The Nearest Point Problem in a Polyhedral Cone and Its Extensions
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2009-08-07
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Abstract
The problem of finding the nearest point in a polyhedral cone to a given point in n-dimensional space can be formulated as a convex quadratic programming problem with special structure. This problem has applications in a wide range of areas, such as robotics, computer graphics, optimal control, and stochastic programming.
In this research we study the geometrical structure of the nearest point problem in a polyhedral cone, and propose an efficient algorithm for solving this problem. We refer to this algorithm as the active index algorithm. In particular, we show that, given the index of one active constraint of the nearest point problem in a polyhedral cone, the order of the problem (number of variables and number of constraints) can be reduced by one. Further, by exploiting the relationship between the nearest point problem in a polyhedral cone and the nearest point problem in a pos cone, we design an efficient procedure to either find the optimal solution to the problem or find one of its active constraints. We also propose several strategies for efficient implementation of this algorithm.
Furthermore, we show how we can use the active index algorithm to solve an instance of the nearest point problem in a polyhedral set. And finally we show how to extend the reach of this algorithm to solve any strictly convex quadratic programming problem with linear inequality constraints.
In addition, we construct a large collection of instances using a random data generator. We then solve those instances using our proposed algorithm as well as the three solvers of Cplex 11 (Barrier solver, Primal Simplex solver, and Dual Simplex solver), and compare the corresponding execution times. Computational results show that for this collection of instances our proposed algorithm has a smaller execution time than any of the three solvers of Cplex 11.
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Active constraint, Projection face, Pos cone, Quadratic Programming
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Degree
PhD
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Operations Research
