23 research outputs found
Multi-Objective Mixed Integer Programming: An Objective Space Algorithm
This paper introduces the first objective space algorithm which can exactly
find all supported and non-supported non-dominated solutions to a mixed-integer
multi-objective linear program with an arbitrary number of objective functions.
This algorithm is presented in three phases. First it builds up a super-set
which contains the Pareto front. This super-set is then modified to not contain
any intersecting polytopes. Once this is achieved, the algorithm efficiently
calculates which portions of the super-set are not part of the Pareto front and
removes them, leaving exactly the Pareto front.Comment: 6 pages, presented at LeGO International Global Optimization
Workshop. At time of submission, no competing algorithm was known, but a
competing algorithm was published between submission and presentation of this
wor
Computing the crosscap number of a knot using integer programming and normal surfaces
The crosscap number of a knot is an invariant describing the non-orientable
surface of smallest genus that the knot bounds. Unlike knot genus (its
orientable counterpart), crosscap numbers are difficult to compute and no
general algorithm is known. We present three methods for computing crosscap
number that offer varying trade-offs between precision and speed: (i) an
algorithm based on Hilbert basis enumeration and (ii) an algorithm based on
exact integer programming, both of which either compute the solution precisely
or reduce it to two possible values, and (iii) a fast but limited precision
integer programming algorithm that bounds the solution from above.
The first two algorithms advance the theoretical state of the art, but remain
intractable for practical use. The third algorithm is fast and effective, which
we show in a practical setting by making significant improvements to the
current knowledge of crosscap numbers in knot tables. Our integer programming
framework is general, with the potential for further applications in
computational geometry and topology.Comment: 19 pages, 7 figures, 1 table; v2: minor revisions; to appear in ACM
Transactions on Mathematical Softwar
Transmission Expansion Planning Considering Energy Storage
In electricity transmission networks, energy storage systems (ESS) provide a
means of upgrade deferral by smoothing supply and matching demand. We develop a
mixed integer programming (MIP) extension to the transmission network expansion
planning (TEP) problem that considers the installation and operation of ESS as
well as additional circuits. The model is demonstrated on the well known
Garver's 6-bus and IEEE 25-bus test circuits for two 24 hour operating
scenarios; a short peak, and a long peak. We show optimal location and capacity
of storage is sensitive not only to cost, but also variability of demand in the
network
Multi-objective integer programming: An improved recursive algorithm
This paper introduces an improved recursive algorithm to generate the set of
all nondominated objective vectors for the Multi-Objective Integer Programming
(MOIP) problem. We significantly improve the earlier recursive algorithm of
\"Ozlen and Azizo\u{g}lu by using the set of already solved subproblems and
their solutions to avoid solving a large number of IPs. A numerical example is
presented to explain the workings of the algorithm, and we conduct a series of
computational experiments to show the savings that can be obtained. As our
experiments show, the improvement becomes more significant as the problems grow
larger in terms of the number of objectives.Comment: 11 pages, 6 tables; v2: added more details and a computational stud