Optimistic Variants of Single-Objective Bilevel Optimization for Evolutionary Algorithms

Single-objective bilevel optimization is a specialized form of constraint optimization problems where one of the constraints is an optimization problem itself. These problems are typically non-convex and strongly NP-Hard. Recently, there has been an increased interest from the evolutionary computation community to model bilevel problems due to its applicability in real-world applications for decision-making problems. In this work, a partial nested evolutionary approach with a local heuristic search has been proposed to solve the benchmark problems and have outstanding results. This approach relies on the concept of intermarriage-crossover in search of feasible regions by exploiting information from the constraints. A new variant has also been proposed to the commonly used convergence approaches, i.e., optimistic and pessimistic. It is called an extreme optimistic approach. The experimental results demonstrate the algorithm converges differently to known optimum solutions with the optimistic variants. Optimistic approach also outperforms pessimistic approach. Comparative statistical analysis of our approach with other recently published partial to complete evolutionary approaches demonstrates very competitive results

An efficient hybrid differential evolutionary algorithm for zbilevel optimisation problems

Bilevel problems are widely used to describe the decision problems with hierarchical upper–lower-level structures in many economic fields. The bilevel optimisation problem (BLOP) is intrinsically NP-hard when its objectives and constraints are complex and the decision variables are large in scale at both levels. An efficient hybrid differential evolutionary algorithm for BLOP (HDEAB) is proposed where the optimal lower level value function mapping method, the differential evolutionary algorithm, k-near- est neighbours (KNN) and a nested local search are hybridised to improve the computational accuracy and efficiency. To show the performance of the HDEAB, numerical studies were conducted on SMD (Sinha, Maro and Deb) instances and an application example of optimising a venture capital staged-financing contract. The results demonstrate that the HDEAB outperforms the BLEAQ (bile- vel evolutionary algorithm based on quadratic approximations) greatly in solving the BLOPs with different scale

Development of Methods for Solving Bilevel Optimization Problems

Bilevel optimization, also referred to as bilevel programming, involves solving an upper level problem subject to the optimality of a corresponding lower level problem. The upper and lower level problems are also referred to as the leader and follower problems, respectively. Both levels have their associated objective(s), variable(s) and constraint(s). Such problems model real-life scenarios of cases where the performance of an upper level authority is realizable/sustainable only if the corresponding lower level objective is optimum. A number of practical applications in the field of engineering, logistics, economics and transportation have inherent nested structure that are suited to this type of modelling. The range of applications as well as a rapid increase in the size and complexity of such problems has prompted active interest in the design of efficient algorithms for bilevel optimization. Bilevel optimization problems present a number of unique and interesting challenges to algorithm design. The nested nature of the problem requires optimization of a lower level problem to evaluate each upper level solution, which makes it computationally exorbitant. Theoretically, an upper level solution is considered valid/feasible only if the corresponding lower level variables are the true global optimum of the lower level problem. Global optimality can be reliably asserted in very limited cases, for example convex and linear problems. In deceptive cases, an inaccurate lower level optimum may result in an objective value better than true optimum at the upper level, which poses a severe challenge for ranking/selection strategies used within any optimization technique. In turn, this also makes the performance evaluation very difficult since the performance cannot be judged based on the objective values alone. While the area of bilevel (or more generally, multilevel) programming itself is not very new, most reports in this direction up until about a decade ago considered solving linear or at most quadratic problems at both levels. Correspondingly, the focus on was on development of exact methods to solve such problems. However, such methods typically require assumptions on mathematical properties, which may not always hold in practical applications. With increasing use of computer simulation-based evaluations in a number of disciplines in science and engineering, there is more need than ever to handle problems that are highly nonlinear or even black-box in nature. Metaheuristic algorithms, such as evolutionary algorithms are more suited to this emerging paradigm. The foray of evolutionary algorithms in bilevel programming is relatively recent and there remains scope of substantial development in the field in terms of addressing the aforementioned challenges. The work presented in this thesis is directed towards improving evolutionary techniques to enable them solve generic bilevel problems more accurately using lower number of function evaluations compared to the existing methods. Three key approaches are investigated towards accomplishing this: (a) e active hybridization of global and local search methods during dierent stages of the overall search; (b) use of surrogate models to guide the search using approximations in lieu of true function evaluations, and (c) use of a non-nested re-formulation of the problem. While most of the work is focused on single-objective problems, preliminary studies are also presented on multi-objective bilevel problems. The performance of the proposed approaches is evaluated on a comprehensive suite of mathematical test problems available in the literature, as well as some practical problems. The proposed approaches are observed to achieve a favourable balance between accuracy and computational expense for solving bilevel optimization problems, and thus exhibit suitability for use in real-life applications