Sensitivity Analysis for Two-Level Value Functions with Applications to Bilevel Programming

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.

This paper addresses a bilevel optimization problem with a guaranteed (pessimistic) solution in a rather general form. The problem is reduced to a family of bilevel problems with optimistic solutions and perturbed follower's problems. For the quadratic-linear bilevel optimization problem, a reduction of the bilevel problem to a series of non-convex one-level optimization problems is proposed using the penalty method. Additionally, a solution method based on Global Search Theory (GST) is developed for the obtained problem. The GST consists of two stages: special local search and Global Search Scheme (GSS), which is based on Global Optimality Conditions (GOCs). The computational testing of the developed approach on specially constructed bilevel problems of high dimensions was successfully carried out. Finally, the computational testing on specially constructed bilevel problems of rather high dimensions was successfully carried out, that substantiate the developed approach.

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.

In this article, we consider bilevel optimization problems with discrete lower level and continuous upper level problems. Taking into account both approaches (optimistic and pessimistic) which have been developed in the literature to deal with this type of problem, we derive some conditions for the existence of solutions. In the case where the lower level is a parametric linear problem, the bilevel problem is transformed into a continuous one. After that, we are able to discuss local optimality conditions using tools of variational analysis for each of the different approaches. Finally, we consider a simple application of our results namely the bilevel programming problem with the minimum spanning tree problem in the lower level.

Presolving linear bilevel optimization problems

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.

This paper presents an optimal bidding strategy for a strategic wind power producer (WPP) in a distribution-level energy market (DLEM). The behavior of the WPP is modelled through a bi-level stochastic optimization problem where the upper-level problem maximizes the profit of the WPP and the lower-level problem describes the clearing processes of the DLEM while considering network constraints. The bi-level problem is a stochastic mathematical program with equilibrium constraints (MPEC) that is formulated as a mixed-integer linear programming (MILP) problem. The main focus of this study is investigating prosumers’ impact on the market power of the strategic WPP in a DLEM structure. In this model, the effect of flexible prosumers from the aspects of demand response (DR) participants and photovoltaic penetration level (PVPL) on the WPP’s offering strategy is investigated. Moreover, the impact of bilateral contract on the market power of the strategic WPP and the cleared prices of the network is addressed. The proposed model is implemented in an IEEE 33-bus and numerical results illustrate how behavior of flexible prosumers and PVPL index affect the decision making of the strategic WPP when network constraints are considered. Numerical results show that by active participation of prosumers in DR programs, the reliance of DLEM on the strategic WPP reduces. Moreover, if the WPP participates in bilateral contracts, its offering to the DELM decreases, and as the result, the cleared prices augment indicating market power of the WPP.

A hierarchy model to use local resources by DSO and TSO in the balancing market

Multilevel optimization problems often arise in various applications (in economics, ecology, power engineering and other areas) when modeling complex systems with a hierarchical structure associated with independent actions of subsystems. The difficulty of analyzing such complex systems requires first of all the study of bilevel models, the management of which would be an integral part of the analysis of more complex systems. In solving bilevel programming problems, an important role is played by the property of partial calmness, the presence of which allows us to reduce the bilevel problem to the classical nonlinear programming problem with a nonsmooth objective function. It is known that linear bilevel programming problems are partially stable. The proof of this property for more complex problems meets difficulties. In particular, our article shows the inaccuracy of some results in this area. The goal of the paper is to obtain some new results in the partial calmness of bilevel programming. In particular, new sufficient conditions for bilevel problems are proved. The results are obtained on the base of Lipschitz-like properties for multivalued mappings. In the paper we propose new sufficient conditions for partial calmness which are based on some modification of the known constraint qualification RCPLD which have been proposed by the researches Andreani, Haeser, Schuverdt and Silva.

The Great Britain (GB) government is paving the path to decarbonisation by actively promoting the integration of wind power into its generation mix. This sharp transition to renewable energy, however, introduces specific challenges. The characteristics of non-synchronous wind turbines have the potential to impact grid stability and frequency security due to the reduction in system inertia. In response to these challenges, the deployment of virtual power plants (VPPs) is envisioned in deregulated power systems, which allow for the aggregation and coordinated control of diverse distributed energy resources, enhancing grid flexibility, reliability, and efficiency. Moreover, VPPs offer the provision of system inertia and frequency response services at the national level. This study drops this assumption and leverages the localised flexibility inherent in VPPs to meet the ancillary service requirements of the low-carbon power system. The formulated problem is constructed as a non-convex bi-level optimisation problem, where the upper-level problem represents the operation of a frequency-constrained unit commitment model, and the lower-level problem embodies the energy dispatches and frequency responses of a group of VPPs, guided by the dual price signals of cleared energy and ancillary services. To address the non-convex nature of the unit commitment problem, a two-fold approach is employed. First, binary commitment status decision variables are relaxed to continuous versions. Subsequently, the duality gap between the original non-convex unit commitment problem and its relaxed dual form is minimised, yielding a solution closely approximating the optimal solution of the original problem. Second, the relaxed bi-level optimisation problem is transformed into a single-level mathematical programs with equilibrium constraints by replacing the lower-level VPP problems with their equivalent Karush-Kuhn–Tucker optimality conditions. The case studies conducted in this work encompass a comprehensive scope. Initially, the study evaluates the effectiveness of VPPs in delivering ancillary services within the projected context of the GB power system for 2030. The findings reveal a substantial 28.28% decrease in system operation costs when compared to the benchmark case lacking VPP flexibility. Additionally, the integration of VPPs leads to significantly lower price signals for inertia, primary frequency response, and enhanced frequency response, reduced by 81.21%, 79.13%, and 86.27%, respectively, in contrast to the benchmark case. The analysis then delves into a sensitivity exploration, investigating the inherent flexibility of VPPs. The profit of local VPPs is expected to increase with VPP flexibility. Finally, the study illustrates how VPPs can adapt to and derive benefits from an increasing level of wind power penetration by providing a higher amount of frequency response.

Linear bilevel programs with multiple objectives at the upper level

A Novel Matheuristic based on bi-level optimization for the multi-Objective design of hydrogen supply chains

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.

Bilevel problems are highly challenging optimization problems that appear in many applications of energy market design, critical infrastructure defense, transportation, pricing, and so on. Often these bilevel models are equipped with integer decisions, which makes the problems even harder to solve. Typically, in such a setting in mathematical optimization, one develops primal heuristics in order to obtain feasible points of good quality quickly or to enhance the search process of exact global methods. However, there are comparably few heuristics for bilevel problems. In this paper, we develop such a primal heuristic for bilevel problems with a mixed-integer linear or quadratic upper level and a linear or quadratic lower level. The heuristic is based on a penalty alternating direction method, which allows for a theoretical analysis. We derive a convergence theory stating that the method converges to a stationary point of an equivalent single-level reformulation of the bilevel problem and extensively test the method on a test set of more than 2,800 instances—which is one of the largest computational test sets ever used in bilevel programming. The study illustrates the very good performance of the proposed method in terms of both running times and solution quality. This renders the method a suitable subroutine in global bilevel solvers as well as a reasonable standalone approach.Summary of Contribution: Bilevel optimization problems form a very important class of optimization problems in the field of operations research, which is mainly due to their capability of modeling hierarchical decision processes. However, real-world bilevel problems are usually very hard to solve—especially in the case in which additional mixed-integer aspects are included in the modeling. Hence, the development of fast and reliable primal heuristics for this class of problems is very important. This paper presents such a method.

Bilevel optimization problems are challenging to solvedue to thecomplex interplay between upper-level and lower-level decision variables.Classical solution methods generally simplify the bilevel problemto a single level problem, whereas more recent methods such as evolutionaryalgorithms and Bayesian optimization take a black-box view that cansuffer from scalability to larger problems. While advantageous forhandling high-dimensional and nonconvex optimization problems, theapplication of gradient-based solution methods to bilevel problemsis impeded by the implicit relationship between the upper-level andlower-level decision variables. Additionally, lack of an equation-orientedrelationship between decision variables and the upper-level objectivecan further impede differentiability. To this end, we present a gradient-basedoptimization framework that leverages implicit function theorem andmodel-free reinforcement learning (RL) to solve bilevel optimizationproblems wherein only zeroth-order observations of the upper-levelobjective are available. Implicit differentiation allows for differentiatingthe optimality conditions of the lower-level problem to enable calculationof gradients of the upper-level objective. Using policy gradient RL,gradient-based updates of the upper-level decisions can then be performedin a scalable manner for high-dimension problems. The proposed frameworkis applied to the bilevel problem of learning optimization-based controlpolicies for uncertain systems. Simulation results on two benchmarkproblems illustrate the effectiveness of the framework for goal-orientedlearning of model predictive control policies. Synergizing derivative-freeoptimization via model-free RL and gradient calculation via implicitfunction differentiation can create new avenues for scalable and efficientsolution of bilevel problems with black-box upper-level objectiveas compared to black-box optimization methods that discard the problemstructure.