ALGORITHMS FOR LINEAR OPTIMIZATION

... MODELLINGTechnical University of DenmarkDK-2800 Lyngby – DenmarkJ. No. ALO1.2.1999HBN/ms ALGORITHMS FOR LINEAR OPTIMIZATION An IntroductionHans Bruun Nielsen2nd EditionIMM ✁✂✄☎✆✂✝✞✟✠✟✡☛☞✟✠✌✟✍✟✌✍✎☞☞✟✡✏☛✍☞✞✟✠☞✑✒✟✡☞✠✎✡☞✞✟✓☛✑✍✠✟✔✕✕✖✗✘✙✚✛✜✛✢✣✚✛✤✥✣✥✦✧✣✚✣★✛✚✚✛✥✩✗✪☞☞✞✟✝✟✓✞✡✎✓✪✫✬✡✎✭✟✍✠✎☞✮☛✏✯✟✡✰✪✍✱✳✲✴☛✵✟✏✑✫✫✮✷✶☞✞✟✮✓✪✡✪✫✠☛✸✟✑✠✟✏✑✫☞☛☛☞✞✟✍✠✲✹✑✺✺✟✠☞✎☛✡✠✏☛✍✎✰✵✍☛✭✟✰✟✡☞✪✍✟✭✟✍✮✌✟✫✓☛✰✟✲✻✮✡✺✸✮✷✶✯✟✓✟✰✸✟✍✼✽✽✾✴✪✡✠✿✍✑✑✡❀✎✟✫✠✟✡✟✰✪✎✫❁❂❃❄❅❆❇❇❈❉❊❋❈❉●✑✍✫❁❂❊❊❍■❏❏▲❑❑❑❈❆❇❇❈❉❊❋❈❉●❏◆▼❂❃❄❖✪✰✺✍✪☞✟✏✑✫☞☛✵✍☛✏✟✠✠☛✍✲◗☛☛✠✏☛✍✵☛✎✡☞✎✡✺☛✑☞✠☛✰✟✟✍✍☛✍✠✶✌✞✎✓✞✞✪✭✟✸✟✟✡✓☛✍✍✟✓☞✟✒✎✡☞✞✎✠✭✟✍✠✎☛✡✷✲✻✮✡✺✸✮✷✶❘✟✸✍✑✪✍✮✼✽✽❙✲✴✪✡✠✿✍✑✑✡❀✎✟✫✠✟✡❚☛✍✟✟✍✍☛✍✠✞✪✭✟✸✟✟✡✏☛✑✡✒✪✡✒✓☛✍✍✟✓☞✟✒✎✡☞✞✎✠✠✟✓☛✡✒✟✒✎☞✎☛✡✷✲✻✮✡✺✸✮✷✶❯✪✡✑✪✍✮✼✽✽✽✲✴✪✡✠✿✍✑✑✡❀✎✟✫✠✟✡ ✁ ✂✄☎✆✝➎▼❖❙◗❙◗❘❳❖▼♣❙❖▲▼✈✂❥✈✂ r▲❘◗◆❙✈✂✮♦▲❘q♣❘◆t❱◆❙❖❙t❙❖▲▼✈①①♦❘◗◗④♣❘❖♣❱❲◗✈❢①✟❢✶✈✶ ♦t❲❲❘♣▼✕✈✂❥✩▲❲❳♣▼✈❪❜✩▲❲t❱✈✂✶s❘♣❖◗▼❙✈❪✶✈ ❢✈✶✶s❘▲q❙❚♦♣②❙▲❘✈✶❥✆♣♣❳♣❘❯❘▲t②❙✈①✈❪❪✆◗◆◆◗▼❱◗❘s♦▲❘❳✈✂✮✆▲t◆◗❚▲❲◗❘✈✮✶✆t❱◗❘✈❢❥❥❖◗❳❯▲❙◗▼❙❳♣❙❘❖➍✈✶✂➌✎✎✎◆❙♣▼♣❘✈ ❖❲❲②▲▼❖❙❖▲▼❖▼s✈①➑✈① ❖▼②▲▼◆❖◆❙◗▼❙◆③◆❙◗❳✈✮✮✈❢❢➑❖▼♦◗♣◆❖❱❲◗❯❘▲❱❲◗❳✈✂✈✮✈❪➑✈➑✶✈①❥✈ ➑✈✂➑✈✂ ❖▼❖❙❖♣❲❙❚❘◗◆❚▲❲✈✮①❖▼❙◗❘❖▲❘❯▲❖▼❙②▲▼❖❙❖▲▼✈❜❜✟❳◗❙❚▲✈❢❥✈❜①✟ ➑✈❢❥①❖▼④◗❘◆◗❳♣❙❘❖➍✈①①❖❙◗❘♣❙❖④◗◆▲❲④◗❘✈①➑✈✮✮✞♣❘❳♣❘✕♣❘✈❢❥✈ ❥✞♣❘t◆❚✈❜❜✕❖▼✕④♣❲t◗✈✮❪✈✮➑✞✞♥◆③◆❙◗❳✈❜❜✈❜①✈➑①✈①✶✞t❚▼✈❜❜❲♣❘s◗t❯♣❙◗✈➑❪❲◗♣◆❙✛◆✏t♣❘◗◆✈❜❢❲❖▼◗◆◗♣❘②❚✈✮❢❲❖▼◗♣❘❯❘▲s❘♣❳❳❖▼s✈✂✈✂❪✈✶❜❲▲s✉❱♣❘❘❖◗❘♦t▼②❙❖▲▼✈❢❢✈❜❪❳♣②❚❖▼◗♣②②t❘♣②③✈ ❳♣❘s❖▼♣❲②▲◆❙✈✶❜✔♣❘✏t♣❘❙✈✮✂❳❖▼❖❳t❳◗s❘◗◗✈①①✟▼▲❘❳◆▲❲t❙❖▲▼✈❢❢➑✔▲❘❘❖◆▲▼✈①✂✠◗q❙▲▼ ◆❳◗❙❚▲✈❜ ✈➑✂✈①✶✈✮❢✈✮✶▼▲▼❱♣◆❖②④♣❘❖♣❱❲◗✈  ▼▲▼❲❖▼◗♣❘▲❯❙❖❳❖➏♣❙❖▲▼✈❪✶✟❯❘▲❱❲◗❳✈✂①▼▲❘❳♣❲◗✏t♣❙❖▲▼◆✈❜❢✈①➑▼t❲❲◆❯♣②◗✈❜❥✈❢❢❜▼t❳◗❘❖②♣❲◆❙♣❱❖❲❖❙③✈①✂✈✂✂✈✮✶▲❱✡◗②❙❖④◗♦t▼②❙❖▲▼✈✂✈❢❜✈➑✶✈✶✂▲❯❙❖❳♣❲❯♣❘❙❖❙❖▲▼✈❪❜✈❪①✈❪ ✈✂①✈❢❥❢✟◆◗❙✈✂✈❢✂✈❪❢✈❪❪✈❪➑▲❘❙❚▲s▲▼♣❲❙❘♣▼◆♦▲❘❳♣❙❖▲▼✈❜❢▲④◗❘◗❙◗❘❳❖▼◗◆③◆❙◗❳✈❜❢❯◗▼♣❲❙③♦♣②❙▲❘✈❢✶✟♦t▼②❙❖▲▼✈✶✂❯◗❘❳t❙♣❙❖▲▼❳♣❙❘❖➍✈①①✈✂✂❯◗❘❙t❘❱◗❯❘▲❱❲◗❳✈✂❪✓❚♣◆◗❢✟❪✈✂❜✈✶❪✈❢❥❢❯❖◗②◗q❖◆◗❯▲❲③▼▲❳❖♣❲✈✂ ✈✶✶✈❢❥❥❯❖④▲❙❖▼s✈①①✈①✂❯▲❲③▼▲❳❖♣❲②▲◆❙✈➑❥❯▲❲③❙▲❯◗✈ ①❯▲◆❖❙❖④◗◗➎▼❖❙◗✈❜❜✈①❜✈✮ ❯▲❙◗▼❙❖♣❲✈ ❪❯❘◗②▲▼❖❙❖▲▼◗❘✈①➑✈✮✮❯❘◗❖②❙▲❘✛②▲❘❘◗②❙▲❘✈➑ ✈①✶❯❘◗❯❘▲②◗◆◆❖▼s✈❢✂❯❘❖②❖▼s④◗②❙▲❘✈✶❜✓❘❖❳♣❲✡t♣❲✔◗❙❚▲✈❜①✓❘❖❳♣❲✗▲s✉✄♣❘✉✔◗❙❚▲✈①✮❯❘▲✡◗②❙❖▲▼✈❜❥✈❢❥❜❯❘▲➍❖❳❖❙③✈❜✶✈➑❢✈➑➑✈➑ ❁✞✟✠✲ ✁✁✡✂♦♣②☛☞✌✍➏♣☛✍☞✎✈✮✂✏t♣✏✌✑☛✍✒✒☞✎✓✔✌✕✔✎✒✔✗✖✘✶✖✙✚✟✛✌☞✕✌✑✜✜✍✎✕✖✢✚✢✌✑✎✕✖✣✣✤✖✢✢✘✌✔✏✥✒✔✏✛✒☞✦☛✖ ✂✌✔✏✥✎✏✑✎✒✧✖✢✂✌✔★✑✒☛☞✌✍✩✑☛✍☞✎✖✶✚✖✣ ✤✖✣✶✌✔✕✥✪✑✌✍✩✑☛✍☞✎✖✢✢✙✌✔✪✑✫✔✏✛✌☞✬✪✔✜✗✖✢✶✌☞✬✥✦☛✎✔✦✦✖✙✶✌☞✥✎✏✍✎✕✔✌✌☞✌✦✖✶✚✖✣✶✌☞✭☞✛✔✌✑☛✍☞✎✖✂✶✟✦✛✑✒✔✗✖✘✚✖✢✢✘✦✑★✔☛✧✓✑✪✓✔✗✖✙✶✦✔✪★✛✏✥✑✪✛✌☞✬✪✔✜✗✖✮✘✤✖✙✶✦✔✎✦✍☛✍✓✍☛✧✖✶✘✯✰✔✌✜✑✎✖✱✂✯✍✜✛✪✔✫✖✶✖ ✱✟✶✱✖✢✚✮✖✢✚✱✦✍✎✕✥✪✑✌✜✑☛✌✍✫✖✣ ✦✕✔✭✛✦✧✜✜✔☛✌✍✒✖✮✘✦✪✑✒✕✓✑✌✍✑✬✪✔✗✖✢✙✤✖✙✢✖ ✙✦✜☞☞☛✰✑✛✛✌☞✫✍✜✑☛✍☞✎✖✶ ✤✖✢✚✚✡☛✞☞✌✖✱✶✖✂✂✦✛✑✌✦✔✜✑☛✌✍✫✖✱✙✤✖✂✂✖✣✣✦✛✍✕✔✗✖✂✶✦✛✪✍☛☛✍✎✕✖✢ ✤✖✮✘✤✖✮✱✖✱✘✤✖ ✱✖✂✱✖✶✮✖✶✣✦☛✑✪✪✍✎✕✖✂✮✦☛✑✎✏✑✌✏★☞✌✜✗✖✂✖✢✙✤✖✢✂✖✮✘✤✖✮✱✖✘✱✖ ✮✖ ✱✦☛✑✌☛✍✎✕✛☞✍✎☛✖✙✂✖✙✣✤✖✱✮✖✂✘✤✖✂✱✖✣✱✦☛✔✔✛✔✦☛✏✔✦✒✔✎☛✖✶✖✘✢✦☛☞✛✛✍✎✕✒✌✍☛✔✌✍✑✖✙✶✦☛✌✍✒☛✒☞✜✛✪✔✜✔✎☛✑✌✍☛✧✖✮✮✦✥✌✛✪✥✦✓✔✒☛☞✌✖✢✚✖✢✙✤✖✮✢✖✶ ☛✰✌✔✦✰☞✪✏✌✔✏✥✒☛✍☞✎✖✣✙✲☞✜✪✍✎✖✂✣☛✌✑✎✦★☞✌✜✑☛✍☞✎✖ ✘✤✖✂✣✲✥✒✳✔✌✖✮✘✤✖✘✘✥✎✬☞✥✎✏✔✏✛✌☞✬✪✔✜✗✖✂✖✢✚✖✮✙✤✖✙✶✖ ✙✤✖ ✂✖✂✚✖✶✙✤✖✣✮✥✛✏✑☛✍✎✕✖✂✶✖✂✣✤✖✣✘✤✖✣ ✤✖✣✶✓✔✌☛✔✫✖✢✚✖ ✱✖✢✚✢✭✑✌✜✦☛✑✌☛✖✂✱✖✶✙✭✔✑✳✏✥✑✪✍☛✧✖✮✢✴☞☞✏✬✥✌✧✖✱✂✼✽✔✕✖✝✎✖✞✠✒✟✗✘✟✒✙✚✎✝✑✡✏❆❇❈❉❊❋●✍■ ■r▲❘❙❚◗❯❘▲❱❲◗❳❖◆②t◆◆◗❖▼✎➍♣❳❯❲◗❢✉❢q◗s♣④◗♣◆❙♣▼♣❘♦▲❘❳t❲♣❙❖▲▼❖▼✎➍♣❳❯❲◗❪✉❜✉✞◆❖▼s❙❚❖◆q◗➎▼❙❚◗t♣❲❯❘▲❱❲◗❳❨✁❬❳♣➍❭✂❵❛❪✂❝❞❵❝✂❵❤❝❣✂❝✒❪✈❤✂❵✒❥✂❵❛✂❝✒❜✈✂❝✒❥❦❧♥❚◗♦◗♣◆❖❱❲◗▲❳♣❖▼✄❖◆◆❚▲q▼❱◗❲▲q✉♥❚◗◆▲❲t❙❖▲▼②♣▼❱◗♦▲t▼♣◆❖◆✛②t◆◆◗❖▼②▲▼▼◗②❙❖▲▼q❖❙❚r❖st❘◗❢✉❜✈◗➍②◗❯❙❙❚♣❙▼▲qq◗❚♣④◗♣❳♣➍❖❳❖➏♣❙❖▲▼❯❘▲❱❲◗❳✉♥❚◗◆▲❲t❙❖▲▼❖◆☛☛✇❨❜✐❥❬q❖❙❚☎✝✆❨☛☛❬✇❜✉...

... values for each iteration [42].3.7. Hyperheuristic strategiesOver the past 50 years, the well-known meta-heuristic algorithms have been used as optimization tool for a wide range of optimization ... strategy, as well as the acceptance strategy.4. Performance metrics for heuristic algorithms evaluationAiming to quantify the performance of the heuristic algorithms in terms of stability andconvergence ... need for quality of solutionsSearch Algorithms for Engineering Optimization9 2Chapter 1Search Algorithm for Image Recognition Based onLearning Algorithm for Multivariate Data AnalysisJuan...

... functions for the given thermodynamic system, and another is to determine the optimal thermodynamic process for the given optimization objectives. The former case belongs to a class of static optimization ... ' T , the heat flux rate 1 q for the pseudo-Newtonian heat transfer law decreases linearly, while that for the radiative heat transfer law decreases non-linearly; for the same Carnot temperature ... i for the optimal strategy is relative uniform. Calculation results show that the total power output of the system for the stage-by-stage optimization strategy is 3 3.71 10 W× , while that for...

... methodology for testing for statistically significant differences between fully 3-D PETreconstruction algorithms, Phys. Med. Biol.,39, 341–354, 1994.c1999 by CRC Press LLCAlgorithmsforComputedTomographyGaborT.HermanUniversityofPennsylvania26.1Introduction26.2TheReconstructionProblem26.3TransformMethods26.4FilteredBackprojection(FBP)26.5TheLinogramMethod26.6SeriesExpansionMethods26.7AlgebraicReconstructionTechniques(ART)26.8ExpectationMaximization(EM)26.9ComparisonofthePerformanceofAlgorithmsReferences26.1 ... algorithm for aparticular task and to compare the efficacy of different algorithms for that task.An approach to evaluating an algorithm is to first start with a specification of the task for whichthe ... reconstruction algorithm, for performing that task. In thenumerical observer approach [11, 16, 17], a task-specific FOM is computed for each image. Basedon the FOMs for all the images produced...

... 0.3−0.0500.050.10.150.20.250.3step 13Figure 7: QR; the blue lines denote the true solution set.1An Interval Method for Linear IVPs for ODEsNed NedialkovDepartment of Computing and SoftwareMcMaster University, Canadanedialk@mcmaster.caJoint ... 1)[e0]. For all yi∈ { b0+ Bα | α ∈ [0, 1]n},y(ti+1; ti, yi) ∈w0+nj=1αj(wj− w0) | αj∈ [0, 1], wj∈ [wj]⊆c0+ Cα + [e] | α ∈ [0, 1]n,12since for α ∈ ... 10Assume that at a point ti, for all y0∈ [y0],y(ti; t0, y0) ∈ { b0+ Bα | α ∈ [0, 1]n},where B ∈ Rn×n, and [0,...

... several tabular algorithms for Tree Adjoining Grammar parsing, creating a continuum from simple pure bottom-up algorithms to complex pre- dictive algorithms and showing what transformations must ... resenting structure. Several parsing algorithms have been proposed for this formalism, most of them based on tabular techniques, ranging from simple bottom-up algorithms (Vijay-Shanker and Joshi, ... 1997). However, it is diffi- cult to inter-relate different parsing algorithms. In this paper we study several tabular algorithms for TAG parsing, showing their common characteris- tics and how...

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... finding all for a given is . Therefore,the total cost is . For all non-empty , we create a new state and for all we set . We create a transition, and for all such that ,we set . For all such ... size of that represen-tation makes it impractical for offline optimizations suchas those used in large-vocabulary speech recognition orgeneral information extraction systems. Most offline rep-resentations ... tech-nique for creating an exact representation of -gram lan-guage models by WFAs whose size is practical for offlineuse even in tasks with a vocabulary size of about 500,000words and for .Class-based...