A Practical Algorithm for General Large Scale Nonlinear Optimization Problems

Boggs, Paul T.; Kearsley, Anthony J.; Tolle, Jon W.

doi:10.1137/s105262349426722x

“…Limited-memory updates 5 working set, and for linearly constrained problems (section 5.4) it can accept a known Cholesky factor R for the reduced Hessian.…”

Section: Problemmentioning

confidence: 99%

SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization

Gill

¹

,

Murray

²

,

Saunders

³

2002

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Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first derivatives are available and that the constraint gradients are sparse.We discuss an SQP algorithm that uses a smooth augmented Lagrangian merit function and makes explicit provision for infeasibility in the original problem and the QP subproblems. SNOPT is a particular implementation that makes use of a semidefinite QP solver. It is based on a limited-memory quasi-Newton approximation to the Hessian of the Lagrangian and uses a reduced-Hessian algorithm (SQOPT) for solving the QP subproblems. It is designed for problems with many thousands of constraints and variables but a moderate number of degrees of freedom (say, up to 2000). An important application is to trajectory optimization in the aerospace industry. Numerical results are given for most problems in the CUTE and COPS test collections (about 900 examples).

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“…Examples are Boggs, Kearsley, and Tolle [8,9] and Sargent and Ding [91]. They typically require an exact or finite-difference Lagrangian Hessian but can accommodate many degrees of freedom.…”

mentioning

confidence: 99%

SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization

Gill¹,

Murray²,

Saunders³

2005

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Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first derivatives are available and that the constraint gradients are sparse. Second derivatives are assumed to be unavailable or too expensive to calculate.We discuss an SQP algorithm that uses a smooth augmented Lagrangian merit function and makes explicit provision for infeasibility in the original problem and the QP subproblems. The Hessian of the Lagrangian is approximated using a limited-memory quasi-Newton method.SNOPT is a particular implementation that uses a reduced-Hessian semidefinite QP solver (SQOPT) for the QP subproblems. It is designed for problems with many thousands of constraints and variables but is best suited for problems with a moderate number of degrees of freedom (say, up to 2000). Numerical results are given for most of the CUTEr and COPS test collections (about 1020 examples of all sizes up to 40000 constraints and variables, and up to 20000 degrees of freedom). become large), SNOPT enters nonlinear elastic mode and solves the problemis called a composite objective, and the penalty parameter γ (γ ≥ 0) may take a finite sequence of increasing values. If (NP) has a feasible solution and γ is sufficiently large, the solutions to (NP) and (NP(γ)) are identical. If (NP) has no feasible solution, (NP(γ)) will tend to determine a "good" infeasible point if γ is again sufficiently large. (If γ were infinite, the nonlinear constraint violations would be minimized subject to the linear constraints and bounds.) A similar 1 formulation of (NP) is used in the SQP method of Tone [98] and is fundamental to the S 1 QP 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 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 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, 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 (Hargraves and Paris [67]) has been...

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“…It is analysed from the point of view of the FEM error estimates (and also of the Spectral Method error estimates) in Amautu and Neittaanmaki, 1998. Other interesting papers on Interior Point Methods and preconditioning are reported in the Table of References (El-Bakry et al, 1996;Freund and Jarre, 1996;Boggs et al, 1999;Coleman and Liu, 1999;Nayakkankuppam and Overton, 1999). Other interesting papers on Interior Point Methods and preconditioning are reported in the Table of References (El-Bakry et al, 1996;Freund and Jarre, 1996;Boggs et al, 1999;Coleman and Liu, 1999;Nayakkankuppam and Overton, 1999).…”

Section: Bibliographical Notes and Remarksmentioning

confidence: 99%

Optimal Control from Theory to Computer Programs

Arnăutu

¹

,

Neittaanmki

²

2003

Solid Mechanics and Its Applications

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The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids.The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design.The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity.

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A Practical Algorithm for General Large Scale Nonlinear Optimization Problems

Cited by 41 publications

References 22 publications

SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization

SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization

SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization

Optimal Control from Theory to Computer Programs

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