63,680 research outputs found
A Global Optimization Algorithm for Generalized Quadratic Programming
We present a global optimization algorithm for solving generalized quadratic programming (GQP), that is, nonconvex quadratic programming with nonconvex quadratic constraints. By utilizing a new linearizing technique, the initial nonconvex programming problem (GQP) is reduced to a sequence of relaxation linear programming problems. To improve the computational efficiency of the algorithm, a range reduction technique is employed in the branch and bound procedure. The proposed algorithm is convergent to the global minimum of the (GQP) by means of the subsequent solutions of a series of relaxation linear programming problems. Finally, numerical results show the robustness and effectiveness of the proposed algorithm
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
Fast interior point solution of quadratic programming problems arising from PDE-constrained optimization
Interior point methods provide an attractive class of approaches for solving linear, quadratic and nonlinear programming problems, due to their excellent efficiency and wide applicability. In this paper, we consider PDE-constrained optimization problems with bound constraints on the state and control variables, and their representation on the discrete level as quadratic programming problems. To tackle complex problems and achieve high accuracy in the solution, one is required to solve matrix systems of huge scale resulting from Newton iteration, and hence fast and robust methods for these systems are required. We present preconditioned iterative techniques for solving a number of these problems using Krylov subspace methods, considering in what circumstances one may predict rapid convergence of the solvers in theory, as well as the solutions observed from practical computations
Sensor Selection Based on Generalized Information Gain for Target Tracking in Large Sensor Networks
In this paper, sensor selection problems for target tracking in large sensor
networks with linear equality or inequality constraints are considered. First,
we derive an equivalent Kalman filter for sensor selection, i.e., generalized
information filter. Then, under a regularity condition, we prove that the
multistage look-ahead policy that minimizes either the final or the average
estimation error covariances of next multiple time steps is equivalent to a
myopic sensor selection policy that maximizes the trace of the generalized
information gain at each time step. Moreover, when the measurement noises are
uncorrelated between sensors, the optimal solution can be obtained analytically
for sensor selection when constraints are temporally separable. When
constraints are temporally inseparable, sensor selections can be obtained by
approximately solving a linear programming problem so that the sensor selection
problem for a large sensor network can be dealt with quickly. Although there is
no guarantee that the gap between the performance of the chosen subset and the
performance bound is always small, numerical examples suggest that the
algorithm is near-optimal in many cases. Finally, when the measurement noises
are correlated between sensors, the sensor selection problem with temporally
inseparable constraints can be relaxed to a Boolean quadratic programming
problem which can be efficiently solved by a Gaussian randomization procedure
along with solving a semi-definite programming problem. Numerical examples show
that the proposed method is much better than the method that ignores dependence
of noises.Comment: 38 pages, 14 figures, submitted to Journa
A binarisation heuristic for non-convex quadratic programming with box constraints
Non-convex quadratic programming with box constraints is a fundamental problem in the global optimization literature, being one of the simplest NP-hard nonlinear programs. We present a new heuristic for this problem, which enables one to obtain solutions of excellent quality in reasonable computing times. The heuristic consists of four phases: binarisation, convexification, branch-and-Bound, and local optimisation. Some very encouraging computational results are given
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