T.J. Carpenter and D.F. Shanno, "An interior point method for quadratic programs based on conjugate projected gradients", Comput. Optim. Appl., 2, pp. 5--28, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....) 0 (5) iii) 0: The idea of the method is then to solve a sequence of problems of type (4) while decreasing the parameter towards zero. Notice that by using problem (4) we have restricted the solution to have positive components only. This follows an interior point approach (cf. [3], 6] 24] in which the iterates are feasible and positive. We shall now introduce a further simplification by substituting the nonlinear barrier problems (4) by quadratically constrained quadratic problems, or trustregion subproblems, where the objective function will be a quadratic ....

T.J. Carpenter and D.F. Shanno. An interior point method for quadratic programs based on conjugate projected gradients, Comput. Optim. Appl., 2:528, 1993.

....coefficient matrix. The combination of a direct method and an iterative method was reported by Karmarkar and Ramakrishnan [8] A low rank correction to the matrix was used by Goldfarb and Liu [6] to prove a low complexity bound for convex quadratic programming. A frequently used iterative method [3, 8, 9] is the preconditioned conjugate gradient method on the normal equations. Throughout this paper we use the following notation: min i or max i is for all i for which the argument is defined. For any matrix A, A ij is the element in the i th row and j th column, A j is the j th column, and A jffl is ....

T.J. Carpenter and D.F. Shanno, An interior point method for quadratic programs based on conjugate projected gradients, Computational Optimization and Applications, Vol. 2, pp. 5-28, 1993.

....Preconditioners 2. 1 Motivation In most implementations of primal dual interior point algorithms for linear programming, the direct method of Cholesky factorization is used to solve the system (4) The use of iterative methods of conjugate gradient type have also been considered (see for example [5, 13, 15]) However, the success of iterative methods has been, at best, very limited for interior point algorithms due to the difficulty in constructing general purpose, effective preconditioners. It is well known that effective preconditioners are critical for accelerating the convergence rate of ....

T.J. Carpenter and D.F. Shanno, An interior point method for quadratic programs based on conjugate projected gradients, Computational Optimization and Applications, Vol. 2, pp. 5-28, 1993.

....GENERAL CONVEX QP PROBLEMS VIA AN EXACT QUADRATIC AUGMENTED LAGRANGIAN WITH BOUND CONSTRAINTS P. SPELLUCCI Abstract Large convex quadratic programs, where constraints are of box type only, can be solved quite efficiently [1], 2] 12] 13] 16] In this paper an exact quadratic augmented Lagrangian with bound constraints is constructed which allows one to use these methods for general constrained convex quadratic programming. This is in contrast to well known exact differentiable penalty functions for this type of ....

....iteration using a projection or elimination technique. This however makes necessary using a QR or LU decomposition of the Jacobian of those equality constraints, which may be very costly. Inequality constraints then are handled either by an active set strategy or by interior point methods, e.g. [1], 5] 6] Using duality theory a general convex QP problem may be transformed into a bound constrained one. This process involves solution of a linear equation with the Hessian as coefficient matrix for any intermediate value of the dual variable. In the general case this will be very costly ....

[Article contains additional citation context not shown here]

Carpenter, T.J.; Shanno, D.F.: An interior point method for quadratic programs based on conjugate projected gradients. Comp. Optim. Appl. 2, (1993), 5-28 .

....(8) Karmarkar and Ramakrishnan [53] used the factorization of the matrix for one value of to precondition the problem when is changed. Mehrotra [65] used an incomplete Cholesky factorization as a preconditioner, recomputing the factorization for each new value. Carpenter and Shanno [16] used diagonal preconditioning, and Portugal, Resende, Veiga, and J udice [79] also used spanning tree preconditioners. The best solution algorithm will surely be a hybrid approach that sometimes chooses direct solvers and sometimes iterative ones. Wang and O Leary [92, 91] proposed an adaptive ....

Tamra J. Carpenter and David F. Shanno. An interior point method for quadratic programs based on conjugate projected gradients. Computational Optimization and Applications, 2:5--28, 1993.

....implementation. At each interior point iteration, an incomplete Cholesky factor was computed and used as the preconditioner. Carpenter and Shanno used a diagonal preconditioner for a conjugate gradient solver for the normal equations in an interior point method for quadratic and linear programs [3]. They also considered recomputing the preconditioner every other iteration. Portugal, Resende, Veiga, and J udice introduced a truncated primalinfeasible dual feasible interior point method, focusing on network flow problems [29] The preconditioned conjugate gradient algorithm was used to solve ....

....be required. Such ill conditioning is inevitable in the end stages of the interior point method. An alternative to computing the Cholesky factorization on every interior point iteration is to use the preconditioner computed for one fixed value of the barrier parameter for several values of [3] [19] This reduces the computational work in forming the factorization. Preconditioner 2 : QR decomposition. Rather than computing the Cholesky factors of the matrix K = AD 2 A T , we may simply compute the n Theta m matrix DA T and factor it as QR, where Q is a n Theta m matrix with ....

Tamra J. Carpenter and David F. Shanno. An interior point method for quadratic programs based on conjugate projected gradients. Computational Optimization and Applications, 2:5--28, 1993.

....conjugate gradient method as a corrector step and is consistently faster than MINOS by orders of magnitude. Further computational experience comparing an interior point method OB1 and a simplex method CPLEX is reported in technical reports Bixby, Gregory, Lustig, Marsten and Shanno (1991) Carpenter and Shanno (1991) and Lustig, Marsten and Shanno (1991) Polak, Higgins and Mayne (1992) have given an algorithm for solving semi infinite minimax problems which bears a resemblance to the interior penalty function methods. They report numerical results which show that the algorithm is extremely robust and its ....

T. J. Carpenter and D. F. Shanno (1991), An interior point method for quadratic programs based on conjugate projected gradients, Technical Report RRR 5591, Rutgers Univ., New Brunswick, NJ.

....for quadratic programs, and in the case of SVM s, is not designed in particular for the special characteristics of the problem. Having as a reference the experience obtained with MINOS 5. 4, new approaches to a tailored solver through, for example, projected Newton [2] or interior point methods [7], should be attempted. At this point it is not clear whether the same type of algorithm is appropriate for all stages of the solution process. To be more specific, it could happen that an algorithm performs well with few non zero variables at early stages, and then is outperformed by others when ....

T. Carpenter and D. Shanno. An interior point method for quadratic programs based on conjugate projected gradients. Computational Optimization and Applications, 2(1):5--28, June 1993.

....or a decomposition of a matrix of a corresponding KKT system, which may be very costly, if there is no favorable sparsity structure of the Jacobian which doesn t interfere with that of the Hessian of the objective. Inequality constraints may be handled by interior point methods too, e.g. [3], 10] If there are only box constraints, the active set and the projected gradient technique can be combined to yield highly efficient methods [5] 18] 19] 24] For problems of very high dimension and or dense constraints these active set methods become untractable however. An alternative ....

....0 V T and B = U BU T ; 8.3) where Sigma is the diagonal matrix of the singular values oe i and B the diagonal matrix of the generated eigenvalues fi i . U and V are made up from Householder reflectors represented by vectors u [1] 2 IR m ; u [2] 2 IR n Gammam Gammaj 0 ; u [3] 2 IR j0 and v 2 IR m of length p 2 each: U = 0 0 0 I j0 Gamma u [3] u [3] T 0 I n Gammam Gammaj 0 Gamma u [2] u [2] T 0 I m Gamma u [1] u [1] T 0 0 1 A and V = I m Gamma vv T : 8.4) For later reference we write U in block form U = 0 O O U III O ....

[Article contains additional citation context not shown here]

Carpenter, T.J.; Shanno, D.F.: An interior point method for quadratic programs based on conjugate projected gradients. Comp. Optim. Appl. 2, (1993), 5--28 .

....of the implementation. At each interior point step, an incomplete Cholesky factor was computed and used as the preconditioner. Carpenter and Shanno used a diagonal preconditioner for a conjugate gradient solver for the normal equations in an interior point method for quadratic and linear programs [3]. They also considered recomputing the preconditioner every other step. Portugal, Resende, Veiga, and J udice introduced a truncated primal infeasible dual feasible interior point method, focusing on network flow problems [32] The preconditioned conjugate gradient algorithm was used to solve the ....

....current matrix, but this requires frequent factorizations. An alternative to computing a new Cholesky factorization on every interior point step is to reuse the preconditioner that was computed for one value of the barrier parameter in order to solve systems for several successive values of [3] [21] This reduces the computational work in forming the factorization. An incomplete Cholesky factorization, originally proposed by Varga [34] could be used in place of the Cholesky if density of the matrix factors is too great, but we do not pursue that idea in our implementations. Rather than ....

Tamra J. Carpenter and David F. Shanno. An interior point method for quadratic programs based on conjugate projected gradients. Computational Optimization and Applications, 2:5--28, 1993.

No context found.

T.J. Carpenter and D.F. Shanno, "An interior point method for quadratic programs based on conjugate projected gradients", Comput. Optim. Appl., 2, pp. 5--28, 1993.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC