127 research outputs found
A Polynomial Optimization Approach to Constant Rebalanced Portfolio Selection
We address the multi-period portfolio optimization problem with the constant rebalancing strategy. This problem is formulated as a polynomial optimization problem (POP) by using a mean-variance criterion. In order to solve the POPs of high degree, we develop a cutting-plane algorithm based on semidefinite programming. Our algorithm can solve problems that can not be handled by any of known polynomial optimization solvers.Multi-period portfolio optimization;Polynomial optimization problem;Constant rebalancing;Semidefinite programming;Mean-variance criterion
Semidefinite programming and eigenvalue bounds for the graph partition problem
The graph partition problem is the problem of partitioning the vertex set of
a graph into a fixed number of sets of given sizes such that the sum of weights
of edges joining different sets is optimized. In this paper we simplify a known
matrix-lifting semidefinite programming relaxation of the graph partition
problem for several classes of graphs and also show how to aggregate additional
triangle and independent set constraints for graphs with symmetry. We present
an eigenvalue bound for the graph partition problem of a strongly regular
graph, extending a similar result for the equipartition problem. We also derive
a linear programming bound of the graph partition problem for certain Johnson
and Kneser graphs. Using what we call the Laplacian algebra of a graph, we
derive an eigenvalue bound for the graph partition problem that is the first
known closed form bound that is applicable to any graph, thereby extending a
well-known result in spectral graph theory. Finally, we strengthen a known
semidefinite programming relaxation of a specific quadratic assignment problem
and the above-mentioned matrix-lifting semidefinite programming relaxation by
adding two constraints that correspond to assigning two vertices of the graph
to different parts of the partition. This strengthening performs well on highly
symmetric graphs when other relaxations provide weak or trivial bounds
New bounds for the max--cut and chromatic number of a graph
We consider several semidefinite programming relaxations for the max--cut
problem, with increasing complexity. The optimal solution of the weakest
presented semidefinite programming relaxation has a closed form expression that
includes the largest Laplacian eigenvalue of the graph under consideration.
This is the first known eigenvalue bound for the max--cut when that is
applicable to any graph. This bound is exploited to derive a new eigenvalue
bound on the chromatic number of a graph. For regular graphs, the new bound on
the chromatic number is the same as the well-known Hoffman bound; however, the
two bounds are incomparable in general. We prove that the eigenvalue bound for
the max--cut is tight for several classes of graphs. We investigate the
presented bounds for specific classes of graphs, such as walk-regular graphs,
strongly regular graphs, and graphs from the Hamming association scheme
On bounding the bandwidth of graphs with symmetry
We derive a new lower bound for the bandwidth of a graph that is based on a
new lower bound for the minimum cut problem. Our new semidefinite programming
relaxation of the minimum cut problem is obtained by strengthening the known
semidefinite programming relaxation for the quadratic assignment problem (or
for the graph partition problem) by fixing two vertices in the graph; one on
each side of the cut. This fixing results in several smaller subproblems that
need to be solved to obtain the new bound. In order to efficiently solve these
subproblems we exploit symmetry in the data; that is, both symmetry in the
min-cut problem and symmetry in the graphs. To obtain upper bounds for the
bandwidth of graphs with symmetry, we develop a heuristic approach based on the
well-known reverse Cuthill-McKee algorithm, and that improves significantly its
performance on the tested graphs. Our approaches result in the best known lower
and upper bounds for the bandwidth of all graphs under consideration, i.e.,
Hamming graphs, 3-dimensional generalized Hamming graphs, Johnson graphs, and
Kneser graphs, with up to 216 vertices
Exploiting Group Symmetry in Semidefinite Programming Relaxations of the Quadratic Assignment Problem
We consider semidefinite programming relaxations of the quadratic assignment problem, and show how to exploit group symmetry in the problem data. Thus we are able to compute the best known lower bounds for several instances of quadratic assignment problems from the problem library: [R.E. Burkard, S.E. Karisch, F. Rendl. QAPLIB — a quadratic assignment problem library. Journal on Global Optimization, 10: 291–403, 1997]. AMS classification: 90C22, 20Cxx, 70-08.quadratic assignment problem;semidefinite programming;group sym- metry
An Aggregated Optimization Model for Multi-Head SMD Placements
In this article we propose an aggregate optimization approach by formulating the multi-head SMD placement optimization problem into a mixed integer program (MIP) with the variables based on batches of components. This MIP is tractable and effective in balancing workload among placement heads, minimizing the number of nozzle exchanges, and improving handling class. The handling class which specifies the traveling speed of the robot arm, to the best of our knowledge, has been for the first time incorporated in an optimization model. While the MIP produces an optimal planning for batches of components, a new sequencing heuristics is developed in order to determine the final sequence of component placements based on the outputs of the MIP. This two-stage approach guarantees a good feasible solution to the multi-head SMD placement optimization problem. The computational performance is examined using real industrial data.Multi-head surface mounting device;Component placement;Variable placement speed
On Semidefinite Programming Relaxations of the Travelling Salesman Problem (Replaced by DP 2008-96)
AMS classification: 90C22, 20Cxx, 70-08traveling salesman problem;semidefinite programming;quadratic as- signment problem
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