  We first prove semi-orthogonal decompositions of derived factorization
categories arising from sums of potentials of gauged Landau-Ginzburg models,
where the sums are not necessarily Thom--Sebastiani type. We then apply the
result to the category ${\rm HMF}^{L_f}(f)$ of maximally graded matrix
factorizations of an invertible polynomial $f$ of chain type, and explicitly
construct a full strong exceptional collection $E_1$,..., $E_{\mu}$ in ${\rm
HMF}^{L_f}(f)$ whose length $\mu$ is the Milnor number of the
Berglund--H\"ubsch transpose $\widetilde{f}$ of $f$. This proves a conjecture,
which postulates that for an invertible polynomial $f$ the category ${\rm
HMF}^{L_f}(f)$ admits a tilting object, in the case when $f$ is a chain
polynomial. Moreover, by careful analysis of morphisms between the exceptional
objects $E_i$, we explicitly determine the quiver with relations $(Q,I)$ which
represents the endomorphism ring of the associated tilting object
$\oplus_{i=1}^{\mu}E_i$ in ${\rm HMF}^{L_f}(f)$, and in particular we obtain an
equivalence ${\rm HMF}^{L_f}(f)\cong {\rm D}^{\rm b}({\rm mod}\, kQ/I)$.
