An algorithm is developed for projecting a point onto a polyhedron. The algorithm solves a dual version of the projection problem and then uses the relationship between the primal and dual to recover the projection. T...
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An algorithm is developed for projecting a point onto a polyhedron. The algorithm solves a dual version of the projection problem and then uses the relationship between the primal and dual to recover the projection. The techniques in the paper exploit sparsity. Sparse reconstruction by separable approximation (SpaRSA) is used to approximately identify active constraints in the polyhedron, and the dual active set algorithm (DASA) is used to compute a high precision solution. A linear convergence result is established for SpaRSA that does not require the strong concavity of the dual to the projection problem, and an earlier R-linear convergence rate is strengthened to a Q-linear convergence property. An algorithmic framework is developed for combining SpaRSA with an asymptotically preferred algorithm such as DASA. It is shown that only the preferred algorithm is executed asymptotically. Numerical results are given using the polyhedra associated with the Netlib LP test set. A comparison is made to the interior point method contained in the general purpose open source software package IPOPT for nonlinear optimization, and to the commercial package CPLEX, which contains an implementation of the barrier method that is targeted to problems with the structure of the polyhedral projection problem.
We present an implementation of the LP dual active set algorithm (LP DASA) based on a quadratic proximal approximation, a strategy for dropping inactive equations from the constraints, and recently developed algorithm...
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We present an implementation of the LP dual active set algorithm (LP DASA) based on a quadratic proximal approximation, a strategy for dropping inactive equations from the constraints, and recently developed algorithms for updating a sparse Cholesky factorization after a low-rank change. Although our main focus is linear programming, the first and second-order proximal techniques that we develop are applicable to general concave-convex Lagrangians and to linear equality and inequality constraints. We use Netlib LP test problems to compare our proximal implementation of LP DASA to Simplex and Barrier algorithms as implemented in CPLEX.
We study the structure of dual optimization problems associated with linear constraints, bounds on the variables, and separable cost. We show how the separability of the dual cost function is related to the sparsity s...
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We study the structure of dual optimization problems associated with linear constraints, bounds on the variables, and separable cost. We show how the separability of the dual cost function is related to the sparsity structure of the linear equations. As a result, techniques for ordering sparse matrices based on nested dissection or graph partitioning can be used to decompose a dual optimization problem into independent subproblems that could be solved in parallel. The performance of a multilevel implementation of the dual active set algorithm is compared with CPLEX Simplex and Barrier codes using Netlib linear programming test problems.
The dual active set algorithm (DASA), presented in Hager, Advances in Optimization and Parallel Computing, P.M. Pardalos (Ed.), North Holland: Amsterdam, 1992, pp. 137-142, for strictly convex optimization problems, i...
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The dual active set algorithm (DASA), presented in Hager, Advances in Optimization and Parallel Computing, P.M. Pardalos (Ed.), North Holland: Amsterdam, 1992, pp. 137-142, for strictly convex optimization problems, is extended to handle linear programming problems. Line search versions of both the DASA and the LPDASA are given.
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