combination of drugs for tasks such as selective perturbation of cellular metabolism. This may encourage the use of the approach also for other cases in which a more realistic modeling is required.

Dynamic traffic assignment (DTA) applications require traffic signal control data that are typically difficult to obtain and cumbersome to code in the required format. In addition, the evaluation of future scenarios requires future traffic signal settings consistent with the forecast demand. These signal settings are not known a priori and are costly to estimate. Intuitively, future signal timings need to be reasonably optimized so as to represent what the traffic management agency will do. In the literature, integration between traffic control and DTA models has been formulated as a bi-level or single-level optimization problem with system or user optimal constraints. Most existing solution procedures require certain nested structure with an inner-loop algorithm solving the problem of user-equilibrium or system-optimal assignment and the outer-loop algorithm searching for the optimal signal-timing settings. Most of these solution approaches remain only research tools without practical use because of computational intractability. This research proposes an efficient solution algorithm to the problem. An adaptive traffic signal control model is embedded in a simulation-based DTA model. For each inbound approach at an intersection of interest, the adaptive model uses upstream information and a dynamic rolling-horizon approach to project traffic flow conditions for a dynamic but short (projection) period. The adaptive model provides the signal settings during the entire process of traffic flow simulation and for every iteration of the solution algorithm. Thus, during the entire solution process, the experienced travel times and resulting traffic assignment flows are based on the adaptive (demand-responsive) signal settings, allowing the DTA flows and the adaptive signal settings to be generated simultaneously in a single-loop algorithmic structure. Simulation experiments illustrate the capabilities of the proposed approach.

Bilevel and multi-objective optimization methods are often useful to spatially target agri-environmental policy throughout a watershed. This type of problem is complex and is comprised of a number of practicalities: (i) a large number of decision variables, (ii) at least two inter-dependent levels of optimization between policy makers and policy followers, and (iii) uncertainty in decision variables and problem parameters. Given agricultural and economic data from the Raccoon watershed in central Iowa, we formulate a bilevel multi-objective optimization problem that accommodates objectives of both policy makers and farmers. The solution procedure then explicitly accounts for the nested nature of farm-level management decisions in response to agri-environmental policy incentives constructed by policy makers. We specifically examine the spatial targeting of a fertilizer-reduction incentive policy while seeking to maximize farm-level productivity while generating mandated water quality improvements using this framework. We test three different evolutionary optimization algorithms - m-BLEAQ, NSGA-II, and SPEA2 - and show that m-BLEAQ is well suited for handling the bilevel optimization problems and the considered practicalities.

A solution to bi/tri-level programming problems using particle swarm optimization

The bilevel optimisation problem (BLP) is a subclass of optimisation problems in which one of the constraints of an optimisation problem is another optimisation problem. BLP is widely used to model hierarchical decision making where the leader and the follower correspond to the upper level and lower level optimisation problem, respectively. In BLP, the optimal solutions to the lower level optimisation problem are the feasible solutions to the upper level problem, which makes it particularly difficult to solve. This paper proposes a novel hardware architecture known as Recursive Pipelined Genetic Propagation (RPGP), to solve BLP efficiently on FPGA. RPGP features a graph of genetic operation nodes which can be scaled to exploit hardware resources. In addition, the topology of the RPGP graph can be changed at run-time to escape from local optima. We evaluate the proposed architecture on an Altera Stratix-V FPGA, using a benchmark bilevel optimisation problem set. Our experimental results show that RPGP can achieve a significant speed-up against previous work.

Digital Twin (DT) and mobile edge computing are two promising solutions for achieving latency-sensitive and computing-intensive applications in Industrial Internet of Things (IIoT). However, existing task offloading schemes with DT empowerment are faced with challenges, such as the spatio-temporal heterogeneity of edge server (ES) resources, resource-constrained ESs, and the explosive growth of data in emerging applications. This paper investigates the issues of task offloading and resource allocation under the assistance of DT and multiple ESs parallel collaboration. One novel scheme, abbreviated as Mes-PCORA, is proposed. With comprehensive information within the digital space, the Mes-PCORA scheme dynamically adjusts task allocation ratios across multiple ESs to achieve collaborative task offloading. The offloading request is formulated as a non-convex problem. To make the non-convex problem solvable in polynomial time, the original problem transformed into a bilevel optimization problem. Then, a bilevel iterative optimization approach is proposed. Specifically, the upper-level optimization problem is formulated as a multi-agent Markov Decision Process, and a deep reinforcement learning-based resource allocation algorithm is designed to solve it. Subsequently, for the lower-level optimization problem, it is solved by the interior point method. Numerical results demonstrate that the proposed scheme reduces the average task completion latency by 27.16%–63.44% and decreases the task offloading failure rate by 35.83%–73.95%, compared to state-of-the-art baselines.

Electronic countermeasure (ECM) has become one of the most significant factors in modern warfare, in the course of combat, the electronic jamming allocation tasks need to be flexibly adjusted with the change of combat stage, which puts forward higher requirements for the modeling and solution method of this kind of problems. To solve the ECM dynamic weapon target assignment (ECM-DWTA) problem, a hybrid multi-target bi-level programming model is established. The upper level takes the sum of the electronic jamming effects in the whole combat stage as an optimization objective, and locally optimizes the ECM weapon (ECM-WP) assignment scheme in each stage. The lower level takes the importance expectation value of the target subjected to interference and combat consumption as double optimization objectives to globally optimize the ECM-WP assignment scheme. Focus on solving this complex model, a hybrid multi-objective bi-level interactive fuzzy programming algorithm (HMOBIF) is proposed, in this method, exponential membership function is used to describe the satisfaction degree of each level. When solving the multi-objective optimization problem composed of membership functions in the upper and lower levels, we use the MOEA/D algorithm to obtain the Pareto Front (PF) solution set, and then each solution in PF is evaluated and selected by the TOPSIS multi-criteria evaluation method. This local and global interactive optimization process of bi-level model is actually the process of executing observation-orientation-decision-action loop in practical combat. According to the current example, we conduct numerical simulation on the parameters in the model and obtain the parameter values suitable for the model solution. The computational experiments on different scale ECM-DWTA problems show that HMOBIF method is superior to four bi-level programming algorithms in terms of performance index, and can better solve ECM-DWTA problems.

In this paper, we propose a set of six test problems for single-objective bilevel optimization. The test-collection represents various difficulties which are commonly encountered in practical bilevel optimization problems. To support experiments with problems of different size, all of the test problems are scalable in terms of the number of variables. The problem set is also accompanied by a construction procedure, which helps to generate new test problems with controlled difficulties in convergence and interaction patterns between the two optimization levels. To provide a baseline result for easy comparisons, we have solved a 10 variable instance for each of the test problems using a simple bilevel evolutionary algorithm. The results presented may be used as a benchmark while evaluating the performance of any bilevel optimization algorithm.

Multilevel optimization problems deals with mathematical programming problems whose feasible set is implicitly determined by a sequence of nested optimization problems. These kind of problems are common in different applications where there is a hierarchy of decision makers exists. Solving such problems has been a challenge especially when they are non linear and non convex. In this paper we introduce a new algorithm, inspired by natural adaptation, using (1+1)-evolutionary strategy iteratively. Suppose there are k level optimization problem. First, the leader’s level will be solved alone for all the variables under all the constraint set. Then that solution will adapt itself according to the objective function in each level going through all the levels down. When a particular level’s optimization problem is solved the solution will be adapted the level’s variable while the other variables remain being a fixed parameter. This updating process of the solution continues until a stopping criterion is met. Bilevel and trilevel optimization problems are used to show how the algorithm works. From the simulation result on the two problems, it is shown that it is promising to uses the proposed metaheuristic algorithm in solving multilevel optimization problems.KeywordsMultilevel optimization(1+1)-Evolutionary strategymetaheuristic algorithmsNatural adaptation