R. H. Byrd, M. E. Hribar and J. Nocedal, An interior point algorithm for large-scale nonlinear programming, SIAM J. Optim., 11 (1999), pp. 877-900.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....SQP method that we had intended including in GALAHAD [18] despite having produced both e ective active set and interior point QP solvers. Our experience has been that without QP truncation, the cost of the QP solution so dominates that other non SQP approaches (such as IPOPT [33] KNITRO [4] and LOQO [32] in which truncation is possible, have made signi cant progress even before our QP code had solved its rst subproblem see also [23] for further evidence that interior point methods appear to scale better than SQP ones. We are more enthusiastic about an SLP QP approach we are ....

R. H. Byrd, M. E. Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming. SIAM Journal on Optimization, 9(4):877-900, 2000.

.... For examples of such methods, see, e.g. Forsgren and Gill [FG98] Gay, Overton and Wright [GOW98] Shanno and Vanderbei [SV99] El Bakry et al. EBTTZ96] and Conn, Gould and Toint [CGT96] Other interior methods use a trust region technique for safeguarding; see, e.g. Byrd, Hribar and Nocedal [BHN99], Byrd, Gilbert and Nocedal [BGN00] Conn, Gould and Toint [CGT00, Chapter 13] and references therein. Recently, interior methods with lters have been studied as well, see Ulbrich, Ulbrich and Vicente [UUV00] and W achter and Biegler [WB01] However, trust region and lter methods do generally ....

R. H. Byrd, M. E. Hribar, and J. Nocedal. An interior point algorithm for large-scale nonlinear programming. SIAM J. Optim., 9, 877-900, 1999.

....NLPs. Our goal is to identify features of these codes that are e#cient and those that are ine#cient on a variety of problem classes. We have worked with three algorithms: loqo, an interior point method code by Vanderbei et al. 15] knitro, a trust region algorithm by Byrd et al. [2] . snopt, a quasi Newton algorithm by Gill et al. 9] A fourth code, lancelot by Conn et al. 4] is designed for largescale nonlinear optimization, but previous work with the code [6] has shown it not competitive with the above codes for the larger problems tested. Thus, we do not include ....

....b , or the infeasibility, ### is reduced. Here, and throughout the paper, all norms are Euclidean, unless otherwise indicated) A su#cient reduction is determined by an Armijo rule. Details of this filter approach to loqo are discussed in [1] 3. KNITRO: A Trust Region Algorithm knitro [2] is an interior point solver developed by Byrd et al. It takes a barrier approach to the problem using sequential quadratic programming (SQP) and trust regions to solve the barrier subproblems at each iteration. The SQP technique is used to handle the nonlinearities in the problem and the trust ....

[Article contains additional citation context not shown here]

R.H. Byrd, M.E. Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming. SIAM J. Opt., 9(4):877--900, 1999.

....Consider the problem s.t. c j (x) 0, j = 1, m e (P ) where f : c j : j = 1, m e and d j : j = 1, m i are smooth. No convexity assumptions are made. A number of primaldual interior point methods have been proposed to tackle such problems; see, e.g. [1, 2, 3, 4, 5, 6, 7, 8]. While all of these methods make use of a search direction generated by a Newton or quasi Newton iteration on a perturbed version of some first order necessary conditions of optimality, they di#er in many respects. For example, some algorithms enforce feasibility of all iterates with respects to ....

....necessary conditions of optimality, they di#er in many respects. For example, some algorithms enforce feasibility of all iterates with respects to inequality constraints [4, 5] while others, sometimes referred to as infeasible , sidestep that requirement via the introduction of slack variables [1, 2, 3, 6, 7, 8]. As for equality constraints, some schemes include them as is in the perturbed optimality conditions [1, 2, 3, 4, 6, 7] while some soften this condition by making use of two sets of slack variables [8] or by introducing a quadratic penalty function, yielding optimality conditions involving a ....

[Article contains additional citation context not shown here]

R.H. Byrd, M.E. Hribar, and J. Nocedal. An interior point algorithm for large-scale nonlinear programming. SIAM J. on Optimization, 9(4):877-- 900, 1999.

....AEexible line search approach implemented in TSQP and the robust, but more complex, trust region (TR) technique. For this reason, we have chosen to compare TSQP with two other TR codes: ETR (an equality constraint solver [18] and Knitro (Version 1. 00, an equality and inequality constraint solver [7, 6, 24]) 4.1 Conditions of the tests Our benchmark is formed of a subset of test problems from the cute collection [3] and some industrial real life test problems provided by Essilor, a lens manufacturing company (see also in [17] Jonsson s contribution to this application) All the selected problems ....

R.H. Byrd, M.E. Hribar, J. Nocedal (1999). An interior point algorithm for large scale nonlinear programming. SIAM Journal on Optimization, 9, 877900.

.... ltermpec SQP trust region method; global convergence through use of a lter; MPEC ampl interface; exact Hessian; feasibility restoration [17] and [18] knitro Trust region interior point algorithm; step decomposed into normal and tangential step; special rules for trust region adjustment [5]. loqo Primal dual interior point method; line search to induce global convergence; exact Hessian [28] snopt SQP line search method; augmented Lagrangian merit function; quasiNewton Hessian, inertia control; elastic mode for inconsistent QP subproblems. 5.1 Failures of NLP solvers All NLP ....

Byrd, R.H. Hribar, M.E. and Nocedal, J. An interior point algorithm for large-scale nonlinear programming. SIAM Journal on Optimization, 9(4):877-900, 1999.

....is then solved for converging to zero. LOQO ond KNITRO ore the two codes thor we consider in this cotegory. LOQO [29] solves the equolity constroined problem using Newton s method to colculote o direction ond then finds o new irer ote using o lineseorch olong the direction. KNI TRO [5] uses o sequentiol quodrotic progromming method with o trust region to colculote the solution to the equolity constroined problem for o fixed . Augmented Logrongion methods reformulote the inequolity constroined problem into o problem with only simple bounds by odding slock voriobles ond ....

R.H. Byrd, M. E. Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming. $'IAM d. Optimiza- tioga, 9(4):877 900, 1999.

....in order to achieve quadratic or superlinear convergence and by using starting points that were su#ciently close to some local minima. To have global convergence from remote starting points, a variety of line search techniques have been developed [31, 141] Sequential quadratic programming (SQP) [41, 79, 35, 178, 107] approximates solutions to the first order necessary conditions by solving a series of quadratic programming (QP) subproblems, each involving the minimization of a quadratic approximation of the objective function, subject to a linear approximation of the constraints. Such a QP problem is then ....

R. H. Byrd, M. E. Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming. Technical Report OTC97/05, Optimization Technology Center, 1997. 183

....of functions; Column 12 shows the main method used by the solver; Column 13 shows the input formats acceptable by the solver; and, column 14 shows any additional requirements needed by the solver. We see that DONLP2 [146, 6] LANCELOT [54, 13, 107] LOQO [154, 20] MINOS [147, 21] KNITRO [51], SNOPT [75] FSQP [5] HQP OMUSES [2] and MOSEK [29, 19] mainly deal with continuous di#erentiable constrained NLPs. LOQO [154, 20] and MOSEK [29, 19] in addition, require the convexity of functions. Genocop [116, 10] and COBYLA2 [127] can solve continuous constrained NLPs whose functions are ....

R. H. Byrd, M. E. Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming. SIAM Journal on Optimization, 9(4):877--900, 1999.

....original problem. We illustrate such phenomenon in this section with respect to deterministic and stochastic optimization problems. 1.3. 1 Constraint relaxation for NLPs with deterministic functions We have investigated a variety of nonlinear programming algorithms, including Lancelot [52] KNITRO [44], SNOPT [71] and CSA [164] Figure 1.1 plots the relationship between relaxation level r and the expected time to find a feasible solution by each algorithm. For CSA, we have used the optimal cooling schedule (that is discussed later) for each relaxation level. The graphs show that in some of the ....

....convexity of functions; Column 15 shows the principal method used in each solver; Column 16 shows the input format, and Column 17 shows some other special requirement for each solver. We can see from the table that DONLP2 [150, 6] LANCELOT [52, 10, 102] LOQO [155, 14] MINOS [151, 15] KNITRO [44], SNOPT [71] FSQP [5] HQP OMUSES [2] and MOSEK [24, 13] are mainly used for solving continuous constrained NLPs with di#erentiable functions. Genocop [111, 9] and COBYLA2 [127] can solve constrained NLPs whose variables are continuous and whose functions are not di#erentiable or continuous. ....

R. H. Byrd, M. E. Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming. SIAM Journal on Optimization, 9(4):877--900, 1999.

....are still a very open topic of research in nonlinear programming. One of the issues is guaranteeing global convergence because there seems to be no ideal merit function. Several approaches for globalizing interior point methods using different merit functions have been proposed. See the references [5,8,10,15,16,22]. On the other hand, the local convergence properties of interior point methods for nonlinear programming are quite well studied in the literature [6,7,10,18,23,28] although difficulties arise when the limit point does not satisfy strict complementarity or linear independence of the gradients of ....

R. H. BYRD, M. E. HRIBAR, AND J. NOCEDAL, An interior point algorithm for large-scale nonlinear programming, SIAM J. Optim., 9 (1999), pp. 877--900.

....It uses a bound constrained augmented Lagrangian method. In general, LANCELOT is recommended for large problems with many degrees of freedom. It complements SNOPT and the other methods discussed above. A comparison between LANCELOT and MINOS has been made in [8, 9] LOQO [77] and KNITRO [16, 15] are examples of large scale optimization packages that treat inequality constraints by a primal dual interior method. Both packages require second derivatives but can accommodate many degrees of freedom. 1.5. Notation. Some important quantities follow: x; s) Primal, dual and slack variables ....

R. H. Byrd, M. E. Hribar, and J. Nocedal, An interior point algorithm for large-scale nonlinear programming, SIAM J. Optim., 9 (1999), pp. 877-900 (electronic). Dedicated to John E. Dennis, Jr., on his 60th birthday.

No context found.

R. H. Byrd, M. E. Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming. SIAM Journal on Optimization, 9(4):877-900, 2000.

No context found.

R.H. Byrd, M.E. Hribar, and J. Nocedal, "An interior point algorithm for large scale nonlinear programming", Technical Report OTC 97/05, Optimization Technology Center, Northwestern University (1997).

....programming. The methods proposed in [1, 7, 12, 13] either start with a feasible point or apply a phase one procedure to compute one, and then generate strictly feasible iterates. Most other implementations of interior methods for nonlinear programming are based, however, on infeasible algorithms [5, 10, 20, 23] which may enter and leave the feasible region during the course of the minimization. In this paper we describe a framework for transforming slack based infeasible methods into feasible methods. In this framework, feasible and infeasible interior algorithms can be considered as variants of the ....

.... method for solving (2:3) that computes steps by applying Newton s method in the variables x; s; h ; g to the system i s i = i 2 I h(x) 0 g(x) Gamma s = 0; which is equivalent to the KKT conditions for (2:3) This system is the basis for primaldual infeasible algorithms; see e.g. [5, 20]. Application of Newton s method gives rise to the linear system B B xx L 0 A h (x) A g (x) 0 0 GammaS C C A 0 B B C C = Gamma B B Gammae S g C C ; 3.9) where S and denote diagonal matrices with s and g on their respective diagonals, and L stands for the ....

[Article contains additional citation context not shown here]

R. H. Byrd, M. E. Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming. SIAM Journal on Optimization, 9(4):877--900, 2000.

....Our testing will be done primarily on small, medium sized and moderately large problems from the CUTE collection. By this we mean problems that have up to 10,000 variables or constraints. A variety of interior (or barrier) methods for nonlinear programming have been proposed in the last few years [1,5,7,12,15,17,28,29], but only a few implementations are currently available for public use. We have chosen to experiment with the software packages LOQO [28] and KNITRO [5] which are available through the NEOS system [9] as well as in machine executable form. These packages implement two distinct interior point ....

....or constraints. A variety of interior (or barrier) methods for nonlinear programming have been proposed in the last few years [1,5,7,12,15,17,28,29] but only a few implementations are currently available for public use. We have chosen to experiment with the software packages LOQO [28] and KNITRO [5], which are available through the NEOS system [9] as well as in machine executable form. These packages implement two distinct interior point approaches: LOQO is a line search algorithm that has much in common with interior algorithms for linear and convex quadratic programming, whereas KNITRO is ....

[Article contains additional citation context not shown here]

R. H. Byrd, M. E. Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming. SIAM Journal on Optimization, 9(4):877--900, 2000.

....Current reports available from www.ece.nwu.edu nocedal. This author was supported by National Science Foundation grant CDA 9726385 and by Department of Energy grant DE FG02 87ER25047 A004. 1. Introduction A variety of algorithms for linearly and nonlinearly constrained optimization (e.g. [8, 13, 14, 35, 36]) use the conjugate gradient (CG) method [28] to solve subproblems of the form q(x) 2 x Hx c x (1.1) subject to Ax = b: 1.2) In nonlinear optimization, the n vector c usually represents the gradient rf of the objective function or the gradient of the Lagrangian, the n Theta n ....

....optimization that handle bound constraints l x u by adding terms of the form Gamma P n i=1 (log(x i Gamma l i ) log(u i Gamma x i ) to the objective function, for some positive barrier parameter . The choice G = I arises in several trust region methods for constrained optimization [8, 14, 15, 27, 35, 39, 46]. These methods include a trust region constraint of the form kZx Z k Delta in the subproblem (2.3) In order to transform it into a spherical constraint, we introduce the change of variables x Z (Z Gamma1=2 x Z whose effect in the CG iteration is identical to that of defining Z GZ = ....

[Article contains additional citation context not shown here]

R. H. Byrd, M. E. Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming. SIAM Journal on Optimization, 9(4):877--900, 1999.

....programming. The methods proposed in [1, 7, 12, 13] either start with a feasible point or apply a phase one procedure to compute one, and then generate strictly feasible iterates. Most other implementations of interior methods for nonlinear programming are based, however, on infeasible algorithms [5, 10, 20, 23] whichmayenter and leave the feasible region during the course of the minimization. In this paper we describe a framework for transforming slack based infeasible methods into feasible methods. In this framework, feasible and infeasible interior algorithms can be considered as variants of the same ....

.... feasible method for solving (2:3) that computes steps by applying Newton s method in the variables x# s# h# g to the system g = i s i = # i 2I ; s = 0# which is equivalent to the KKT conditions for (2:3) This system is the basis for primaldual infeasible algorithms# see e.g. [5, 20]. Application of Newton s method gives rise to the linear system B B xx L 0 A h 0 0 ;S C C A 0 B B C C = B B L(x# ) S C C (3.9) where S and denote diagonal matrices with s g on their respective diagonals, and L stands for the Lagrangian of ....

[Article contains additional citation context not shown here]

R. H. Byrd, M. E. Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming. SIAM Journal on Optimization,9(4)" 2000.

.... Delta 0 = 1 and Delta 0 = 0:8 Delta 0 = p n, where n is the number of problem variables. 11 Numerical Tests In order to assess the potential of the SLP EQP approach taken in Slique, we test it here on the CUTEr [1, 15] set of problems and compare it with the state of the art codes Knitro [3, 23] and Snopt [11] Slique 1.0 implements the algorithm outlined in the previous section. In all results reported in this section, Slique 1.0 uses the commercial LP software package ILOG CPLEX 8.0 [17] running the default dual simplex approach to solve the LP subproblems. Knitro 2.1 implements a ....

R. H. Byrd, M. E. Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming. SIAM Journal on Optimization, 9(4):877--900, 1999.

No context found.

R. H. Byrd, M. E. Hribar and J. Nocedal, An interior point algorithm for large-scale nonlinear programming, SIAM J. Optim., 11 (1999), pp. 877-900.

No context found.

R. H. Byrd, M. E. Hribar, J. Nocedal, An interior point algorithm for large-scale nonlinear programming, SIAM Journal on Optimization 9 (4) (1999) 877--900.

No context found.

R. H. Byrd, M. E. Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming. SIAM Journal on Optimization, 9(4):877-900, 1999.

No context found.

R. H. Byrd, M. E. Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming. SIAM J. Optimization, 9(4):877-900, 1999.

No context found.

R.H. Byrd, M.E. Hribar, and J. Nocedal. An interior point algorithm for large-scale nonlinear programming. SIAM J. on Optimization, 9(4):877-- 900, 1999.

No context found.

- Byrd, R. H., Hribar, M. E., Nocedal, J. 2000: An interior point algorithm for large scale nonlinear programming. SIAM Journal on Optimization, volume 9, no. 4, pp. 877--900.

First 50 documents  Next 50

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