A parametric integer programming algorithm for bilevel mixed integer programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader's variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case it yields a ``better than fully polynomial time'' approximation scheme with running time polynomial in the logarithm of the relative precision. For the pure integer case where the leader's variables are integer, and hence optimal solutions are guaranteed to exist, we present two algorithms which run in polynomial time when the total number of variables is fixed.Comment: 11 page

A Stackelberg game theoretic model for optimizing product family architecting with supply chain consideration

Planning of an optimal product family architecture (PFA) plays a critical role in defining an organization's product platforms for product variant configuration while leveraging commonality and variety. The focus of PFA planning has been traditionally limited to the product design stage, yet with limited consideration of the downstream supply chain-related issues. Decisions of supply chain configuration have a profound impact on not only the end cost of product family fulfillment, but also how to design the architecture of module configuration within a product family. It is imperative for product family architecting to be optimized in conjunction with supply chain configuration decisions. This paper formulates joint optimization of PFA planning and supply chain configuration as a Stackelberg game. A nonlinear, mixed integer bilevel programming model is developed to deal with the leader–follower game decisions between product family architecting and supply chain configuration. The PFA decision making is represented as an upper-level optimization problem for optimal selection of the base modules and compound modules. A lower-level optimization problem copes with supply chain decisions in accordance with the upper-level decisions of product variant configuration. Consistent with the bilevel optimization model, a nested genetic algorithm is developed to derive near optimal solutions for PFA and the corresponding supply chain network. A case study of joint PFA and supply chain decisions for power transformers is reported to demonstrate the feasibility and potential of the proposed Stackelberg game theoretic joint optimization of PFA and supply chain decisions

Co-evolutionary Hybrid Bi-level Optimization

Multi-level optimization stems from the need to tackle complex problems involving multiple decision makers. Two-level optimization, referred as ``Bi-level optimization'', occurs when two decision makers only control part of the decision variables but impact each other (e.g., objective value, feasibility). Bi-level problems are sequential by nature and can be represented as nested optimization problems in which one problem (the ``upper-level'') is constrained by another one (the ``lower-level''). The nested structure is a real obstacle that can be highly time consuming when the lower-level is $\mathcal{NP}-hard$. Consequently, classical nested optimization should be avoided. Some surrogate-based approaches have been proposed to approximate the lower-level objective value function (or variables) to reduce the number of times the lower-level is globally optimized. Unfortunately, such a methodology is not applicable for large-scale and combinatorial bi-level problems. After a deep study of theoretical properties and a survey of the existing applications being bi-level by nature, problems which can benefit from a bi-level reformulation are investigated. A first contribution of this work has been to propose a novel bi-level clustering approach. Extending the well-know ``uncapacitated k-median problem'', it has been shown that clustering can be easily modeled as a two-level optimization problem using decomposition techniques. The resulting two-level problem is then turned into a bi-level problem offering the possibility to combine distance metrics in a hierarchical manner. The novel bi-level clustering problem has a very interesting property that enable us to tackle it with classical nested approaches. Indeed, its lower-level problem can be solved in polynomial time. In cooperation with the Luxembourg Centre for Systems Biomedicine (LCSB), this new clustering model has been applied on real datasets such as disease maps (e.g. Parkinson, Alzheimer). Using a novel hybrid and parallel genetic algorithm as optimization approach, the results obtained after a campaign of experiments have the ability to produce new knowledge compared to classical clustering techniques combining distance metrics in a classical manner. The previous bi-level clustering model has the advantage that the lower-level can be solved in polynomial time although the global problem is by definition $\mathcal{NP}$-hard. Therefore, next investigations have been undertaken to tackle more general bi-level problems in which the lower-level problem does not present any specific advantageous properties. Since the lower-level problem can be very expensive to solve, the focus has been turned to surrogate-based approaches and hyper-parameter optimization techniques with the aim of approximating the lower-level problem and reduce the number of global lower-level optimizations. Adapting the well-know bayesian optimization algorithm to solve general bi-level problems, the expensive lower-level optimizations have been dramatically reduced while obtaining very accurate solutions. The resulting solutions and the number of spared lower-level optimizations have been compared to the bi-level evolutionary algorithm based on quadratic approximations (BLEAQ) results after a campaign of experiments on official bi-level benchmarks. Although both approaches are very accurate, the bi-level bayesian version required less lower-level objective function calls. Surrogate-based approaches are restricted to small-scale and continuous bi-level problems although many real applications are combinatorial by nature. As for continuous problems, a study has been performed to apply some machine learning strategies. Instead of approximating the lower-level solution value, new approximation algorithms for the discrete/combinatorial case have been designed. Using the principle employed in GP hyper-heuristics, heuristics are trained in order to tackle efficiently the $\mathcal{NP}-hard$ lower-level of bi-level problems. This automatic generation of heuristics permits to break the nested structure into two separated phases: \emph{training lower-level heuristics} and \emph{solving the upper-level problem with the new heuristics}. At this occasion, a second modeling contribution has been introduced through a novel large-scale and mixed-integer bi-level problem dealing with pricing in the cloud, i.e., the Bi-level Cloud Pricing Optimization Problem (BCPOP). After a series of experiments that consisted in training heuristics on various lower-level instances of the BCPOP and using them to tackle the bi-level problem itself, the obtained results are compared to the ``cooperative coevolutionary algorithm for bi-level optimization'' (COBRA). Although training heuristics enables to \emph{break the nested structure}, a two phase optimization is still required. Therefore, the emphasis has been put on training heuristics while optimizing the upper-level problem using competitive co-evolution. Instead of adopting the classical decomposition scheme as done by COBRA which suffers from the strong epistatic links between lower-level and upper-level variables, co-evolving the solution and the mean to get to it can cope with these epistatic link issues. The ``CARBON'' algorithm developed in this thesis is a competitive and hybrid co-evolutionary algorithm designed for this purpose. In order to validate the potential of CARBON, numerical experiments have been designed and results have been compared to state-of-the-art algorithms. These results demonstrate that ``CARBON'' makes possible to address nested optimization efficiently

Comparing Machine Learning and Optimization Approaches for the N − k Interdiction Problem Considering Load Variability

Power grids must be operated, protected, and maintained such that a small number of line failures will not result in significant load shedding. To identify problematic combinations of failures, we consider an N-k interdiction problem that seeks the set of k failed lines (out of N total lines) that result in the largest load shed. This is naturally formulated as a bilevel optimization problem with an upper level representing the attacker that selects line failures and a lower level modeling the defender's generator redispatch to minimize the load shedding. Compounding the difficulties inherent to the bilevel nature of interdiction problems, we consider a nonlinear AC power flow model that makes this problem intractable with traditional solution approaches. Furthermore, since the solutions found at a particular load condition may not generalize to other loading conditions, operators may need to quickly recompute these worst-case failures online to protect against them during operations. To address these challenges, we formulate and compare the performance of three simplified methods for solving the N-k interdiction problem: a state-of-the-art optimization approach based on a network-flow relaxation of the power flow equations and two newly developed machine learning algorithms that predict load sheds given the state of the network

Multi-objective constrained optimization for energy applications via tree ensembles

Energy systems optimization problems are complex due to strongly non-linear system behavior and multiple competing objectives, e.g. economic gain vs. environmental impact. Moreover, a large number of input variables and different variable types, e.g. continuous and categorical, are challenges commonly present in real-world applications. In some cases, proposed optimal solutions need to obey explicit input constraints related to physical properties or safety-critical operating conditions. This paper proposes a novel data-driven strategy using tree ensembles for constrained multi-objective optimization of black-box problems with heterogeneous variable spaces for which underlying system dynamics are either too complex to model or unknown. In an extensive case study comprised of synthetic benchmarks and relevant energy applications we demonstrate the competitive performance and sampling efficiency of the proposed algorithm compared to other state-of-the-art tools, making it a useful all-in-one solution for real-world applications with limited evaluation budgets

A new differential evolution using a bilevel optimization model for solving generalized multi-point dynamic aggregation problems

The multi-point dynamic aggregation problem (MPDAP) comes mainly from real-world applications, which is characterized by dynamic task assignation and routing optimization with limited resources. Due to the dynamic allocation of tasks, more than one optimization objective, limited resources, and other factors involved, the computational complexity of both route programming and resource allocation optimization is a growing problem. In this manuscript, a task scheduling problem of fire-fighting robots is investigated and solved, and serves as a representative multi-point dynamic aggregation problem. First, in terms of two optimized objectives, the cost and completion time, a new bilevel programming model is presented, in which the task cost is taken as the leader's objective. In addition, in order to effectively solve the bilevel model, a differential evolution is developed based on a new matrix coding scheme. Moreover, some percentage of high-quality solutions are applied in mutation and selection operations, which helps to generate potentially better solutions and keep them into the next generation of population. Finally, the experimental results show that the proposed algorithm is feasible and effective in dealing with the multi-point dynamic aggregation problem

Bilevel facility location problems: theory and applications.

In this doctoral thesis we focus on studying facility location problems considering customer preferences. In these problems, there is a set of customers or users who demand a service or product that must be supplied by one or more facilities. By facilities it is understood some object or structure that offers some service to customers. One of the most important assumptions is that customers have established their own preferences over the facilities and should be taken into account in the customer-facility assignment. In real life, customers choose facilities based on costs, preferences, a predetermined contract, or a loyalty coefficient, among others. That is, they are free to choose the facilities that will serve them. The situation described above is commonly modeled by bilevel programming, where the upper level corresponds to location decisions to optimize a predefined criteria, such as, minimize location and distribution costs or maximize the demand covered by the facilities; and the lower level is associated to -customer allocation- to optimize customer preferences. The hierarchy among both levels is justified because the decision taken in the upper level directly affects the decision’s space in the lower level

Evolutionary Algorithms in Engineering Design Optimization

Evolutionary algorithms (EAs) are population-based global optimizers, which, due to their characteristics, have allowed us to solve, in a straightforward way, many real world optimization problems in the last three decades, particularly in engineering fields. Their main advantages are the following: they do not require any requisite to the objective/fitness evaluation function (continuity, derivability, convexity, etc.); they are not limited by the appearance of discrete and/or mixed variables or by the requirement of uncertainty quantification in the search. Moreover, they can deal with more than one objective function simultaneously through the use of evolutionary multi-objective optimization algorithms. This set of advantages, and the continuously increased computing capability of modern computers, has enhanced their application in research and industry. From the application point of view, in this Special Issue, all engineering fields are welcomed, such as aerospace and aeronautical, biomedical, civil, chemical and materials science, electronic and telecommunications, energy and electrical, manufacturing, logistics and transportation, mechanical, naval architecture, reliability, robotics, structural, etc. Within the EA field, the integration of innovative and improvement aspects in the algorithms for solving real world engineering design problems, in the abovementioned application fields, are welcomed and encouraged, such as the following: parallel EAs, surrogate modelling, hybridization with other optimization techniques, multi-objective and many-objective optimization, etc