maxF (x, y) subject to Ax + By y = Y (x)

=

c^{T }x + d^{T }y

≤

r

(P 3)

Then, if the point (x, y) is the solution of the following problem

maxF (x, y)

=

c^{T }x + d^{T }y

subject to

Ax + By

≤

r

and y = Y (x), then (x, y) is the solution of the problem (P 1).

# De nition 1: The point (x, y) is feasible if (x, y) ∈ ψ_{f }(S)

De nition 2: The feasible point (x^{∗}, y^{∗}) ∈ ψ_{f }(S) is the optimal solution of the BLPP (solution for short) if F (x^{∗}, y^{∗}) ≥ F (x, y) for each point (x, y) ∈ ψ_{f }(S) .

The paper will discuss the numerical method of BLPP under the definition.

# 3

# Design of the GA for BLPP

It is not easy to know the upper-level objective function of BLPP has no explicit formulation, since it is compounded by the lower-level solution function which has no explicit formulation. Thus, it is hard to express the definition of the derivation of the function in common sense. And it is difficult to discuss the conditions and the algorithms of the optimal solution with the definition. We concerned the GA is a numerical algorithm compatible for the optimization problem since it has no special requirements for the differentiability of the function. Hence the paper solves BLPP by GA.

The basic idea solving BLPP by GA is: firstly, choose the initial population satisfying the constraints, then the lower-level decision maker makes the corresponding optimal reaction and evaluate the individuals according to the fitness function constructed by the feasible degree, until the optimal solution is searched by the genetic operation over and over.

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Coding and Constraints

At present, the coding often used are binary vector coding and floating vector coding. But the latter is more near the space of the problem compared with the former and experiments show the latter converges faster and has higher computing precision[23] . The paper adopts the floating vector coding. Hence the individual is expressed by: v_{k }= (v_{k1}, v_{k2}, . . . , v_{km}).

The individuals of the initial population are generally randomly generated in GA, which tends to generate off-springs who are not in the constraint region. Hence, we must deal with them.

3