On the Effects of Smoothing Rugged Landscape by Different Toy Problems: A Case Study on UBQP

The hardness of the Unconstrained Binary Quadratic Program (UBQP) problem is due its rugged landscape. Various algorithms have been proposed for UBQP, including the Landscape Smoothing Iterated Local Search (LSILS). Different from other UBQP algorithms, LSILS tries to smooth the rugged landscape by building a convex combination of the original UBQP and a toy UBQP. In this paper, our study further investigates the impact of smoothing rugged landscapes using different toy UBQP problems, including a toy UBQP with matrix
$\hat{\boldsymbol{Q}}^{1}$
(construct by “
$+/-1$
‘), a toy UBQP with matrix
$\hat{\boldsymbol{Q}}^{2}$
(construct by “
$+/-\mathrm{i}$
’) and a toy UBQP with matrix
$\hat{\boldsymbol{Q}}^{3}$
(construct randomly). We first assess the landscape flatness of the three toy UBQPs. Subsequently, we test the efficiency of LSILS with different toy UBQPs. Results reveal that the toy UBQP with
$\hat{\boldsymbol{Q}}^{1}$
(construct by “
$+/-1$
”) exhibits the flattest landscape among the three, while the toy UBQP with
$\hat{Q}^{3}$
(construct randomly) presents the most non-flat landscape. Notably, LSILS using the toy UBQP with
$\hat{\boldsymbol{Q}}^{2}$
(construct by “
$+/\cdot \mathbf{i})$
emerges as the most effective, while
$\hat{\boldsymbol{Q}}^{3}$
(construct randomly) has the poorest result. These findings contribute to a detailed understanding of landscape smoothing techniques in optimizing UBQP.