An algorithm for nonlinear optimization using linear programming and equality constrained subproblems

Byrd, Richard H.; Gould, Nicholas I. M.; Nocedal, Jorge; Waltz, Richard A.

doi:10.1007/s10107-003-0485-4

Cited by 85 publications

(110 citation statements)

References 15 publications

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“…In addition, the incorporation of second derivative information in SQP methods has proved to be difficult. We use, instead a sequential linear-quadratic programming (SLQP) method [5,9,16] that computes a step in two stages, each of which scales up well with the number of variables. First, a linear program (LP) is solved to identify a working set.…”

Section: Active-set Sequential Linear-quadratic Programmingmentioning

confidence: 99%

“…We now outline the SLQP approach implemented in Knitro-Active. This algorithm is an implementation of the algorithm Slique described in [5].…”

Section: Active-set Sequential Linear-quadratic Programmingmentioning

confidence: 99%

“…Mosek [1] is a primal-dual interior-point method for convex optimization, and Pennon [25] follows an augmented Lagrangian approach. New active-set methods based on Sequential Linear-Quadratic Programming (SLQP) have recently been studied by Chin and Fletcher [9] and Byrd et al [5]. Unlike SQP methods, which combine the active-set identification and the step computation in one quadratic subproblem, SLQP methods decouple these tasks into two subproblems.…”

Section: Introductionmentioning

confidence: 99%

“…Unlike SQP methods, which combine the active-set identification and the step computation in one quadratic subproblem, SLQP methods decouple these tasks into two subproblems. The active-set algorithm in Knitro, implements the SLQP method described in [5].…”

Section: Introductionmentioning

confidence: 99%

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Knitro: An Integrated Package for Nonlinear Optimization

Byrd

Nocedal

Waltz

2006

Nonconvex Optimization and Its Applications

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754

518

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This paper describes Knitro 5.0, a C-package for nonlinear optimization that combines complementary approaches to nonlinear optimization to achieve robust performance over a wide range of application requirements. The package is designed for solving large-scale, smooth nonlinear programming problems, and it is also effective for the following special cases: unconstrained optimization, nonlinear systems of equations, least squares, and linear and quadratic programming. Various algorithmic options are available, including two interior methods and an active-set method. The package provides crossover techniques between algorithmic options as well as automatic selection of options and settings.

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Section: Active-set Sequential Linear-quadratic Programmingmentioning

confidence: 99%

“…We now outline the SLQP approach implemented in Knitro-Active. This algorithm is an implementation of the algorithm Slique described in [5].…”

Section: Active-set Sequential Linear-quadratic Programmingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Knitro: An Integrated Package for Nonlinear Optimization

Byrd

Nocedal

Waltz

2006

Nonconvex Optimization and Its Applications

Self Cite

754

518

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“…This constitutes our LP-based heuristic, Section 4. It can be related to Successive Linear Programming (SLP) methods which are well-known for non-linear programming problems [Kel60,PGLE82,ZKL85,SW10], also in combination with SQP methods [BGNW04]. SLP methods are rather efficient in practice [MF09].…”

Section: Conservative Scales and Bounds For Oppmentioning

confidence: 99%

Conservative scales in packing problems

et al. 2011

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Packing problems (sometimes also called cutting problems) are combinatorial optimization problems concerned with placement of objects (items) in one or several containers. Some packing problems are special cases of several other problems such as resource-constrained scheduling, capacitated vehicle routing, etc. In this paper we consider a bounding technique for one-and higher-dimensional orthogonal packing problems, called conservative scales (CS) (in the scheduling terminology, redundant resources). CS are related to the possible structure of resource consumption: filling of a bin, distribution of the resource to the jobs, etc. In terms of packing, CS are modified item sizes such that the set of feasible packings is not reduced. In fact, every CS represents a valid inequality for a certain binary knapsack polyhedron.CS correspond to dual variables of the set-partitioning model of a special 1D cutting-stock problem. Some CS can be constructed by (data-dependent) dualfeasible functions ((D)DFFs). We discuss the relation of CS to DFFs: CS assume that at most 1 copy of each object can appear in a combination, whereas DFFs allow several copies. The literature has investigated so-called extremal maximal DFFs (EMDFFs) which should provide very strong CS. Analogously, we introduce the notions of maximal CS (MCS) and extremal maximal CS (EMCS) and show that EMDFFs do not necessarily produce (E)MCS. We propose fast greedy methods to "maximize" a given CS. Using the fact that EMCS define facets of the binary knapsack polyhedron, we use lifted cover inequalities as EMCS. For higherdimensional orthogonal packing, we propose a Sequential LP (SLP) method over the set of CS and investigate its convergence. Numerical results are presented.

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