A basis-deficiency-allowing variation of the simplex method for linear programming

Pan, Ping-Qi

doi:10.1016/s0898-1221(98)00127-8

Cited by 59 publications

(25 citation statements)

References 15 publications

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“…GBP builds up a solution to a linear programming problem; several linear programming methods adopt a similar strategy, solving increasingly complex problems as constraints or variables are iteratively introduced [98,92,81]; see also [60]. We remark that one method, an interior point method called the gravitational method [77,18], can be shown to be equivalent to GBP when applied to the problem dual to (4).…”

Section: Linear Programmingmentioning

confidence: 99%

Diffusion Maps and Geometric Harmonics for Automatic Target Recognition (ATR). Volume 2. Appendices

Zucker¹,

Coifman²

2007

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The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Department of Defense, Washington Headquarters Services, Directorate for Information , 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS. REPORT DATE (DD-MM-YY)2. REPORT TYPE 3. DATES COVERED (From -To) SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORING AGENCY ACRONYM(S)AFRL SPONSORING/MONITORING AGENCY REPORT NUMBER(S) AFRL-RY-WP-TP-2007-1223, V2 DISTRIBUTION/AVAILABILITY STATEMENTApproved for public release; distribution unlimited. SUPPLEMENTARY NOTES Report contains color. See also Volume 1 (AFRL-RY-WP-TM-2007-1223, V1). ABSTRACTGeometric harmonics provides a framework for taking data in high-dimensional measurement spaces and embedding them in low dimensional Euclidean space according to a similarity measure. Euclidean coordinates then characterize the "manifold" on (or near) which the data live. Our goal in this project is to develop this manifold as a mechanism for integrating data from different sensors to facilitate automatic recognition. During the tenure of this research grant, we were able to formulate and complete the first series of experiments on embedded fusion. The resulting experiment on integrating voice and audio streams was extremely successful. This definitely revealed the potential for this approach and set the stage for further experiments. The problem formulation has been completed and confirmed with an experiment on the integration of audio and video streams. Researchers at AFRL, Wright-Patterson Air Force Base, have received a first version of the software, and are attempting to apply it to radar signal interpretation. SUBJECT TERMSgeometric, algorithms, target recognition, ATR Harmonic analysis on manifolds and graphs has recently led to mathematical developments in the field of data analysis. The resulting new tools can be used to compress and analyze large and complex data sets, such as those derived from sensor networks or neuronal activity datasets, obtained in the laboratory or through computer modeling. The nature of the algorithms (based on diffusion maps and connectivity strengths on graphs) possesses a certain analogy with neural information processing, and has the potential to provide inspiration for modeling and understanding biological organization in perception and memory format...

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Section: Linear Programmingmentioning

confidence: 99%

Diffusion Maps and Geometric Harmonics for Automatic Target Recognition (ATR). Volume 2. Appendices

Zucker¹,

Coifman²

2007

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“…A new trend appears in the late nineties with the paper of Pan (1998) who formulates a generic basis for degenerate solutions. Embedding this concept in a column generation scheme led to the Dynamic Constraint Aggregation (DCA) algorithm of Elhallaoui et al (2005Elhallaoui et al ( , 2008 for the set partitioning problem.…”

Section: Introductionmentioning

confidence: 99%

Tools for primal degenerate linear programs: IPS, DCA, and PE

Gauthier

Desrosiers

Lübbecke

2016

EURO Journal on Transportation and Logistics

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“…The first milestone is the well-known simplex method founded by Dantzig [2], in which the philosophy is to move on the underlying polyhedron, from vertex to an adjacent vertex, until reaching an optimal vertex. Since then, extensive theoretical analysis, numerical implementations as well as numerous variants (see e.g., [1,3,4,6,7,9,[13][14][15][16][17][18][20][21][22][23][24]) have been developed. To date, the simplex algorithm is accepted as one of the most famous and widely used mathematical tools in the world [13].…”

Section: Introductionmentioning

confidence: 99%

On an Efficient Implementation of the Face Algorithm for Linear Programming

Zhang

Yang

Liao³

2013

JCM

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In this paper, we consider the solution of the standard linear programming (LP). A remarkable result in LP claims that all optimal solutions form an optimal face of the underlying polyhedron. In practice, many real-world problems have infinitely many optimal solutions and pursuing the optimal face, not just an optimal vertex, is quite desirable. The face algorithm proposed by Pan [19] targets at the optimal face by iterating from face to face, along an orthogonal projection of the negative objective gradient onto a relevant null space. The algorithm exhibits a favorable numerical performance by comparing the simplex method. In this paper, we further investigate the face algorithm by proposing an improved implementation. In exact arithmetic computation, the new algorithm generates the same sequence as Pan's face algorithm, but uses less computational costs per iteration, and enjoys favorable properties for sparse problems.Mathematics subject classification: 62H20, 15A12, 65F10, 65K05.

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A basis-deficiency-allowing variation of the simplex method for linear programming

Cited by 59 publications

References 15 publications

Diffusion Maps and Geometric Harmonics for Automatic Target Recognition (ATR). Volume 2. Appendices

Diffusion Maps and Geometric Harmonics for Automatic Target Recognition (ATR). Volume 2. Appendices

Tools for primal degenerate linear programs: IPS, DCA, and PE

On an Efficient Implementation of the Face Algorithm for Linear Programming

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