Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for this type of system. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

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Numerical solution of saddle point problems

A methodfor restoringan optical imagewhich is subjectedto low-passfrequency filtering is presented. It is assumedthat the objectwhoseimageis restoredis of finite spatialextent.The problemis treatedasanalgebraicimage-restorationproblemwhich is thensolvedasaquadraticprogrammingproblem with boundedvariables.The regularizationtechniquefor the ill-posedsystemis to replacethe consistent systemof thequadraticprogrammingproblemby anapproximatesystemof smallerrank.Therank which givesa bestor near-bestsolutionis estimated.This methodis a novel one,andit comparesfavorablywith otherknown methods.Computer -simulatedexamplesarepresented. Commentsandconclusionsaregiven. N. N. AbdelmalekandT. Kasvand. Digital imagerestorationusingquadraticprogramming. AppliedOptics, 19(19),3407–3415, 1980. Abstract. The problemof digital imagerestorationis consideredby obtainingan approximatesolutionto the Fredholmintegral equationof the first kind in two variables.The systemof linear equationsresulting from thediscretizationof the integral equationis convertedto a consistentsystemof linearequations.The problemis thensolved asa quadraticprogrammingproblemwith boundedvariableswherethe unknown solutionis minimizedin theL2 norm.In thismethodminimumcomputerstorageis needed, andtherepeated solutionsareobtainedin anefficient way. Also therankof theconsistentsystemwhich givesa bestor near bestsolutionis estimated. Computersimulatedexamplesusingspatiallyseparablepointspreadfunctionsare presented. Commentsandconclusionsaregiven. The problemof digital imagerestorationis consideredby obtainingan approximatesolutionto the Fredholmintegral equationof the first kind in two variables.The systemof linear equationsresulting from thediscretizationof the integral equationis convertedto a consistentsystemof linearequations.The problemis thensolved asa quadraticprogrammingproblemwith boundedvariableswherethe unknown solutionis minimizedin theL2 norm.In thismethodminimumcomputerstorageis needed, andtherepeated solutionsareobtainedin anefficient way. Also therankof theconsistentsystemwhich givesa bestor near bestsolutionis estimated. Computersimulatedexamplesusingspatiallyseparablepointspreadfunctionsare presented. Commentsandconclusionsaregiven. R. A. Abramsand A. Ben Israel. A duality theoremfor complex quadraticprogramming. Journalof OptimizationTheoryandApplications, 4(4), 245–252,1969. Abstract. A duality theoryfor complex quadraticprogrammingover polyhedralconesis developed,following Dorn,by usinglinearduality theory. A duality theoryfor complex quadraticprogrammingover polyhedralconesis developed,following Dorn,by usinglinearduality theory. J. W. Adams. Quadraticprogrammingapproachesto new optimal windows andantennaarrays. ConferenceRecord. TwentyFourth AsilomarConferenceon Signals,Systemsand Computer sMaplePress,SanJose, CA,USA, 1, 69–72,1990a. Abstract. New window designproblemsareformulatedin termsof quadraticprogramming.Thenew windowspermitthedesignerto controlthetradeof betweenthepeaksidelobelevel andthetotalsidelobeenergy. In addition,linearconstraintscanbeimposedon thedesignproblem.Theproposedmethodsareapplicable to applicationsin thefieldsof signalprocessingandantennaarrays. New window designproblemsareformulatedin termsof quadraticprogramming.Thenew windowspermitthedesignerto controlthetradeof betweenthepeaksidelobelevel andthetotalsidelobeenergy. In addition,linearconstraintscanbeimposedon thedesignproblem.Theproposedmethodsareapplicable to applicationsin thefieldsof signalprocessingandantennaarrays. J.W. Adams.Quadraticprogrammingapproaches to new problemsin digital filter design.ConferenceRecord. TwentyFourthAsilomarConferenceonSignals,SystemsandComputer s MaplePress,SanJose, CA,USA, 1, 307–310,1990b. Abstract. New digital filter designproblemsareformulatedin termsof quadraticprogramming.The new filters permit the designerto control the tradeof betweenthe peakerrorsand the total squarederrors.In New digital filter designproblemsareformulatedin termsof quadraticprogramming.The new filters permit the designerto control the tradeof betweenthe peakerrorsand the total squarederrors.In 2 A QUADRATIC PROGRAMMING BIBLIOGRAPHY particular , themean-squared errorcanbeminimizedsubjectto peakerrorsconstraints,asrequiredin many practicaldesignproblems.In addition,equalityconstraintscanbeimposedfor specialapplications. J. W. Adams,P. Kruethong,R. Hashemi,J. L. Sullivan, andD. R. Gleeson. New quadratic programmingalgorithmsfor designingFIR digital filters. ConferenceRecord of The TwentySeventhAsilomarConferenceonSignals,SystemsandComputer s.IEEEComput. SocPress,LosAlamitos,CA,USA, 2, 1206–1210, 1993. Abstract. TheParks-McClellanalgorithm(1973)is verypopularfor designingFIR digital filters. It is based on a linear programmingalgorithmcalledthe Remezexchange.Our new algorithmis basedon quadratic programming,which includeslinear programmingas a specialcase.The filters in this paperpermit the designerto controlthetradeof betweenthepeakerrorandthetotal squarederror. Thesefilters aredesigned accordingto thepeak-constrained least-squares (PCLS)optimalitycriterion. TheParks-McClellanalgorithm(1973)is verypopularfor designingFIR digital filters. It is based on a linear programmingalgorithmcalledthe Remezexchange.Our new algorithmis basedon quadratic programming,which includeslinear programmingas a specialcase.The filters in this paperpermit the designerto controlthetradeof betweenthepeakerrorandthetotal squarederror. Thesefilters aredesigned accordingto thepeak-constrained least-squares (PCLS)optimalitycriterion. J. W. Adams,J. L. Sullivan, D. R. Gleeson,P. H. Chang,andR. Hashemi. Application of quadraticprogrammingto FIR digital filter designproblems.ConferenceRecord of the TwentyEighthAsilomarConferenceon Signals,SystemsandComputer s. IEEE Comput. SocPress,LosAlamitos,CA,USA, 1, 314–318,1994. Abstract. Quadraticprogrammingproblemshavelongbeenof interestin thebusinesscommunity. Quadratic programmingis often usedasthe basisfor ”programtrading” wherestocksareautomaticallyboughtand soldby mutual fundsto optimizeprofits.Quadraticprogrammingalgorithmscanalsobe usedto optimize digital filters, asdiscussedin this paper . We presentthe generalizedmultiple exchange(GME), simplified generalizedmultipleexchange(SGME)andmodifiedgeneralizedmultipleexchange(MGME) algorithmsfor designingconstrainedleast-squares (CLS)filters.TheCLSfiltersaregeneralizationsof thepopularminimax andleast-squares filters.TheCLS filters areimportantnot only becauseof their generality, but alsobecause they areneededfor many practicalapplications. Quadraticprogrammingproblemshavelongbeenof interestin thebusinesscommunity. Quadratic programmingis often usedasthe basisfor ”programtrading” wherestocksareautomaticallyboughtand soldby mutual fundsto optimizeprofits.Quadraticprogrammingalgorithmscanalsobe usedto optimize digital filters, asdiscussedin this paper . We presentthe generalizedmultiple exchange(GME), simplified generalizedmultipleexchange(SGME)andmodifiedgeneralizedmultipleexchange(MGME) algorithmsfor designingconstrainedleast-squares (CLS)filters.TheCLSfiltersaregeneralizationsof thepopularminimax andleast-squares filters.TheCLS filters areimportantnot only becauseof their generality, but alsobecause they areneededfor many practicalapplications. W. P. AdamsandP. M. Dearing. On the equivalencebetweenroof duality andLagrangian duality for unconstrained0–1quadraticprogrammingproblems.DiscreteAppliedMathematics, 48(1), 1–20,1994. Abstract. Considerstechniquesfor computingupperboundson theoptimalobjecti ve functionvalueto any unconstrained0–1 quadraticprogrammingproblem(maximization).In particular , the authorsstudy three methodsfor obtainingupperboundsaspresentedin arecentpaperby Hammer , Hansen,andSimeone(1984) (1) generatingtwo classesof upper-boundinglinearfunctionsreferredto aspavedupperplanesandroofs,(2) solving thecontinuousrelaxationof a mixed-integer linear problemby Rhys(1970),and(3) the quadratic complementationof variableswhich resultsin a boundcalled the height.The authorsshow that all three methodsdirectly result from standardpropertiesof a reformulationof the quadraticproblemasa mixedinteger linear program,with methods(1) and (3) resultingfrom a Lagrangiandual of this reformulation. Basedon this reformulation,they expanduponthepublishedresults. Considerstechniquesfor computingupperboundson theoptimalobjecti ve functionvalueto any unconstrained0–1 quadraticprogrammingproblem(maximization).In particular , the authorsstudy three methodsfor obtainingupperboundsaspresentedin arecentpaperby Hammer , Hansen,andSimeone(1984) (1) generatingtwo classesof upper-boundinglinearfunctionsreferredto aspavedupperplanesandroofs,(2) solving thecontinuousrelaxationof a mixed-integer linear problemby Rhys(1970),and(3) the quadratic complementationof variableswhich resultsin a boundcalled the height.The authorsshow that all three methodsdirectly result from standardpropertiesof a reformulationof the quadraticproblemasa mixedinteger linear program,with methods(1) and (3) resultingfrom a Lagrangiandual of this reformulation. Basedon this reformulation,they expanduponthepublishedresults. W. P. Adamsand H. D. Sherali. A tight linearizationand an algorithm for 0–1 quadratic programmingproblems.ManagementScience, 32(10),1274–1290, 1986. Abstract. Thepaperis concernedwith thesolutionof linearlyconstrained0–1quadraticprogrammingproblems.Problemsof thiskind arisein numerouseconomic,locationdecision,andstrategic planningsituations, includingcapitalbudgeting,facility location,quadraticassignment, mediaselection,anddynamicsetcovering.A new linearizationtechniqueis presentedfor this problemwhich is demonstratedto yield a tighter continuousor linearprogrammingrelaxationthanis availablethroughothermethods.An implicit enumeration algorithmwhichusesLagrangianrelaxation,Benders’cuttingplanes,andlocalexplorationsis designed to exploit thestrengthof this linearization.Computational experienceis providedto demonstratetheusefulnessof 

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A Quadratic Programming Bibliography

Nonlinear two-point boundary-value problems (TPBVP) can be reduced to the iterative solution of a sequence of linear problems by means of quasilinearization techniques. Therefore, the efficient solution of linear problems is the key to the efficient solution of nonlinear problems.

Parallel computation of two-point boundary-value problems via particular solutions

The main role of the suspension system is to achieve ride comfort by reducing vibrations generated by the road roughness. The active damper is getting much attention due to its reduced cost and ability to enhance ride comfort especially when the road ahead is measurable by an environment sensor. In this study a preview active suspension control system was developed in order to improve ride comfort when the vehicle is passing over a speed bump. The control system consists of a feedback controller based on the skyhook logic and a feedforward controller for canceling out the road disturbance. The performance limit for the active suspension control system was computed via trajectory optimization to provide a measure against which to compare and validate the performance of the developed controller. The simulation results indicated that the controller of this study could enhance ride comfort significantly over the active suspension control system employing only the skyhook feedback control logic. Also the developed controller, by displaying similar control pattern as the trajectory optimization during significant time portions, proved that its control policy is legitimate.

Development of preview active suspension control system and performance limit analysis by trajectory optimization

The efficiency of a parallel shooting method on a parallel computer for solving a variety of optimal control guidance problems is studied. Several examples are considered to demonstrate that a speedup of nearly 7 to 1 is achieved with the use of 16 processors. It is suggested that further improvements in performance can be achieved by parallelizing in the state domain. 10 refs.

Trajectory optimization using parallel shooting method on parallel computer

A key algorithmic element of a real-time trajectory optimization hardware/software implementation is presented, the search step solver. This is one piece of an algorithm whose overall goal is to make nonlinear trajectory optimization fast enough to provide real-time commands during guidance of a vehicle such as an aeromaneuvering orbiter or the National Aerospace Plane. Many methods of nonlinear programming require the solution of a quadratic program (QP) at each iteration to determine the search step. In the trajectory optimization case, the QP has a special dynamic programming structure. The algorithm exploits this special structure with a divide- and conquer type of parallel implementation. The algorithm solves a (p.N)-stage problem on N processors in O(p + log2 N) operations. The algorithm yields a factor of 8 speed-up over the fastest known serial algorithm when solving a 1024-stage test problem on 32 processors.

Parallel solver for trajectory optimization search directions

An algorithm has been developed to solve quadratic programs that have a dynamic programming structure. It has been developed for use as part of a parallel trajectory optimization algorithm and aims to achieve significant speed without sacrificing numerical stability. the algorithm makes use of the dynamic programming problem structure and the domain decomposition approach. It parallelizes the orthogonal factorization null-space method of quadratic programming by developing a parallel orthogonal factorization and a parallel Cholesky factorization. Tests of the algorithm on a 32-node INTEL iPSC/2 hypercube demonstrate speedup factors as large as 10 in comparison to the fastest known equivalent serial algorithm.

Parallel orthogonal factorization null-space method for dynamic quadratic programming

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Parallel solver for trajectory optimization search directions

Parallel orthogonal factorization null-space method for dynamic quadratic programming