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## SNOPT: An SQP Algorithm For Large-Scale Constrained Optimization (2002)

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3220 | Numerical optimization
- NOCEDAL, WRIGHT
- 1999
(Show Context)
Citation Context ...timization. The approach was popularized mainly by Biggs [7], Han [66], and Powell [85, 87] for general nonlinear constraints. Further history of SQP methods and extensive bibliographies are given in =-=[61, 39, 73, 78, 28]-=-. For a survey of recent results, see Gould and Toint [65]. Several general-purpose SQP solvers have proved reliable and efficient during the last 20 years. For example, under mild conditions the solv... |

1941 |
Practical Optimization
- Gill, Murray, et al.
- 1981
(Show Context)
Citation Context ...self an iterative procedure, with the minor iterations of an SQP method being the iterations of the QP method. For an overview of SQP methods, see, for example, Fletcher [19], Gill, Murray and Wright =-=[29], Murray-=- [36], and Powell [46]. 2.3. The modified Lagrangian. Let x k and �� k be estimates of x and �� . For several reasons, our SQP algorithm is based on the modified Lagrangian associated with GNP... |

1567 |
Practical Methods of Optimization
- Fletcher
- 1987
(Show Context)
Citation Context ...timization. The approach was popularized mainly by Biggs [7], Han [66], and Powell [85, 87] for general nonlinear constraints. Further history of SQP methods and extensive bibliographies are given in =-=[61, 39, 73, 78, 28]-=-. For a survey of recent results, see Gould and Toint [65]. Several general-purpose SQP solvers have proved reliable and efficient during the last 20 years. For example, under mild conditions the solv... |

759 | On the limited memory BFGS method for large scale optimization
- Liu, Nocedal
- 1989
(Show Context)
Citation Context ...e is then “reset” by discarding the previous updates. (Similar schemes are described by Buckley and LeNir [17, 18] and Gilbert and Lemaréchal [49]. More elaborate schemes are given by Liu and Nocedal =-=[70]-=-, Byrd, Nocedal, and Schnabel [24], and Gill and Leonard [51], and some have been evaluated by Morales [71]. However, as already indicated, these schemes would require refactorization of the reduced H... |

644 | LSQR: An algorithm for sparse linear equations and sparse least squares
- Paige, Saunders
- 1982
(Show Context)
Citation Context ...2 � (8.3) min � GkZ t ��� � dZ + , t = Gk(x − xk), r = Z dZ δI r/δ 2 T gk, where t and r both become small as the SQP method converges. This formulation would allow the use of the CG-type solver LSQR =-=[84]-=-, which has effective stopping rules to control the accuracy of dZ. 8.3. Schur-Complement Updates. For limited-memory Hessians of the form Hk = H0 + UUT − VVT (8.1), system (4.1) is equivalent to ⎛ H0... |

548 | A limited memory algorithm for bound constrained minimization
- Byrd, Lu, et al.
- 1995
(Show Context)
Citation Context ...oblems have no constraints or only simple bounds, and in these cases, the large number of major iterations is consistent with results obtained by other limited-memory quasi-Newton methods (see, e.g., =-=[22, 51]-=-). If the infeasible LC problems, the unbounded problems, and the 2 (conjectured) infeasible problems are counted as successes, SNOPT solved a grand total of 870 of the 1007 problems attempted. In ano... |

447 |
Solutions of sparse indefinite systems of linear equations
- Paige, Saunders
- 1975
(Show Context)
Citation Context ...specifies the maximum size of the dense reduced Hessian available for SQOPT. If the number of superbasics exceeds this value during the QP solution, SQOPT solves (4.5) using the CG-type solver SYMMLQ =-=[83]-=-. Violation limit sets τ V in section 2.7 to define an expanded feasible region in which the objective is expected to be bounded below.s120 PHILIP E. GILL,WALTER MURRAY, AND MICHAEL A. SAUNDERS BEGIN ... |

371 | Benchmarking optimization software with performance profiles
- Dolan, Moré
(Show Context)
Citation Context ...le strategy for treating infeasibility of the original problem and the QP subproblems. 7.3. Results on the COPS 3.0 Test Set. Next we describe tests on the 22 problems in the COPS 3.0 test collection =-=[10, 31, 32]-=- implemented in the AMPL modeling language [45, 46, 1]. The dimension of a particular instance of a COPS problem is determined by one or more parameters assigned in its AMPL data file. For each of the... |

242 | Nonlinear programming without a penalty function
- Fletcher, Leyffer
(Show Context)
Citation Context ..., using quasi-Newton updates. For large problems with general inequality constraints as in problem (NP), SQP methods have been proposed by Eldersveld [36], Tjoa and Biegler [97], Fletcher and Leyffer =-=[40, 41]-=-, and Betts and Frank [5]. The first three approaches are also reducedHessian methods. Eldersveld forms a full Hessian approximation from the reduced Hessian, and his implementation LSSQP solves the s... |

228 |
A Fast Algorithm for Nonlinearly Constrained Optimization Calculations, volume 630
- Powell
- 1978
(Show Context)
Citation Context ... solution. 1.2. The SQP Approach. An SQP method was first suggested by Wilson [102] for the special case of convex optimization. The approach was popularized mainly by Biggs [7], Han [66], and Powell =-=[85, 87]-=- for general nonlinear constraints. Further history of SQP methods and extensive bibliographies are given in [61, 39, 73, 78, 28]. For a survey of recent results, see Gould and Toint [65]. Several gen... |

181 | CUTE: constrained and unconstrained testing environment
- Bongartz, Conn, et al.
- 1995
(Show Context)
Citation Context ...tion 3 and 14 P. E. GILL, W. MURRAY AND M. A. SAUNDERS Table 6.1 Sets of test problems. Problems Reference bt Boggs and Tolle [4, 5] hs Hock and Schittkowski [34] CUTE Bongartz, Conn, Gould and Toint =-=[8]-=- Spring Murtagh and Saunders [40] Min-time Hargraves and Paris [32] Table 6.2 Notation in tables of results. QP The problem is a quadratic program. LC The objective is nonlinear but the constraints ar... |

158 |
Methods for modifying matrix factorizations
- Gill, Golub, et al.
- 1974
(Show Context)
Citation Context ...here w = RQ ffi Q (see Goldfarb [30], Dennis and Schnabel [15]). This rank-one modification of RQ could be restored to upper-triangular form by applying two sequences of plane rotations from the left =-=[21]. To-=- simplify the notation we write (5.2) as RQ +uv T , where RQ is an �� n \Theta n uppertrapezoidal matrix, u = w=kwk and v = p ` k yQ \Gamma R T Q u. Let v Z be the first n Z elements of v. The fol... |

158 | Representations of quasi-Newton matrices and their use in limited memory methods, Mathematical Programming 63
- Byrd, Nocedal, et al.
- 1994
(Show Context)
Citation Context ...e previous updates. (Similar schemes are described by Buckley and LeNir [17, 18] and Gilbert and Lemaréchal [49]. More elaborate schemes are given by Liu and Nocedal [70], Byrd, Nocedal, and Schnabel =-=[24]-=-, and Gill and Leonard [51], and some have been evaluated by Morales [71]. However, as already indicated, these schemes would require refactorization of the reduced Hessian in the linearly constrained... |

152 | A trust region method based on interior point techniques for nonlinear programming
- Byrd, Gilbert, et al.
(Show Context)
Citation Context ...able 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 [12, 13]. LOQO [100], KNITRO =-=[21, 20]-=-, and IPOPT [101] are examples of large-scale optimization packages that treat inequality constraints by a primal-dual interior method. They require second derivatives but can accommodate many degrees... |

148 |
NLPQL: A FORTRAN Subroutine for Solving Constrained Nonlinear Programming Problems,” ed
- Schittkowski
- 1986
(Show Context)
Citation Context ...rvey of recent results, see Gould and Toint [65]. Several general-purpose SQP solvers have proved reliable and efficient during the last 20 years. For example, under mild conditions the solvers NLPQL =-=[92]-=-, NPSOL [57, 60], and DONLP [95] typically find a (local) optimum from an arbitrary starting point, and they require relatively few evaluations of the problem functions and gradients compared to tradi... |

136 |
Direct Trajectory Optimization Using Nonlinear Programming and Collocation
- Hargraves, Paris
- 1987
(Show Context)
Citation Context ...rge (say, less than 1000). Important examples are control problems such as those arising in optimal trajectory calculations. For several years, the optimal trajectory system OTIS (Hargraves and Paris =-=[32]-=-) has been applied successfully within the aerospace industry, using NPSOL to solve the associated optimization problems. Although NPSOL has solved examples with over a thousand variables and constrai... |

133 |
CUTE: constrained and unconstrained testing environment
- Toint
- 1995
(Show Context)
Citation Context ...as well as translation into C via f 2c [37] (though these features are not used here). We give the results of applying SNOPT 7.1 of January 2005 to problems in the CUTEr and COPS 3.0 test collections =-=[11, 10, 31, 33]-=-. Function and gradient values were used throughout (but not second derivatives). All runs were made on a Linux PC with 2GB of RAM and two 3.06GHz Xeon processors (only one being used for each problem... |

114 |
Some numerical experiments with variable-storage quasi-Newton algorithms
- Gilbert, Lemaréchal
- 1989
(Show Context)
Citation Context ...rm the next positive-definite H r (with r = k + 1). Storage is then "reset" by discarding the previous updates. (Similar schemes are suggested by Buckley and LeNir [9, 10] and Gilbert and Le=-=mar'echal [20].) If �� n ! n, -=-H k has the form (3.1) and the same procedure is applied to �� H k . Note that the vectors y j and q j have length �� n---a benefit when �� n �� n. The modified Lagrangian LA (2.7) ret... |

113 |
User’s Guide for NPSOL (Version 4.0): a Fortran package for nonlinear programming
- Gill, Murray, et al.
- 1986
(Show Context)
Citation Context ...nt results, see Gould and Toint [65]. Several general-purpose SQP solvers have proved reliable and efficient during the last 20 years. For example, under mild conditions the solvers NLPQL [92], NPSOL =-=[57, 60]-=-, and DONLP [95] typically find a (local) optimum from an arbitrary starting point, and they require relatively few evaluations of the problem functions and gradients compared to traditional solvers s... |

109 |
LANCELOT: A Fortran Package for LargeScale Nonlinear Optimization
- Toint
- 1992
(Show Context)
Citation Context ...tabilized form of MINOS named Knossos has recently been developed [47] (it makes use of MINOS or SNOPT as subproblem solvers), and CONOPT version 3 [2] is now able to use second derivatives. LANCELOT =-=[27, 64]-=- is another widely used package for large-scale constrained optimization. It uses a bound constrained augmented Lagrangian method, is effective with either first or second derivatives, and is suitable... |

108 | Large–scale linearly constrained optimization
- Murtagh, Saunders
- 1978
(Show Context)
Citation Context ...95] typically find a (local) optimum from an arbitrary starting point, and they require relatively few evaluations of the problem functions and gradients compared to traditional solvers such as MINOS =-=[75, 76, 77]-=- and CONOPT [34, 2]. SQP methods have been particularly successful in solving the optimization problems arising in optimal trajectory calculations. For many years, the optimal trajectory system OTIS (... |

105 |
LANCELOT: a Fortran package for large-scale nonlinear optimization (Release A
- Conn, Gould, et al.
- 1992
(Show Context)
Citation Context ...te. Relative to MINOS, an added advantage is the existence of a merit function to ensure global convergence. This is especially important when the constraints are highly nonlinear. LANCELOT Release A =-=[14]-=- is another widely used package in the area of largescale constrained optimization. It uses a sequential augmented Lagrangian (SAL) method. All constraints other than simple bounds are included in an ... |

93 |
An interior point algorithm for large-scale nonlinear programming
- Byrd, Hribar, et al.
- 1999
(Show Context)
Citation Context ...able 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 [12, 13]. LOQO [100], KNITRO =-=[21, 20]-=-, and IPOPT [101] are examples of large-scale optimization packages that treat inequality constraints by a primal-dual interior method. They require second derivatives but can accommodate many degrees... |

88 |
A Projected Lagrangian Algorithm and Its Implementation for Sparse Nonlinear
- Murtagh, Saunders
- 1982
(Show Context)
Citation Context ...95] typically find a (local) optimum from an arbitrary starting point, and they require relatively few evaluations of the problem functions and gradients compared to traditional solvers such as MINOS =-=[75, 76, 77]-=- and CONOPT [34, 2]. SQP methods have been particularly successful in solving the optimization problems arising in optimal trajectory calculations. For many years, the optimal trajectory system OTIS (... |

86 |
Trust–Region Methods
- Toint
- 2000
(Show Context)
Citation Context ...timization. The approach was popularized mainly by Biggs [7], Han [66], and Powell [85, 87] for general nonlinear constraints. Further history of SQP methods and extensive bibliographies are given in =-=[61, 39, 73, 78, 28]-=-. For a survey of recent results, see Gould and Toint [65]. Several general-purpose SQP solvers have proved reliable and efficient during the last 20 years. For example, under mild conditions the solv... |

77 |
The Convergence of Variable Metric Methods for Nonlinearly Constrained Optimization Calculations, in Nonlinear Programming 3
- Powell
- 1978
(Show Context)
Citation Context ... solution. 1.2. The SQP Approach. An SQP method was first suggested by Wilson [102] for the special case of convex optimization. The approach was popularized mainly by Biggs [7], Han [66], and Powell =-=[85, 87]-=- for general nonlinear constraints. Further history of SQP methods and extensive bibliographies are given in [61, 39, 73, 78, 28]. For a survey of recent results, see Gould and Toint [65]. Several gen... |

73 |
Test Examples for Nonlinear Programming
- Hock, Schittkowski
(Show Context)
Citation Context ...nal parameters were used, such as Crash option 3 and 14 P. E. GILL, W. MURRAY AND M. A. SAUNDERS Table 6.1 Sets of test problems. Problems Reference bt Boggs and Tolle [4, 5] hs Hock and Schittkowski =-=[34]-=- CUTE Bongartz, Conn, Gould and Toint [8] Spring Murtagh and Saunders [40] Min-time Hargraves and Paris [32] Table 6.2 Notation in tables of results. QP The problem is a quadratic program. LC The obje... |

71 |
MINOS 5.5 User’s Guide
- Murtagh, Saunders
- 1998
(Show Context)
Citation Context ...95] typically find a (local) optimum from an arbitrary starting point, and they require relatively few evaluations of the problem functions and gradients compared to traditional solvers such as MINOS =-=[75, 76, 77]-=- and CONOPT [34, 2]. SQP methods have been particularly successful in solving the optimization problems arising in optimal trajectory calculations. For many years, the optimal trajectory system OTIS (... |

69 | An SQP method for general nonlinear programs using only equality constrained subproblems
- Spellucci
- 1998
(Show Context)
Citation Context ...d and Toint [65]. Several general-purpose SQP solvers have proved reliable and efficient during the last 20 years. For example, under mild conditions the solvers NLPQL [92], NPSOL [57, 60], and DONLP =-=[95]-=- typically find a (local) optimum from an arbitrary starting point, and they require relatively few evaluations of the problem functions and gradients compared to traditional solvers such as MINOS [75... |

64 | CUTEr and SifDec: A constrained and unconstrained testing environment, revisited - Toint |

59 |
Superlinearly convergent variable metric algorithms for general nonlinear programming problems
- Han
- 1976
(Show Context)
Citation Context ...ion exists, or if k!k is very large, no update is made. The approach just described is related to the idea of updating an approximation of the Hessian of the augmented Lagrangian, as suggested by Han =-=[31]-=- and Tapia [49]. However, we emphasize that the second modification is not required in the neighborhood of a solution because as x ! x , r 2 LA converges to r 2 L and the first modification will alrea... |

57 |
CONOPT: A GRG code for large sparse dynamic nonlinear optimization problems
- Drud
- 1985
(Show Context)
Citation Context ...orks with sparse KKT factorizations (see x7), the method is not restricted to problems with few degrees of freedom. 1.4. Other large-scale methods. Two existing packages MINOS [39, 40, 41] and CONOPT =-=[16]-=- are designed for large problems with a modest number of degrees of freedom. MINOS uses a projected Lagrangian or sequential linearly constrained (SLC) method, whose subproblems require frequent evalu... |

56 |
MA28 a set of Fortran subroutines for sparse unsymmetric matrices. Harwell Report AERE R.8730
- Duff
- 1977
(Show Context)
Citation Context ...j| ≤τL (1 <τL ≤ 100 say), Al = the remaining matrix to be factored after l steps. For most factorizations, LUSOL uses a threshold partial pivoting (TPP) strategy similar to that in LA05 [88] and MA28 =-=[35]-=-. A classical Markowitz criterion is used to choose an entry apq from a sparse row and column of Al to become the next pivot element (the next diagonal of U). To be acceptable, apq must be sufficientl... |

56 |
Algorithms for nonlinear constraints that use Lagrangian functions
- Powell
- 1978
(Show Context)
Citation Context ...s very large), the objective is considered “unbounded” in the expanded region. Elastic mode is entered (or continued) as described in section 4.7. 2.9. The Approximate Hessian. As suggested by Powell =-=[86]-=-, we maintain a positive-definite approximate Hessian Hk. On completion of the line search, let the change in x and the gradient of the modified Lagrangian be δk = xk+1 − xk and yk = ∇L(xk+1,xk,π) −∇L... |

47 | On the implementation of an algorithm for large-scale equality constrained optimization
- Lalee, Nocedal, et al.
- 1998
(Show Context)
Citation Context ...P methods to the largescale case (sometimes using exact second derivatives). Some of this work has focused on problems with nonlinear equality constraints. The method of Lalee, Nocedal, and Plantenga =-=[69]-=-, related to the trust-region method of Byrd [19] and Omojokun [79], uses either the exact Lagrangian Hessian or a limited-memory quasi-Newton approximation defined by the method of Zhu et al. [103]. ... |

46 | A reduced Hessian method for large-scale constrained optimization
- BIEGLER, NOCEDAL, et al.
- 1995
(Show Context)
Citation Context ...of Byrd and Omojokun [42], uses either the exact Lagrangian Hessian or a limited-memory quasi-Newton approximation defined by the method of Zhu et al. [52]. The method of Biegler, Nocedal and Schmidt =-=[3]-=- is in the class of reduced-Hessian methods , which maintain a dense approximation to the reduced Hessian, using quasi-Newton updates. For large problems with general inequality constraints as in Prob... |

44 |
Maintaining LU factors of a general sparse matrix, Linear Algebra and its
- Gill, Murray, et al.
- 1987
(Show Context)
Citation Context ...he one-norm of the constraint violations. 5. Basis Handling in SQOPT. The null-space methods in sections 4.3–4.5 require frequent solution of systems involving B and BT . SQOPT uses the package LUSOL =-=[58]-=- for four purposes: • to obtain sparse LU factors of a given basis B; • to replace certain columns of B when it appears singular or ill-conditioned; • to find a better-conditioned B by reordering the ... |

41 |
A sparse nonlinear optimization algorithm
- BETTS, FRANK
- 1994
(Show Context)
Citation Context ...ian, using quasi-Newton updates. For large problems with general inequality constraints as in Problem NP, SQP methods have been proposed by Eldersveld [17], Tjoa and Biegler [50], and Betts and Frank =-=[2]-=-. The first two approaches are also reduced-Hessian methods. In [17], a full but structured Hessian approximation is formed from the reduced Hessian. The implementation LSSQP solves the same class of ... |

40 |
A Sparsity-Exploiting Variant of the Bartels-Golub Decomposition for Linear Programming
- Reid
- 1975
(Show Context)
Citation Context ...quires replacement of a column of B, the LU factors must be updated in a stable way. LUSOL uses the approach suggested by Bartels and Golub [3]. The sparse implementation is analogous to that of Reid =-=[88, 89]-=-. 6. SQP Algorithm Details. A practical SQP algorithm requires many features to achieve reliability and efficiency. We discuss some more of them here before summarizing the main algorithmic steps. 6.1... |

38 | Inertia-controlling methods for general quadratic programming
- Gill, Murray, et al.
- 1991
(Show Context)
Citation Context ...d well, as already mentioned. 4.6. Inertia Control. If (NP) contains linear variables, Hk in (3.1) is positive semidefinite. In SQOPT, only the last diagonal of R in (4.4) is allowed to be zero. (See =-=[59]-=- for discussion of a similar strategy for indefinite QP.) If the initial R is singular, enough temporary constraints are added to the working set to give a nonsingular R. Thereafter, R can become sing... |

36 |
On the local convergence of quasi-Newton methods for constrained optimization
- Boggs, T, et al.
- 1982
(Show Context)
Citation Context ...d x1.4). The default MINOS optional parameters were used, such as Crash option 3 and 14 P. E. GILL, W. MURRAY AND M. A. SAUNDERS Table 6.1 Sets of test problems. Problems Reference bt Boggs and Tolle =-=[4, 5]-=- hs Hock and Schittkowski [34] CUTE Bongartz, Conn, Gould and Toint [8] Spring Murtagh and Saunders [40] Min-time Hargraves and Paris [32] Table 6.2 Notation in tables of results. QP The problem is a ... |

35 |
An `1 penalty method for nonlinear constraints
- Fletcher
- 1985
(Show Context)
Citation Context ...ere infinite, the nonlinear constraint violations would be minimized subject to the linear constraints and bounds.) A similar ` 1 formulation of NP is fundamental to the S` 1 QP algorithm of Fletcher =-=[18]-=-. See also Conn [12]. LARGE-SCALE SQP 3 1.3. Other work on large-scale SQP. There has been considerable interest elsewhere in extending SQP methods to the large-scale case. Some of this work has focus... |

35 |
GALAHAD—a library of thread-safe fortran 90 packages for large-scale nonlinear optimization
- Toint
(Show Context)
Citation Context ...tabilized form of MINOS named Knossos has recently been developed [47] (it makes use of MINOS or SNOPT as subproblem solvers), and CONOPT version 3 [2] is now able to use second derivatives. LANCELOT =-=[27, 64]-=- is another widely used package for large-scale constrained optimization. It uses a bound constrained augmented Lagrangian method, is effective with either first or second derivatives, and is suitable... |

30 | A sequential quadratic programming algorithm using an incomplete solution of the subproblem
- Murray, Prieto
- 1995
(Show Context)
Citation Context ...)}, where τV is a specified constant, e.g., τV = 10. This defines a region in which the objective is expected to be defined and bounded below. (A similar condition is used in [93].) Murray and Prieto =-=[74]-=- show that under certain conditions, convergence can be assured if the line search enforces (2.5). If the objective is bounded below in Rn , then b may be any large positive vector. If αk is essential... |

27 | A practical algorithm for general large scale nonlinear optimization problems
- Boggs, Kearsley, et al.
- 1999
(Show Context)
Citation Context ... the QP subproblems by an interior method. They typically require an exact or finite-difference Lagrangian Hessian but can accommodate many degrees of freedom. Examples are Boggs, Kearsley, and Tolle =-=[8, 9]-=- and Sargent and Ding [91]. 1.3. Other Large-Scale Methods. MINOS and versions 1 and 2 of CONOPT are reduced-Hessian methods for general large-scale optimization. Like SNOPT, they use first derivative... |

27 | Some Theoretical Properties of an Augmented Lagrangian Function
- Gill, Murray, et al.
- 1986
(Show Context)
Citation Context ...nt results, see Gould and Toint [65]. Several general-purpose SQP solvers have proved reliable and efficient during the last 20 years. For example, under mild conditions the solvers NLPQL [92], NPSOL =-=[57, 60]-=-, and DONLP [95] typically find a (local) optimum from an arbitrary starting point, and they require relatively few evaluations of the problem functions and gradients compared to traditional solvers s... |

26 |
An analysis of reduced Hessian methods for constrained optimization
- BYRD, NOCEDAL
- 1990
(Show Context)
Citation Context ...positive semi-definite at a minimizer of GNP. The assumption that the QP working set does not change once z k is known is always justified for problems with equality constraints (see Byrd and Nocedal =-=[11] for-=- a similar scheme in this context). With inequality constraints, we observe that W k pN �� 0, particularly during later major iterations, when the working set has settled down. This modification e... |

25 |
A Stabilization of the Simplex Method
- BARTELS
(Show Context)
Citation Context ... to be singular.) 5.4. Basis Updates. When a QP iteration requires replacement of a column of B, the LU factors must be updated in a stable way. LUSOL uses the approach suggested by Bartels and Golub =-=[3]-=-. The sparse implementation is analogous to that of Reid [88, 89]. 6. SQP Algorithm Details. A practical SQP algorithm requires many features to achieve reliability and efficiency. We discuss some mor... |

25 | 2005), `A globally convergent linearly constrained Lagrangian method for nonlinear optimization
- Friedlander, Saunders
(Show Context)
Citation Context ...rgence, and explicit provision for infeasible subproblems. This is especially important when the constraints are highly nonlinear. A stabilized form of MINOS named Knossos has recently been developed =-=[47]-=- (it makes use of MINOS or SNOPT as subproblem solvers), and CONOPT version 3 [2] is now able to use second derivatives. LANCELOT [27, 64] is another widely used package for large-scale constrained op... |

24 |
QN-like variable storage conjugate gradients
- BUCKLEY, LENIR
- 1983
(Show Context)
Citation Context ... computed from (3.2) and saved to form the next positive-definite H r (with r = k + 1). Storage is then "reset" by discarding the previous updates. (Similar schemes are suggested by Buckley =-=and LeNir [9, 10] and Gilbert and Lem-=-ar'echal [20].) If �� n ! n, H k has the form (3.1) and the same procedure is applied to �� H k . Note that the vectors y j and q j have length �� n---a benefit when �� n �� n. The... |

22 |
Robust trust region methods for constrained optimization
- Byrd
- 1987
(Show Context)
Citation Context ... exact second derivatives). Some of this work has focused on problems with nonlinear equality constraints. The method of Lalee, Nocedal, and Plantenga [69], related to the trust-region method of Byrd =-=[19]-=- and Omojokun [79], uses either the exact Lagrangian Hessian or a limited-memory quasi-Newton approximation defined by the method of Zhu et al. [103]. The method of Biegler,s102 PHILIP E. GILL,WALTER ... |

21 | COPS: Large-scale nonlinearly constrained optimization problems
- Bondarenko, Bortz, et al.
- 1998
(Show Context)
Citation Context ...as well as translation into C via f 2c [37] (though these features are not used here). We give the results of applying SNOPT 7.1 of January 2005 to problems in the CUTEr and COPS 3.0 test collections =-=[11, 10, 31, 33]-=-. Function and gradient values were used throughout (but not second derivatives). All runs were made on a Linux PC with 2GB of RAM and two 3.06GHz Xeon processors (only one being used for each problem... |

21 |
FORTRAN Subroutines for Handling Sparse Linear Programming Bases," Report AERE-R8269, Computer Science and Systems Division, AERE
- Reid
- 1976
(Show Context)
Citation Context ... such that |Lij| ≤τL (1 <τL ≤ 100 say), Al = the remaining matrix to be factored after l steps. For most factorizations, LUSOL uses a threshold partial pivoting (TPP) strategy similar to that in LA05 =-=[88]-=- and MA28 [35]. A classical Markowitz criterion is used to choose an entry apq from a sparse row and column of Al to become the next pivot element (the next diagonal of U). To be acceptable, apq must ... |

20 |
Large-scale sequential quadratic programming algorithms
- Eldersveld
- 1992
(Show Context)
Citation Context ...aintain a dense approximation to the reduced Hessian, using quasi-Newton updates. For large problems with general inequality constraints as in Problem NP, SQP methods have been proposed by Eldersveld =-=[17]-=-, Tjoa and Biegler [50], and Betts and Frank [2]. The first two approaches are also reduced-Hessian methods. In [17], a full but structured Hessian approximation is formed from the reduced Hessian. Th... |

20 | User manual for filterSQP
- Fletcher, Leyffer
- 1998
(Show Context)
Citation Context ..., using quasi-Newton updates. For large problems with general inequality constraints as in problem (NP), SQP methods have been proposed by Eldersveld [36], Tjoa and Biegler [97], Fletcher and Leyffer =-=[40, 41]-=-, and Betts and Frank [5]. The first three approaches are also reducedHessian methods. Eldersveld forms a full Hessian approximation from the reduced Hessian, and his implementation LSSQP solves the s... |

20 |
A simplex algorithm for piecewise-linear programming III: Computational analysis and applications
- Fourer
- 1989
(Show Context)
Citation Context ... than the Minor feasibility tolerance, it is given a linear penalty term γ × infeasibility in the objective function. This is a much-simplified but useful form of piecewise linear programming (Fourer =-=[42, 43, 44]-=-). SNOPT uses elastic bounds in three different ways: • to solve problem (FLP) (section 1.1) if the linear constraints are infeasible; • to solve problem (PP1) (section 6.1); • to solve the QP subprob... |

19 |
Benchmarking optimization software with
- Dolan, Moré
(Show Context)
Citation Context ...as well as translation into C via f 2c [37] (though these features are not used here). We give the results of applying SNOPT 7.1 of January 2005 to problems in the CUTEr and COPS 3.0 test collections =-=[11, 10, 31, 33]-=-. Function and gradient values were used throughout (but not second derivatives). All runs were made on a Linux PC with 2GB of RAM and two 3.06GHz Xeon processors (only one being used for each problem... |

19 |
Lemaréchal, C.: Some numerical experiments with variable-storage quasi-Newton algorithms
- Gilbert
- 1989
(Show Context)
Citation Context ...orm the next positive-definite Hr (with r = k + 1). Storage is then “reset” by discarding the previous updates. (Similar schemes are described by Buckley and LeNir [17, 18] and Gilbert and Lemaréchal =-=[49]-=-. More elaborate schemes are given by Liu and Nocedal [70], Byrd, Nocedal, and Schnabel [24], and Gill and Leonard [51], and some have been evaluated by Morales [71]. However, as already indicated, th... |

18 | Sparse matrix methods in optimization - Gill, Murray, et al. - 1984 |

18 |
A Fortran-to-C Converter. Computing Science
- Feldman, Gay, et al.
- 1991
(Show Context)
Citation Context ...tions 2–6. The Fortran 77 coding is compatible with Fortran 90 and 95 compilers and permits recursive calls, or re-entrant calls in a multithreaded environment, as well as translation into C via f 2c =-=[37]-=- (though these features are not used here). We give the results of applying SNOPT 7.1 of January 2005 to problems in the CUTEr and COPS 3.0 test collections [11, 10, 31, 33]. Function and gradient val... |

17 |
Constrained optimization using a nondifferentiable penalty function
- Conn
- 1973
(Show Context)
Citation Context ...nlinear constraint violations would be minimized subject to the linear constraints and bounds.) A similar ` 1 formulation of NP is fundamental to the S` 1 QP algorithm of Fletcher [18]. See also Conn =-=[12]-=-. LARGE-SCALE SQP 3 1.3. Other work on large-scale SQP. There has been considerable interest elsewhere in extending SQP methods to the large-scale case. Some of this work has focused on problems with ... |

17 |
Constrained minimization using recursive equality quadratic programming
- Biggs
- 1972
(Show Context)
Citation Context ...rees of freedom at each QP solution. 1.2. The SQP Approach. An SQP method was first suggested by Wilson [102] for the special case of convex optimization. The approach was popularized mainly by Biggs =-=[7]-=-, Han [66], and Powell [85, 87] for general nonlinear constraints. Further history of SQP methods and extensive bibliographies are given in [61, 39, 73, 78, 28]. For a survey of recent results, see Go... |

17 |
Elastic Programming: A New Approach to Large-Scale Mixed-Integer Optimizatiion
- Brown, Graves
- 1975
(Show Context)
Citation Context ...mplicitly. This is equivalent to minimizing the one-norm of the general linear constraint violations subject to the simple bounds—often called elastic programming in the linear programming literature =-=[15]-=-. Elastic programming has long been a feature of the XS system of Brown and Graves [16]. Other algorithms based on minimizing one-norms of infeasibilities are given by Conn [26] and Bartels [4]. If th... |

14 |
A quadratically-convergent algorithm for general nonlinear programming problems
- Robinson
- 1972
(Show Context)
Citation Context ...L(x, xk,πk) =f(x) − π T k d L(x, xk), which is defined in terms of the constraint linearization and the departure from linearity: cL(x, xk) =ck + Jk(x − xk), dL(x, xk) =c(x) − cL(x, xk); see Robinson =-=[90]-=- and Van der Hoek [99]. The first and second derivatives of the modified Lagrangian with respect to x are ∇L(x, xk,πk) =g(x) − (J(x) − Jk) T πk, ∇ 2 L(x, xk,πk) =∇ 2 f(x) − � i (πk)i∇ 2 ci(x).s104 PHI... |

11 |
Factorized variable metric methods for unconstrained optimization
- GOLDFARB
- 1976
(Show Context)
Citation Context ...e to RQ implied by (5.1). Given ffi k and y k , if we had the Cholesky factor RQ , it could be updated directly as RQ + w kwk ` p ` k yQ \Gamma R T Q w kwk 'T ; (5.2) where w = RQ ffi Q (see Goldfarb =-=[30]-=-, Dennis and Schnabel [15]). This rank-one modification of RQ could be restored to upper-triangular form by applying two sequences of plane rotations from the left [21]. To simplify the notation we wr... |

11 |
Optimization for
- Wright
- 2012
(Show Context)
Citation Context |

11 | Sqp methods for large-scale nonlinear programming
- Gould, Toint
- 2000
(Show Context)
Citation Context ...d Powell [85, 87] for general nonlinear constraints. Further history of SQP methods and extensive bibliographies are given in [61, 39, 73, 78, 28]. For a survey of recent results, see Gould and Toint =-=[65]-=-. Several general-purpose SQP solvers have proved reliable and efficient during the last 20 years. For example, under mild conditions the solvers NLPQL [92], NPSOL [57, 60], and DONLP [95] typically f... |

9 |
Rank-one and rank-two corrections to positive definite matrices expressed in product form
- Brodlie, Gourlay, et al.
- 1973
(Show Context)
Citation Context ...merically to write Hk in the form Hk = G T k Gk, where Gk is the product of elementary matrices (3.3) Gk = H 1/2 r (I + δrv T r )(I + δr+1v T r+1) ···(I + δk−1v T k−1), with vj = ±(θjφj) 1/2 yj −φjqj =-=[14]-=-. (The sign may be chosen to minimize the rounding error in computing vj.) The quantities (δj,vj) are stored for each j. During major iteration k, the QP solver accesses Hk by requesting products of t... |

8 |
1981]. A new derivation of symmetric positive definite secant updates, Nonlinear Programming 4
- SCHNABEL
(Show Context)
Citation Context ...Given ffi k and y k , if we had the Cholesky factor RQ , it could be updated directly as RQ + w kwk ` p ` k yQ \Gamma R T Q w kwk 'T ; (5.2) where w = RQ ffi Q (see Goldfarb [30], Dennis and Schnabel =-=[15]-=-). This rank-one modification of RQ could be restored to upper-triangular form by applying two sequences of plane rotations from the left [21]. To simplify the notation we write (5.2) as RQ +uv T , wh... |

8 | Limited-memory reduced-Hessian methods for large-scale unconstrained optimization
- Gill, Leonard
- 2000
(Show Context)
Citation Context ...r schemes are described by Buckley and LeNir [17, 18] and Gilbert and Lemaréchal [49]. More elaborate schemes are given by Liu and Nocedal [70], Byrd, Nocedal, and Schnabel [24], and Gill and Leonard =-=[51]-=-, and some have been evaluated by Morales [71]. However, as already indicated, these schemes would require refactorization of the reduced Hessian in the linearly constrained case.) An alternative appr... |

8 |
A new technique for inconsistent QP problems
- Spellucci
- 1998
(Show Context)
Citation Context ...he linear constraints and bounds.) A similar ℓ1 formulation of (NP) is used in the SQP method of Tone [98] and is fundamental to the Sℓ1QP algorithm of Fletcher [38]. See also Conn [25] and Spellucci =-=[94]-=-. An attractive feature is that only linear terms are added to (NP), giving no increase in the expected degrees of freedom at each QP solution. 1.2. The SQP Approach. An SQP method was first suggested... |

7 | A numerical comparison between the LANCELOT and MINOS packages for large-scale nonlinear optimization: The complete results
- Bongartz, Conn, et al.
- 1997
(Show Context)
Citation Context ... 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 =-=[6, 7]-=-. 2. The SQP iteration. Here we discuss the main features of an SQP method for solving a generic nonlinear program. All features are readily specialized to the more general constraints in Problem NP. ... |

7 |
BBVSCG—A variable storage algorithm for function minimization
- Buckley, LeNir
- 1985
(Show Context)
Citation Context ...eration k = r + ℓ, these diagonals form the next positive-definite Hr (with r = k + 1). Storage is then “reset” by discarding the previous updates. (Similar schemes are described by Buckley and LeNir =-=[17, 18]-=- and Gilbert and Lemaréchal [49]. More elaborate schemes are given by Liu and Nocedal [70], Byrd, Nocedal, and Schnabel [24], and Gill and Leonard [51], and some have been evaluated by Morales [71]. H... |

6 |
A numerical comparison between the LANCELOT and MINOS packages for large-scale constrained optimization
- Toint
- 1997
(Show Context)
Citation Context ...cond derivatives, and is suitable 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 =-=[12, 13]-=-. LOQO [100], KNITRO [21, 20], and IPOPT [101] are examples of large-scale optimization packages that treat inequality constraints by a primal-dual interior method. They require second derivatives but... |

6 |
AMPL: A Modeling Language for
- Kernighan
- 1993
(Show Context)
Citation Context ...problem and the QP subproblems. 7.3. Results on the COPS 3.0 Test Set. Next we describe tests on the 22 problems in the COPS 3.0 test collection [10, 31, 32] implemented in the AMPL modeling language =-=[45, 46, 1]-=-. The dimension of a particular instance of a COPS problem is determined by one or more parameters assigned in its AMPL data file. For each of the 22 COPS problems, [33] gives the results of several o... |

5 |
Sequential quadratic programming methods for large problems
- Murray
- 1997
(Show Context)
Citation Context |

5 |
Sparse rank-revealing LU factorization (via threshold complete pivoting and threshold rook pivoting
- O’SULLIVAN, SAUNDERS
- 2002
(Show Context)
Citation Context ...etection during the optimization of Markov decision chains [80] and within SNOPT [54]. In some cases its strict pivot test leads to rather dense LU factors. The TRP option is typically more efficient =-=[82]-=- and in practice its rank-revealing properties are essentially as good as for TCP. (Note that all options can fail to detect near-singularity in certain matrices with regular structure. A classic exam... |

4 |
guide for NPSOL (Version 4.0): A FORTRAN package for nonlinear programming
- User’s
- 1986
(Show Context)
Citation Context ...linear and nonlinear inequality constraints. SQP methods have proved reliable and efficient for many such problems. For example, under mild conditions the general-purpose solvers NLPQL [48] and NPSOL =-=[25, 28]-=- typically find a (local) optimum from an arbitrary starting point, and they require relatively few evaluations of the problem functions and gradients compared to other solvers. 1.1. The optimization ... |

4 |
Linear programming via a nondifferentiable penalty function
- Conn
- 1976
(Show Context)
Citation Context ...programming literature [15]. Elastic programming has long been a feature of the XS system of Brown and Graves [16]. Other algorithms based on minimizing one-norms of infeasibilities are given by Conn =-=[26]-=- and Bartels [4]. If the linear constraints are infeasible (v �= 0 or w �= 0), SNOPT terminates without computing the nonlinear functions. Otherwise, all subsequent iterates satisfy the linear constra... |

3 |
A penalty linear programming method using reduced-gradient basisexchange techniques, Linear Algebra and its Applics
- Bartels
- 1980
(Show Context)
Citation Context ... bounds. (In the linear programming literature, the approach is often called elastic programming . Other algorithms based on minimizing one-norms of infeasibilities are given by Conn [13] and Bartels =-=[1]-=-.) If the linear constraints are infeasible (v 6= 0 or w 6= 0), SNOPT terminates without computing the nonlinear functions. Otherwise, all subsequent iterates satisfy the linear constraints. (As with ... |

3 |
The computation of Lagrange multiplier estimates for constrained minimization
- Gill, Murray
- 1979
(Show Context)
Citation Context ...ences of a negative or small value of y T k ffi k are discussed in the next section. There are several choices for ��, including the QP multipliers b �� k+1 and leastsquares multiplierssk (see=-=, e.g., [22]). H-=-ere we use the updated multipliers �� k+1 from the line search, because they are responsive to short steps in the search and they are available at no cost. The definition of L (2.2) yields y k = r... |

3 |
SQOPT: An algorithm for largescale quadratic programming
- Gill, Murray, et al.
(Show Context)
Citation Context ...e infeasible, we redefine QP k (and all subsequent subproblems) to correspond to the linearization of NP(fl). SNOPT is then in elastic mode thereafter. At present, QP k is solved by the package SQOPT =-=[23]-=-, which employs a two-phase active-set algorithm and implements elastic programming implicitly when necessary. SQOPT can treat any of the variable and constraint bounds as elastic, but SNOPT uses this... |

3 |
A numerical study of limited memory
- Morales
- 2002
(Show Context)
Citation Context ...17, 18] and Gilbert and Lemaréchal [49]. More elaborate schemes are given by Liu and Nocedal [70], Byrd, Nocedal, and Schnabel [24], and Gill and Leonard [51], and some have been evaluated by Morales =-=[71]-=-. However, as already indicated, these schemes would require refactorization of the reduced Hessian in the linearly constrained case.) An alternative approach is to store the quantities (yj,qj,θj,φj) ... |

3 |
Trust Region Algorithms for Nonlinear Equality and Inequality Constraints
- Omojokun
(Show Context)
Citation Context ...vatives). Some of this work has focused on problems with nonlinear equality constraints. The method of Lalee, Nocedal, and Plantenga [69], related to the trust-region method of Byrd [19] and Omojokun =-=[79]-=-, uses either the exact Lagrangian Hessian or a limited-memory quasi-Newton approximation defined by the method of Zhu et al. [103]. The method of Biegler,s102 PHILIP E. GILL,WALTER MURRAY, AND MICHAE... |

3 |
Han's method without solving QP
- Spellucci
- 1981
(Show Context)
Citation Context ...use bi = τV max{1, −ci(x0)}, where τV is a specified constant, e.g., τV = 10. This defines a region in which the objective is expected to be defined and bounded below. (A similar condition is used in =-=[93]-=-.) Murray and Prieto [74] show that under certain conditions, convergence can be assured if the line search enforces (2.5). If the objective is bounded below in Rn , then b may be any large positive v... |

2 |
SNOPT: An SQP Algorithme for large-scale constrained optimization
- Saunders
- 2005
(Show Context)
Citation Context ...ll nonzeros in Al: |apq| ≥maxi,j |aij|/τL. The TCP option was implemented first [81] and has proved valuable for rank-detection during the optimization of Markov decision chains [80] and within SNOPT =-=[54]-=-. In some cases its strict pivot test leads to rather dense LU factors. The TRP option is typically more efficient [82] and in practice its rank-revealing properties are essentially as good as for TCP... |

2 |
andJ. Nocedal, Automatic preconditioning by limited memory quasi-Newton updating
- Morales
(Show Context)
Citation Context ...ubspace. A frozen superbasic variable is then interchanged with one from Rr, and the process is repeated. In SQOPT, our aim has been to use R in (4.6) as a preconditioner for (4.5) [61, pp. 151–153], =-=[72]-=-. To date, this has produced mixed results, but fortunately the CG option without preconditioning has performed well, as already mentioned. 4.6. Inertia Control. If (NP) contains linear variables, Hk ... |

2 |
New Methods for Dynamic Programming over an Infinite Time Horizon
- O’Sullivan
- 2001
(Show Context)
Citation Context ...ly large compared to all nonzeros in Al: |apq| ≥maxi,j |aij|/τL. The TCP option was implemented first [81] and has proved valuable for rank-detection during the optimization of Markov decision chains =-=[80]-=- and within SNOPT [54]. In some cases its strict pivot test leads to rather dense LU factors. The TRP option is typically more efficient [82] and in practice its rank-revealing properties are essentia... |

2 |
A new SQP algorithm for large-scale nonlinear programming
- Sargent, Ding
(Show Context)
Citation Context ...terior method. They typically require an exact or finite-difference Lagrangian Hessian but can accommodate many degrees of freedom. Examples are Boggs, Kearsley, and Tolle [8, 9] and Sargent and Ding =-=[91]-=-. 1.3. Other Large-Scale Methods. MINOS and versions 1 and 2 of CONOPT are reduced-Hessian methods for general large-scale optimization. Like SNOPT, they use first derivatives and were originally desi... |

1 |
An implementation of a quasi-Newton method for constrained optimization
- Boggs, Tolle
- 1981
(Show Context)
Citation Context ...d x1.4). The default MINOS optional parameters were used, such as Crash option 3 and 14 P. E. GILL, W. MURRAY AND M. A. SAUNDERS Table 6.1 Sets of test problems. Problems Reference bt Boggs and Tolle =-=[4, 5]-=- hs Hock and Schittkowski [34] CUTE Bongartz, Conn, Gould and Toint [8] Spring Murtagh and Saunders [40] Min-time Hargraves and Paris [32] Table 6.2 Notation in tables of results. QP The problem is a ... |

1 |
programming via a nondifferentiable penalty function
- Linear
- 1976
(Show Context)
Citation Context ...ect to the simple bounds. (In the linear programming literature, the approach is often called elastic programming . Other algorithms based on minimizing one-norms of infeasibilities are given by Conn =-=[13]-=- and Bartels [1].) If the linear constraints are infeasible (v 6= 0 or w 6= 0), SNOPT terminates without computing the nonlinear functions. Otherwise, all subsequent iterates satisfy the linear constr... |

1 |
Optimal Trajectories by Implicit Integration
- OTIS
- 1988
(Show Context)
Citation Context ... full code optimization. Table 6.10 gives results for 9 optimization problems, each involving a finer level of discretization of the underlying continuous problem. The problems were generated by OTIS =-=[33]-=-, and the constraint gradients are approximated by a sparse finite-difference scheme. In the table, SNOPT is compared with the code NZOPT, which implements an SQP method based on a dense reduced Hessi... |

1 |
A global convergence analysis of an algorithm for large-scale nonlinear optimization problems
- Tolle
- 1999
(Show Context)
Citation Context ... the QP subproblems by an interior method. They typically require an exact or finite-difference Lagrangian Hessian but can accommodate many degrees of freedom. Examples are Boggs, Kearsley, and Tolle =-=[8, 9]-=- and Sargent and Ding [91]. 1.3. Other Large-Scale Methods. MINOS and versions 1 and 2 of CONOPT are reduced-Hessian methods for general large-scale optimization. Like SNOPT, they use first derivative... |

1 |
The XS Mathematical Programming System, working paper
- Graves
- 1975
(Show Context)
Citation Context ...nt violations subject to the simple bounds—often called elastic programming in the linear programming literature [15]. Elastic programming has long been a feature of the XS system of Brown and Graves =-=[16]-=-. Other algorithms based on minimizing one-norms of infeasibilities are given by Conn [26] and Bartels [4]. If the linear constraints are infeasible (v �= 0 or w �= 0), SNOPT terminates without comput... |

1 |
Nocedal, An analysis of reduced Hessian methods for constrained optimization
- ByrdandJ
- 1991
(Show Context)
Citation Context ...ositive semidefinite at a minimizer of (GNP). The assumption that the QP working set does not change once zk is known is always justified for problems with equality constraints. (See Byrd and Nocedal =-=[23]-=- for a similar scheme in this context.) With inequality constraints, we observe that WkpN ≈ 0, particularly during later major iterations, when the working set has settled down. This modification expl... |

1 |
Methods for modifying matrix factorizations
- Saunders
- 1974
(Show Context)
Citation Context ...θky Q − R T Qu (see Goldfarb [62] and Dennis and Schnabel [30]). This rank-one modification of RQ could be restored to upper-triangular form by applying two sequences of plane rotations from the left =-=[50]-=-. The same sequences of rotations can be generated even though not all of RQ is present. Let vZ be the first nZ elements of v. The following algorithm determines the Cholesky factor ¯ R of the first n... |

1 |
User’s Guide for SQOPT 5.3: A Fortran Package for Large-Scale Linear and Quadratic Programming, Numerical Analysis Report
- Saunders
- 1997
(Show Context)
Citation Context ... replaced by its current linearization: (QPk) minimize fk + g x T 1 k (x − xk)+ 2 (x − xk) THk(x − xk) ⎛ ⎞ x subject to l ≤ ⎝ck + Jk(x − xk) ⎠ ≤ u. Ax At present, (QPk) is solved by the package SQOPT =-=[53]-=-, which employs a two-phase active-set algorithm and implements elastic programming implicitly when necessary. The Hessian Hk may be positive semidefinite and is defined by a routine for forming produ... |

1 |
User’s Guide for SNOPT 7.1: A Fortran Package for Large-Scale Nonlinear Programming, Numerical Analysis Report
- Saunders
- 2004
(Show Context)
Citation Context ...are linearizations of the constraints in the original problem, and the objective function of the subproblem is a quadratic approximation to the Lagrangian function. SNOPT (Sparse Nonlinear OPTimizer) =-=[55]-=- is the implementation of a particular SQP algorithm that exploits sparsity in the constraint Jacobian and maintains a limited-memory quasi-Newton approximation Hk to the Hessian of the Lagrangian. A ... |

1 |
FORTRANcodes for estimating the one-norm of a real or complex matrix, with applications to condition estimation
- Higham
- 1988
(Show Context)
Citation Context ...ors display no small diagonals in U, yet the BR factors reveal a large number of dependent columns. Thus, although condition estimators are known that could tell us “this B is ill-conditioned” (e.g., =-=[68]-=-), LUSOL’s RRLU options are more useful in telling us which columns are causing the poor condition, and which slacks should replace them. 5.3. Basis Repair (Rectangular Case). When superbasic variable... |

1 |
Sparse Rank-Revealing LU Factorization.Presented at the
- Saunders
- 2002
(Show Context)
Citation Context .../τL and |apq| ≥maxj |apj|/τL. • Threshold complete pivoting (TCP), in which apq must be sufficiently large compared to all nonzeros in Al: |apq| ≥maxi,j |aij|/τL. The TCP option was implemented first =-=[81]-=- and has proved valuable for rank-detection during the optimization of Markov decision chains [80] and within SNOPT [54]. In some cases its strict pivot test leads to rather dense LU factors. The TRP ... |