In this paper, we consider the solution of the standard linear programming [Lt＇）. A remarkable result in LP claims that all optimal solutions form an optimal face of the underlying polyhedron. In practice, many real...In this paper, we consider the solution of the standard linear programming [Lt＇）. A remarkable result in LP claims that all optimal solutions form an optimal face of the underlying polyhedron. In practice, many real-world problems have infinitely many optimal solutions and pursuing the optimal face, not just an optimal vertex, is quite desirable. The face algorithm proposed by Pan [19] targets at the optimal face by iterating from face to face, along an orthogonal projection of the negative objective gradient onto a relevant null space. The algorithm exhibits a favorable numerical performance by comparing the simplex method. In this paper, we further investigate the face algorithm by proposing an improved implementation. In exact arithmetic computation, the new algorithm generates the same sequence as Pan＇s face algorithm, but uses less computational costs per iteration, and enjoys favorable properties for sparse problems.展开更多
In this paper, we present a block Lanczos met hod for solving the large-scale CDT subproblem. During the algorithm, the original CDT subproblem is projected to a smallscale one, and then some classical method is emplo...In this paper, we present a block Lanczos met hod for solving the large-scale CDT subproblem. During the algorithm, the original CDT subproblem is projected to a smallscale one, and then some classical method is employed to solve this small-scale CDT subproblem to get a solution, which can be used to derive an approximate solution of the original CDT subproblem. Theoretical analysis of the error bounds for both the optimal value and the optimal solution is also proposed. Numerical experiments are carried out, and it is demonstrated that the block Lanczos method is effective and can achieve high accuracy for large-scale CDT subproblems.展开更多
文摘In this paper, we consider the solution of the standard linear programming [Lt＇）. A remarkable result in LP claims that all optimal solutions form an optimal face of the underlying polyhedron. In practice, many real-world problems have infinitely many optimal solutions and pursuing the optimal face, not just an optimal vertex, is quite desirable. The face algorithm proposed by Pan [19] targets at the optimal face by iterating from face to face, along an orthogonal projection of the negative objective gradient onto a relevant null space. The algorithm exhibits a favorable numerical performance by comparing the simplex method. In this paper, we further investigate the face algorithm by proposing an improved implementation. In exact arithmetic computation, the new algorithm generates the same sequence as Pan＇s face algorithm, but uses less computational costs per iteration, and enjoys favorable properties for sparse problems.
文摘In this paper, we present a block Lanczos met hod for solving the large-scale CDT subproblem. During the algorithm, the original CDT subproblem is projected to a smallscale one, and then some classical method is employed to solve this small-scale CDT subproblem to get a solution, which can be used to derive an approximate solution of the original CDT subproblem. Theoretical analysis of the error bounds for both the optimal value and the optimal solution is also proposed. Numerical experiments are carried out, and it is demonstrated that the block Lanczos method is effective and can achieve high accuracy for large-scale CDT subproblems.
