Newton Methods For Large-Scale Linear Inequality-Constrained Minimization

Forsgren, Anders; Murray, Walter

doi:10.1137/s1052623494279122

Cited by 35 publications

(10 citation statements)

References 22 publications

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“…Algorithms which use such negative curvature directions can be made to converge globally to a second-order critical point using either a linesearch (see, e.g. [4,7,8,13,15 -171) or a trust region (see, e.g. [5,181) approach.…”

Section: Introductionmentioning

confidence: 99%

“…At each iteration, such algorithms determine a pair of descent directions, (sk, dk) where, loosely speaking, sk represents a direction calculated from positive curvature information given by the Hessian matrix, and dk is a negative curvature direction. These two directions are cornbined to define trajectories of the form NEGATIVE CURVATURE DIRECTIONS 77 [7,8]. A new point is determined by taking a "suitable" step along the relevant trajectory.…”

Section: Introductionmentioning

confidence: 99%

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Exploiting negative curvature directions in linesearch methods for unconstrained optimization

Gould

Lucidi

Roma

et al. 2000

Optimization Methods and Software

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In this paper we propose efficient new linesearch algorithms for solving large scale unconstrained optimization problems which exploit any local nonconvexity of the objective function. Current algorithms in this class typically compute a pair of search directions at every iteration: a Newton-type direction, which ensures both global and fast asymptotic convergence, and a negative curvature direction, which enables the iterates to escape from the region of local non-convexity. A new point is generated by performing a search along a line or a curve obtained by combining these two directions. However, in almost all if these algorithms, the relative scaling of the directions is not taken into account.We propose a new algorithm which accounts for the relative scaling of the two directions. To do this, only the most promising of the two directions is selected at any given iteration, and a linesearch is performed along the chosen direction. The appropriate direction is selected by estimating the rate of decrease of the quadratic model of the objective function in both candidate directions. We prove global convergence to second-order critical points for the new algorithm, and report some preliminary numerical results.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Exploiting negative curvature directions in linesearch methods for unconstrained optimization

Gould

Lucidi

Roma

et al. 2000

Optimization Methods and Software

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“…The extension of good box-constraint or linear-constraint solvers to the case in which the objective function has the PHR form is an interesting subject of research. Especially interesting should be the extension of the box-constraint conjugate-gradient solver described in [25], the linearly-constrained minimization algorithm of Forsgren and Murray [21] and the extension of interior point box-constraint approaches. We believe that taking profit in a clever way of second-order information will cause general algorithmic improvements, independently of convergence to second-order criticality.…”

Section: Algencanmentioning

confidence: 99%

Second-order negative-curvature methods for box-constrained and general constrained optimization

et al. 2009

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A Nonlinear Programming algorithm that converges to second-order stationary points is introduced in this paper. The main tool is a second-order negative-curvature method for box-constrained minimization of a certain class of functions that do not possess continuous second derivatives. This method is used to define an Augmented Lagrangian algorithm of PHR (Powell-Hestenes-Rockafellar) type. Convergence proofs under weak constraint qualifications are given. Numerical examples showing that the new method converges to second-order stationary points in situations in which first-order methods fail are exhibited.

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“…An example of an SQP which uses an active-set method for the subproblem is SNOPT (Gill et al, 2005). Active-set methods for linearly constrained optimization problems with a general nonlinear objective function are presented in the works of Murtagh and Saunders (1978) and Forsgren and Murray (1997).…”

Section: Introductionmentioning

confidence: 99%

A working-set approach for sizing optimization of frame-structures subjected to time-dependent constraints

Verbart

Stolpe

2018

Struct Multidisc Optim

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Document Version Peer reviewed versionLink back to DTU Orbit Citation (APA): Verbart, A., & Stolpe, M. (2018). A working-set approach for sizing optimization of frame-structures subjected to time-dependent constraints. Structural and Multidisciplinary Optimization, 58(4), 1367-1382. CITATION 1 READS 118 2 authors, including: Some of the authors of this publication are also working on these related projects: Structural optimization of offshore windturbine support structures View project Topology Optimization with Stress Constraints View project Alexander Verbart Ramboll 8 PUBLICATIONS 70 CITATIONS SEE PROFILE All content following this page was uploaded by Alexander Verbart on 03 September 2018. The user has requested enhancement of the downloaded file.Structural and Multidisciplinary Optimization: Post-print. The final publication is available via http://dx. AbstractIn this paper, we propose a working-set approach for sizing optimization of structures subjected to time-dependent loads. The optimization problems we consider have a very large number of constraints while relatively few design variables and degrees of freedom. Instead of solving the original problem directly, we solve a sequence of smaller sub-problems. The sub-problems consider only constraints in the working set, which is a small sub-set of all constraints. After each sub-problem, we compute all constraint function values for the current design and add critical constraints to the working set. The algorithm terminates once an optimal point to a sub-problem is found that satisfies all constraints of the original problem. We tested the approach on several reproducible problem instances and demonstrate that the approach finds optimal points to the original problem by only considering a very small fraction of all constraints. The proposed approach drastically reduces the memory storage requirements and computational expenses of the linear algebra in the optimization solver and the computational cost of the design sensitivity analysis. Consequently, the approach can efficiently solve large-scale optimization problems with several hundred millions of constraints.

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Newton Methods For Large-Scale Linear Inequality-Constrained Minimization

Cited by 35 publications

References 22 publications

Exploiting negative curvature directions in linesearch methods for unconstrained optimization

Exploiting negative curvature directions in linesearch methods for unconstrained optimization

Second-order negative-curvature methods for box-constrained and general constrained optimization

A working-set approach for sizing optimization of frame-structures subjected to time-dependent constraints

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