A new method for strong-weak linear bilevel programming problem

Easier than We Thought - A Practical Scheme to Compute Pessimistic Bilevel Optimization Problem

“…Noting that the solution to the lower level problem might not be unique, the strong-weak bilevel optimization problem, also known as partially cooperative bilevel model, has been proposed and investigated in [1,11,31]. It integrates the optimistic and pessimistic formulations through a weighted summation, where the weight coefficient can be interpreted as the cooperative probability of the lower level DM.…”

“…where S(x) is defined as in (12). Remark: It is shown in [11,31] that SW − PBL has an optimal solution if x are continuous, i.e., n d = 0. Again, through Branch-and-Bound argument, this result extends to a more general case where x are mixed integer variables.…”

“…To solve SW − PBL problem, a couple of penalty methods have been developed in [11,31], which iteratively adjust penalty coefficients and compute the resulting relaxations to ensure the convergence to an optimal solution. Again, using arguments from Section 2.1 and KKT reformulation, we provide its tight R-PBL relaxation and the computationally friendly single level reformulation.…”

“…As demonstrated next, it is not always the case that we can neglect the correction step. (30)(31)(32)(33)(34) where the objective function in (30) is changed to min 0x 1 + 0x 2 + max(−2y 1 − 3y 2 − 2y 3 − 16y 4 ). By repeating the same solution procedure, we obtain an optimal solution with x * 1 = x * 2 = 0 and y * j = 0 for j = 1, .…”

“…Consider the next instance that is extended from (30)(31)(32)(33)(34) with a new constraint on (x 1 , x 2 ) in the upper level and an integer variable restriction on y 3 in the lower level. So, the lower level problem is a mixed integer program.…”