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Chapter 12
Conclusions & directions for further
research
12.1 Conclusions
In Chapter 4 of this thesis a global optimization method is worked out to compute the
global optimum of a Minkowski dominated polynomial. This method is based on the
Stetter-Möller matrix method which transforms the problem of finding the solutions
of the system of first-order conditions into an eigenvalue problem involving a large
and sparse matrix. This method is described in [81] and a similar approach in [50]
and [48]. A drawback of this approach is that the involved matrices are usually very
large. The efficiency of this method is improved in this thesis by using a matrix-free
implementation of the matrix-vector products and by using an iterative eigenvalue
solver instead of a direct solver.
In Chapter 5 we have developed a routine that computes the action of the large
and sparse matrix, provided by the Stetter-Möller matrix method, on a given vector
without having this matrix explicitly at hand. To avoid building the large matrix
the system of first-order conditions are associated with an nD-system of difference
equations. Such an nD-system yields a matrix-free routine and such a routine is all
that is needed for modern iterative eigenproblem solvers as input. It will turn out
that this works perfectly together which proves the computational feasibility of this
approach.
Section 5.3 focuses on improving the efficiency of computing the action of such a
matrix on a vector. One way to compute the action of the matrix by an nD-system
efficiently, is by setting up a corresponding shortest path problem (SPP) and solve it
as described in Section 5.3.2. A drawback here is that this SPP will quickly become
huge and difficult to solve. However, it it is possible to set up a relaxation of the
shortest path problem (RSPP) which is considerably smaller and easier to solve. In
the 2D-case the RSPP is solved analytically and its solution happens to solve the
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