An improved bilevel evolutionary algorithm based on Quadratic Approximations

Evolutionary multitasking in bi-level optimization

“…With regard to future works, we will focus on further improving the algorithm to tackle realtime bi-level optimization problems, such as those occurring in security games. One possible means of achieving further speed up of the M-BLEA is the incorporation of quadratic approximation models (or other surrogate modeling techniques) at the lower level, instead of repeatedly having to solve the lower level optimization task in an evolutionary manner, as has very recently been proposed in [28].…”

“…For a summary of the N-BLEA, the reader is referred to Algorithm 3. It must be kept in mind that a standard EA is used at both levels of the N-BLEA, with no non-traditional algorithmic artifacts as have been incorporated in some other previously proposed methodologies for evolutionary bi-level optimization (such as the use of biased lower level population initialization in [31] or the use of a quadratic approximation model in [28]). This prevents any adulteration of the observed results by factors that are not of interest to the present study.…”

“…The formulation of a single-objective bi-level optimization problem (BLOP) is special in the sense that one optimization task (often referred to as the lower level problem or the follower's problem) is nested within another (which is labeled as the upper level problem or the leader's problem). The two levels of this single-act hierarchical optimization setting together comprise a pair of objective functions, namely, F : R u × R l → R and f : R u × R l → R [7,26,28,29]. The relationship between the two functions can be stated according to Eqs.…”

BOLIB: Bilevel Optimization LIBrary of Test Problems

“…Description: ShimizuEtal1997a is defined as follows F(x, y) := (x − 5) 2 + (2y + 1) 2 f(x, y) := (y − 1) 2 − 1.5xy g(x, y) :=   −3x + y + 3 x − 0.5y − 4 x + y − Comment: A possible solution is (5, 2).Problem name: ShimizuEtal1997b Source:[47] Description: ShimizuEtal1997b is defined as followsF(x, y) := 16x 2 + 9y 2 G(x, y) := −4x + y −x f(x, y) := (x + y − 20) 4 g(x, y) := 4x + y − 50 −yComment:(11.25, 5) is the global optimal solution of the problem and (7.2, 12.8) is a local optimal solution[47].Problem name: SinhaMaloDeb2014TP3 Source: [48] Description: SinhaMaloDeb2014TP3 is defined as follows 2y 1 − y 2 − 3 −x 2 − 3y 1 + 4y 2 + 4 The known best values of the upper-level and lower-level objective values are −18.6787 and −1.0156, respectively; cf. [48].Problem name: SinhaMaloDeb2014TP6Source:[48] Description: SinhaMaloDeb2014TP6 is defined as followsF(x, y) := (x − 1) 2 + 2y 1 − 2x G(x, y) := −x f(x, y) := (2y 1 − 4) 2 + (2y 2 − 1) 2 + xy 4y 2 − 12 4y 2 − 4x − 5y 1 + 4 4x − 4y 1 + 5y 2 − 4 4y 1 − 4x + 5y 2 − 4The known best values of the upper-level and lower-level objective values are −1.2091 and 7.6145, respectively; cf [48][48] Description: SinhaMaloDeb2014TP7 is defined as followsF(x, y) := − (x 1 +y 1 )(x 2 +y 2 )The known best values of the upper-level and lower-level objective values are −1.96 and 1.96, respectively; cf [48]…”

Analyzing the Performance of a Hybrid Heuristic for Solving a Bilevel Location Problem under Different Approaches to Tackle the Lower Level

“…For example, recently in [29], an attempt for approximating the lower level optimal solutions without explicitly solving it is presented. The approximation consists in creating quadratic functions based on a set of initial bilevel feasible solutions.…”