  Combinatorial optimization problem (COP) is difficult to solve because of the
massive number of local optimal solutions in his solution space. Various
methods have been put forward to smooth the solution space of COPs, including
homotopic convex (HC) transformation for the traveling salesman problem (TSP).
This paper extends the HC transformation approach to unconstrained binary
quadratic programming (UBQP) by proposing a method to construct a unimodal toy
UBQP of any size. We theoretically prove the unimodality of the constructed toy
UBQP. After that, we apply this unimodal toy UBQP to smooth the original UBQP
by using the HC transformation framework and empirically verify the smoothing
effects. Subsequently, we introduce an iterative algorithmic framework
incorporating HC transformation, referred as landscape smoothing iterated local
search (LSILS). Our experimental analyses, conducted on various UBQP instances
show the effectiveness of LSILS. Furthermore, this paper proposes a parallel
cooperative variant of LSILS, denoted as PC-LSILS and apply it to both the UBQP
and the TSP. Our experimental findings highlight that PC-LSILS improves the
smoothing performance of the HC transformation, and further improves the
overall performance of the algorithm.
