Browsing by Author "Li, Guangye"

Now showing 1 - 8 of 8
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    A Diagonal-Secant Update Technique for Sparse Unconstrained Optimization
    (1992-01) Li, Guangye
    This paper presents a diagonal-secant modification of the successive element correction method, a finite-difference based method, for sparse unconstrained optimization. This new method uses the gradient values more efficiently in forming the approximate Hessian than the successive element correction method. It is shown that the new method has at least the same local convergence rates as the successive element correction method for general problems and that it has better q-convergence and f-convergence rates than the successive element correction method for problems with band structures. The numerical results show that the new method may be competitive with most of the existing methods for some problems.
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    Algorithms for Solving Sparse Nonlinear Systems of Equations
    (1986-04) Li, Guangye
    In this thesis, we present four algorithms for solving sparse nonlinear systems of equations: the partitioned secant algorithm, the CM-successive displacement algorithm, the modified CM-successive displacement algorithm and the combined secant algorithm. The partitioned secant algorithm is a combination of a finite difference algorithm and a secant algorithm which requires one less function evaluation at each iteration than Curtis, Powell and Reid's algorithm (the CPR algorithm). The combined secant algorithm is a combination of the partitioned secant algorithm and Schubert's algorithm which incorporates the advantages of both algorithms by considering some special structure of the Jacobians to further reduce the number of function evaluations. The CM-successive displacement algorithm is based on Coleman and Moré's partitioning algorithm and a column update algorithm, and it needs only two function values at each iteration. The modified CM-successive displacement algorithm is a combination of the CM-successive displacement algorithm and Schubert's algorithm. It also needs only two function values at each iteration but it uses the information at every step more effectively. The locally q-superlinear convergence results, the r-convergence order estimates and the Kantorovich-type analyses show that these four algorithms have good local convergence properties. The numerical results indicate that the partitioned secant algorithm and the modified CM-successive displacement algorithm are probably more efficient than the CPR algorithm and Schubert's algorithm.
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    An Implementation of a Parallel Primal-Dual Interior Point Method for Multicommodity Flow Problems
    (1992-01) Li, Guangye; Lustig, Irvin J.
    An implementation of the primal-dual predictor-corrector interior point method is specialized to solve linear multicommodity flow problems. The block structure of the constraint matrix is exploited via parallel computation. The bundling constraints require the Cholesky factorization of a dense matrix, where a method that exploits parallelism for the dense Cholesky factorization is used. The resulting implementation is 65 to 90 percent efficient, depending on the problem instance. For a problem with K commodities, an approximate speedup for the interior point method of 0.8K is realized.
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    Column-Secant Update Technique for Solving Systems of Nonlinear Equations
    (1991-10) Li, Guangye; Liu, Tingzhan
    This paper presents a QR update implementation of the successive column correction (SCC) method and a column-secant modification of the SCC method, which is called the CSSCC method. The computational cost of the QR update technique for the SCC method is much less than that for Broyden's method. The CSSCC method uses function values more efficiently than the SCC method, and it shown that the CSSCC method has better local q-convergence and r-convergence rates than the SCC method. The numerical results show that the SCC method and the CSSCC method with the QR update technique are competitive with some well known methods for some standard test problems.
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    Successive Column Correction Algorithms for Solving Sparse Nonlinear Systems of Equations
    (1986-05) Li, Guangye
    This paper presents two algorithms for solving sparse nonlinear systems of equations: the CM-successive column correction algorithm and the modified CM-successive column correction algorithm. A q-superlinear convergence theorem and an r-convergence order estimate are given for both algorithms. The numerical results indicate that these two algorithms, especially the modified algorithm are probably more efficient than some currently used algorithms.
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    Successive Element Correction Algorithms for Sparse Unconstrained Optimization
    (1991-10) Li, Guangye
    This paper presents a successive element correction algorithm and a secant modification of this algorithm. The new algorithms are designed to use the gradient evaluations as efficiently as possible in forming the approximate Hessian. The estimate of the q-convergence and r-convergence rates show that the new algorithms may have good local convergence properties. Some restricted numerical results and comparisons with some previously established algorithms suggest the new algorithms have some promise to be efficient in practice.
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    The Combined Schubert/Secant/Finite Difference Algorithm for Solving Sparse Nonlinear Systems of Equations
    (1986-05) Dennis, J.E. Jr.; Li, Guangye
    This paper presents an algorithm, the combined Schubert/secant/finite difference algorithm, for solving sparse nonlinear systems of equations. This algorithm is based on dividing the columns of the Jacobian into two parts, and using different algorithms on each part. This algorithm incorporates advantages of both algorithms by exploiting some special structure of the Jacobian to obtain a good approximation to the Jacobian by using a little effort as possible. Kantorovich-type analysis and a locally q-superlinear convergence results for this algorithm are given.
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    The Secant/Finite Difference Alogorithm for Solving Sparse Nonlinear Systems of Equations
    (1986-03) Li, Guangye
    This paper presents an algorithm, the secant/finite difference algorithm, for solving sparse nonlinear systems of equations. This algorithm is a combination of a finite difference method and a secant method. A q-superlinear convergence result and an r-convergence rate estimate show that this algorithm has good local convergence properties. The numerical results indicate that this algorithm is probably more efficient than some currently used algorithms.