A literature review: Solving constrained non-linear Bi-level optimization problems with classical methods

Bi-level optimization is an emerging scope of research which consists of two optimization problems, where the lower-level optimization problem is nested into the upper-level problem as a constraint. Bi-level programming has gained much attention recently for practical applications. Bi-level Programming Problems (BLPP) can be solved with classical and heuristic optimization methods. However, applying heuristic methods, though easier to formulate for realistic complex design, are likely to be too computationally expensive for solving bi-level problems, especially when the problem has high function evaluation cost associated with handling large number of constraint functions. Thus, classical approaches are investigated in this paper. As we present, there appears to be no universally best classical method for solving any kind of NP-hard BLPP problem in terms of accuracy to finding true optimal solutions and minimal computational costs. This could cause a dilemma to the researcher in choosing an appropriate classical approach to solve a BLPP in different domains and levels of complexities. Therefore, this motivates us to provide a detailed literature review and a comparative study of the work done to date on applying different classical approaches in solving constrained non-linear, bi-level optimization problems considering continuous design variables and no discontinuity in functions.