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Two extreme points of a polymatroid are adjacent if and only if they either differ in exactly one component or differ in exactly two components with the two components satisfying a certain ordering relation. A polynomial algorithm generates and lists all extreme points adjacent to a given extreme point of a polymatroid. Similar results hold for the core of a convex game.",{"EN":266},"Adjacency on polymatroids",{"VOID":268},"[]",{"VOID":270},"E. Balas and M.W. Padberg, “On the set-covering problem”,Operations Research 20 (1972) 1152–1161.\nE. Balas and M. Padberg, “On the set-covering problem: II. an algorithm for set partitioning”,Operations Research 23 (1975) 74–90.\nE. Balas and M.W. Padberg, “Adjacent vertices of the all 0–1 programming polytope”,RAIRO-Operations Research 13 (1979) 3–12.\nM.L. Balinski and A. Russakoff, “Some properties of the assignment polytope”,Mathematical Programming 3 (1972) 257–258.\nM.L. Balinski and A Russakoff, “On the assignment polytope”,SIAM Review 16 (1974) 516–525.\nR.E. Bixby, W.H. Cunningham and D.M. Topkis, “The partial order of a polymatroid extreme point” (to appear inMathematics of Operations Research).\nV. Chvátal, “On certain polytopes associated with graphs”,Journal of Combinatorial Theory (B) 18 (1975) 138–154.\nV. Chvátal, “Rational behaviour and computational complexity”, Technical Report SOCS-78.9, School of Computer Science, McGill University (1978).\nW.H. Cunningham, “Testing membership in matroid polyhedra”, Working Paper WP-81207, Department of Operations Research, University of Bonn (1981).\nJ. Edmonds, “Submodular functions, matroids, and certain polyhedra”, in R. Guy et al., eds.,Combinatorial structures and their applications (Proceedings Calgary International Conference, 1969) (Gordon and Breach, New York, 1970) pp. 69–87.\nS. Fujishige, “Algorithms for solving the independent-flow problems”,Journal of the Operations Research Society of Japan 21 (1978) 189–204.\nP. Gaiha and S.K. Gupta, “Adjacent vertices on a permutohedron”,SIAM Journal on Applied Mathematics 32 (1977) 323–327.\nR. Giles, “Adjacency on the postman polyhedron”,SIAM Journal on Algebraic and Discrete Methods 2 (1981) 172–175.\nR. Giles and D. Hausmann, “Characterizations of adjacency on the branching polyhedron”,Discrete Mathematics 26 (1979) 219–226.\nM. Grötschel, L. Lovász and A. Schrijver, “The ellipsoid method and its consequences in combinatorial optimization”,Combinatorica 1 (1981) 169–197.\nD. Hausmann, “Adjacency on polytopes in combinatorial optimization”, Mathematical Systems in Economics 49, Oelgeschlager, Gunn & Hain (Cambridge, Massachusetts, 1979).\nD. Hausmann, “Complexity of the testing of adjacency on theb-matching polyhedron”,Operations Research Verfahren\u002FMethods of Operations Research 32 (1979) 133–141.\nD. Hausmann, “Adjacent vertices on theb-matching polyhedron”,Discrete Mathematics 33 (1981) 37–51.\nD. Hausmann and B. Korte, “Colouring criteria for adjacency on 0–1 polyhedra”,Mathematical Programming Study 8 (1978) 106–127.\nP.G. McKeown, “Determining adjacent vertices on assignment polytopes”,Naval Research Logistics Quarterly 23 (1976) 455–460.\nP.G. McKeown and D.S. Rubin, “Adjacent vertices on transportation polytopes”,Naval Research Logistics Quarterly 22 (1975) 365–374.\nK.G. Murty, “On the tours of a traveling salesman”,SIAM Journal on Control 7 (1969), 122–131.\nK.G. Murty, “Adjacency on convex polyhedra”,SIAM Review 13 (1971) 377–386.\nM.W. Padberg and M.R. Rao, “The traveling salesman problem and a class of polyhedra of diameter two”,Mathematical Programming 7 (1974) 32–45.\nC.H. Papadimitriou, “The adjacency relation on the traveling salesman polytope is NP-complete”,Mathematical Programming 14 (1978) 312–324.\nM.R. Rao, “Adjacency of the traveling salesman tours and 0–1 vertices”,SIAM Journal on Applied Mathematics 30 (1976) 191–198.\nL.S. Shapley, “Cores of convex games”,International Journal of Game Theory 1 (1971) 11–26.\nD.M. Topkis, “Activity selection games and the minimum-cut problem”,Networks 13 (1983) 93–105.\nH.P. Young, “On permutations and permutation polytopes”,Mathematical Programming Study 8 (1978) 128–140.",{"VOID":272},"10.1007\u002FBF02591887","PUBLICATION","VERIFIED","2024-09-05T07:41:08.685+00:00","Auto Verify","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF02591887",[279],{"id":280,"sortIndex":21,"researcher":20,"roles":281,"affiliations":283,"properties":292,"displayName":294,"givenName":20,"familyName":20},"a38210aa-4495-4c55-b8ce-04e2b5279ccb",[282],"AUTHOR",[284],{"id":285,"sortIndex":21,"affiliation":286,"properties":20},"140b05d1-b08f-4e81-85bb-aec0c46080f8",{"id":285,"createTime":20,"updateTime":20,"relativeEntities":287,"slug":20,"properties":288,"entityType":20,"verifyStatus":20,"verifyTime":20,"verifyNote":20,"languages":20,"translateLanguages":20,"viewCount":20,"url":20,"parentIds":291,"statistic":20},[],{"title":289},{"VI":290},"AT&T Bell Laboratories, Holmdel, USA",[],{"title":293},{"VI":294},"Donald M. 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-non-Hamilton-tour-decision-problem-for-directed-graphs",{"abstract":361,"title":363,"gsPaper":365,"references":366,"doi":368},{"EN":362},"A truncated permutation matrix polytope is defined as the convex hull of a proper subset of n-permutations represented as 0\u002F1 matrices. We present a linear system that models the coNP-complete non-Hamilton tour decision problem based upon constructing the convex hull of a set of truncated permutation matrix polytopes. Define polytope Pn−1 as the convex hull of all n-1 by n-1 permutation matrices. Each extreme point of Pn−1 is placed in correspondence (a bijection) with each Hamilton tour of a complete directed graph on n vertices. Given any n vertex graph G\n                  n\n                , a polynomial sized linear system F(n) is constructed for which the image of its solution set, under an orthogonal projection, is the convex hull of the complete set of extrema of a subset of truncated permutation matrix polytopes, where each extreme point is in correspondence with each Hamilton tour not in G\n                  n\n                . The non-Hamilton tour decision problem is modeled by F(n) such that G\n                  n\n                 is non-Hamiltonian if and only if, under an orthogonal projection, the image of the solution set of F(n) is P\n                  n\n                −1. The decision problem ‘Is the projection of the solution set of F(n)=P\n                  n\n                −1?’ is therefore coNP-complete, and this particular model of the non-Hamilton tour problem appears to be new.",{"EN":364},"A model of the coNP-complete non-Hamilton tour decision problem for directed graphs",{"VOID":268},{"VOID":367},"Bachem, A., Grotschel, M.:, New Aspects of Polyhedral Theory. In: Korte, B., ed., Modern Applied Mathematics: Optimization and Operations Research, North-Holland Publishing Company, Amsterdam, 1979 pp. 51–106\nBalas, E.: Projection and lifting in combinatorial optimization, Computational combinatorial optimization (Scholb Dagstuhl, 2000), Lecture Notes in Computer Science, 2241 Springer, Berlin, 2001 pp. 26–56\nBarahona, F., Mahjoub, A.R.: On two-connected subgraph polytopes. Disc. Math. 147, 19–34 (1995)\nBartels, H.G., Bartels, S.G.: The Facets of the Asymmetric 5-City traveling salesman polytope. Meth. Models of Opera. Res. 33, 193–197 (1989)\nBook, R.V.: Relativizations of the P =? NP problem and other problems: some developments in Structural Complexity theory, Algorithms and Computation. In: Goos, G., Hartmans, J., eds., Lecture Notes in Computer Science 650, Springer-Verlag, Nagoya, 1992, pp. 175–186\nBoyd, S.C., Cunningham, W.H.: Small travelling salesman polytopes. Math. Oper. Res. 16, 259–271 (1991)\nCarr, R.D.: Separating Clique Tree and Bipartition Inequalities in Polynomial Time. In: Hamacher, H., (ed.), Lecture Notes in Computer Science 920, Springer, Copenhagen, 1995, pp. 40–49\nChopra, S.: Polyhedra of the equivalent subgraph problem and some edge connectivity problems. SIAM J. Disc. Math. 5, 321–337 (1992)\nChopra, S., Gilboa, I., Sastry, S.: Algorithms and extended formulations for one and two facility network design, Integer programming and combinatorial optimization (Vancouver, BC, 1996), Lecture Notes in Computer Science 1084, Springer, Berlin, 1996, pp. 44–57\nChopra, S., Owen, J.H.: Extended formulations for the A-cut problem. Math. Program., Ser. A 73, 7–30 (1996)\nChvatal, V.: Linear Programming. W.H. Freeman and Company, 1980\nCoullard, C.R., Gamble, A.B.: The k-walk polyhedron, Advances in Optimization and Approximation. Nonconvex Optimization, Appl. 1, Kluwer Acad. Publ., Dordrecht, 1994, pp. 9–29\nGismondi, S.J.: Dueling Cubes. Utilitas Mathematica 54, 241–251 (1998)\nGismondi, S.J.: An O(n3) sized external representation of a factorial faceted, factorial extreme point polytope. Utilitas Mathematica 63, 109–114 (2003)\nGismondi, S.J., Swart, E.R.: A Factorial Faceted, Factorial Extreme Point Polytope Crafted from the Assignment Polytope. Utilitas Mathematica 60, 181–192 (2001)\nGrotschel, M., Padberg, M.W.: Polyhedral Theory. In: Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys D.B., eds., The Traveling Salesman Problem, John Wiley and Sons, New York, 1985, pp. 251–306\nOnn, S.: Geometry, Complexity, and Combinatorics of Permutations. J. Combinatorial Theory, Series A 64, 31–49 (1991)\nYannakakis, M.: Expressing Combinatorial Optimization Problems by Linear Programs. 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In each iteration the method uses two previous iterates for an approximation of the local Lipschitz constant without running a linesearch. Thus, every iteration of the method requires only one evaluation of a monotone operator F and a proximal mapping g. The operator F need not be Lipschitz continuous, which also makes the algorithm interesting in the area of composite minimization. The method exhibits an ergodic O(1 \u002F k) convergence rate and R-linear rate under an error bound condition. We discuss possible applications of the method to fixed point problems as well as its different generalizations.",{"EN":611},"Golden ratio algorithms for variational inequalities",{"VOID":613},"[\"3222042286532909959\"]",{"VOID":615},"Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9(1–2), 3–11 (2001)\nAntipin, A.S.: Minimization of convex functions on convex sets by means of differential equations. Differ. Equ. 30(9), 1365–1375 (1994)\nArrow, K.J., Hurwicz, L., Uzawa, H.: Studies in Linear and Non-linear Programming. Stanford University Press, Redwood City (1958)\nAttouch, H., Chbani, Z., Peypouquet, J., Redont, P.: Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity. Math. Program. 168(1–2), 123–175 (2018)\nAttouch, H., Cominetti, R.: A dynamical approach to convex minimization coupling approximation with the steepest descent method. J. Differ. Equ. 128(2), 519–540 (1996)\nBaes, M., Bürgisser, M., Nemirovski, A.: A randomized mirror-prox method for solving structured large-scale matrix saddle-point problems. SIAM J. Optim. 23(2), 934–962 (2013)\nBanert, S., Boţ, R.I.: A forward-backward-forward differential equation and its asymptotic properties. J. Convex Anal. 25(2), 371–388 (2018)\nBauschke, H.H., Bolte, J., Teboulle, M.: A descent lemma beyond Lipschitz gradient continuity: first-order methods revisited and applications. Math. Oper. Res. 42(2), 330–348 (2016)\nBauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367–426 (1996)\nBauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)\nBeck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problem. SIAM J. Imaging Sci. 2(1), 183–202 (2009)\nBello Cruz, J., Díaz Millán, R.: A variant of forward-backward splitting method for the sum of two monotone operators with a new search strategy. Optimization 64(7), 1471–1486 (2015)\nBerinde, V.: Iterative Approximation of Fixed Points, vol. 1912. Springer, Berlin (2007)\nBoţ, R.I., Csetnek, E.R.: An inertial forward-backward-forward primal-dual splitting algorithm for solving monotone inclusion problems. Numer. Algorithms 71(3), 519–540 (2016)\nCensor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optitm. Theory Appl. 148, 318–335 (2011)\nChambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)\nChambolle, A., Pock, T.: On the ergodic convergence rates of a first-order primal-dual algorithm. Math. Program. 159(1–2), 253–287 (2016)\nChang, C.C., Lin, C.J.: LIBSVM: a library for support vector machines. ACM Trans. Intell. Syst. Technol. (TIST) 2(3), 27 (2011)\nChen, G.H., Rockafellar, R.T.: Convergence rates in forward-backward splitting. SIAM J. Optim. 7(2), 421–444 (1997)\nCombettes, P.: The convex feasibility problem in image recovery. Adv. Imaging Electron Phys. 95, 155–270 (1996)\nFacchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, Volume I and Volume II. Springer, New York (2003)\nGiselsson, P., Fält, M., Boyd, S.: Line search for averaged operator iteration. In: 2016 IEEE 55th Conference on Decision and Control (CDC), pp. 1015–1022. IEEE (2016)\nHarker, P.T.: A variational inequality approach for the determination of oligopolistic market equilibrium. Math. Program. 30(1), 105–111 (1984)\nHe, B.: A new method for a class of linear variational inequalities. Math. Program. 66(1–3), 137–144 (1994)\nIshikawa, S.: Fixed points by a new iteration method. Proc. Am. Math. Soc. 44(1), 147–150 (1974)\nIusem, A.N., Svaiter, B.F.: A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization 42, 309–321 (1997)\nJuditsky, A., Nemirovski, A., Tauvel, C.: Solving variational inequalities with stochastic mirror-prox algorithm. Stoch. Syst. 1(1), 17–58 (2011)\nKhobotov, E.N.: Modification of the extragradient method for solving variational inequalities and certain optimization problems. USSR Comput. Math. Math. Phys. 27, 120–127 (1989)\nKinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications, vol. 31. SIAM, University City (1980)\nKonnov, I.: A class of combined iterative methods for solving variational inequalities. J. Optim. Theory Appl. 94(3), 677–693 (1997)\nKorpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomika i Matematicheskie Metody 12(4), 747–756 (1976)\nLorenz, D., Pock, T.: An inertial forward-backward algorithm for monotone inclusions. J. Math. Imaging Vis. 51(2), 311–325 (2015)\nLu, H., Freund, R.M., Nesterov, Y.: Relatively smooth convex optimization by first-order methods, and applications. SIAM J. Optim. 28(1), 333–354 (2018)\nLuke, R.D., Thao, N.H., Tam, M.K.: Quantitative convergence analysis of iterated expansive, set-valued mappings. Math. Oper. Res. 43(4), 1143–1176 (2018)\nLyashko, S.I., Semenov, V.V., Voitova, T.A.: Low-cost modification of Korpelevich’s method for monotone equilibrium problems. Cybernet. Syst. Anal. 47, 631–639 (2011)\nMalitsky, Y.: Reflected projected gradient method for solving monotone variational inequalities. SIAM J. Optim. 25(1), 502–520 (2015)\nMalitsky, Y.: Proximal extrapolated gradient methods for variational inequalities. Optim. Methods Softw. 33(1), 140–164 (2018)\nMalitsky, Y.V., Semenov, V.V.: An extragradient algorithm for monotone variational inequalities. Cybernet. Syst. Anal. 50(2), 271–277 (2014)\nMonteiro, R.D., Svaiter, B.F.: Complexity of variants of Tseng’s modified FB splitting and Korpelevich’s methods for hemivariational inequalities with applications to saddle-point and convex optimization problems. SIAM J. Optim. 21(4), 1688–1720 (2011)\nMoudafi, A., Oliny, M.: Convergence of a splitting inertial proximal method for monotone operators. J. Comput. Appl. Math. 155(2), 447–454 (2003)\nMurphy, F.H., Sherali, H.D., Soyster, A.L.: A mathematical programming approach for determining oligopolistic market equilibrium. Math. Program. 24(1), 92–106 (1982)\nNaimpally, S., Singh, K.: Extensions of some fixed point theorems of Rhoades. J. Math. Anal. Appl. 96(2), 437–446 (1983)\nNemirovski, A.: Prox-method with rate of convergence \\({O}(1\u002Ft)\\) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems. SIAM J. Optim. 15(1), 229–251 (2004)\nNemirovsky, A.: Information-based complexity of linear operator equations. J. Complex. 8(2), 153–175 (1992)\nNesterov, Y.: Dual extrapolation and its applications to solving variational inequalities and related problems. Math. Program. 109(2–3), 319–344 (2007)\nPang, J.S.: Error bounds in mathematical programming. Math. Program. 79(1–3), 299–332 (1997)\nPolyak, B.: Some methods of speeding up the convergence of iteration methods. U.S.S.R. Comput. Math. Math. Phys. 4(5), 1–17 (1967)\nPopov, L.D.: A modification of the Arrow–Hurwicz method for finding saddle points. Math. Notes 28(5), 845–848 (1980)\nQihou, L.: On Naimpally and Singh’s open questions. J. Math. Anal. Appl. 124(1), 157–164 (1987)\nSolodov, M.V.: Convergence rate analysis of iteractive algorithms for solving variational inequality problems. Math. Program. 96(3), 513–528 (2003)\nSolodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37(3), 765–776 (1999)\nSolodov, M.V., Tseng, P.: Modified projection-type methods for monotone variational inequalities. SIAM J. Control Optim. 34(5), 1814–1830 (1996)\nSu, W., Boyd, S., Candes, E.: A differential equation for modeling Nesterov’s accelerated gradient method: Theory and insights. In: Ghahramani, Z., Welling, M., Cortes, C., Lawrence, C.D., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems, pp. 2510–2518. Curran Associates, Inc. (2014). http:\u002F\u002Fpapers.nips.cc\u002Fpaper\u002F5322-a-differential-equation-for-modeling-nesterovs-accelerated-gradient-method-theory-and-insights.pdf\nThemelis, A., Patrinos, P.: Supermann: a superlinearly convergent algorithm for finding fixed points of nonexpansive operators. IEEE Trans. Autom. Control (2019). https:\u002F\u002Fieeexplore.ieee.org\u002Fdocument\u002F8675506\nTran-Dinh, Q., Kyrillidis, A., Cevher, V.: Composite self-concordant minimization. J. Mach. Learn. Res. 16(1), 371–416 (2015)\nTseng, P.: On linear convergence of iterative methods for the variational inequality problem. J. Comput. Appl. 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Comput. 10(2), 270–283 (1981)",{"id":20,"text":1024,"url":20,"identifiers":20},"Frederickson, G.N., JáJá, J.: On the relationship between the biconnectivity augmentation and traveling salesman problems. Theor. Comput. Sci. 19, 189–201 (1982)",{"id":20,"text":1026,"url":20,"identifiers":20},"Gabow, H.N., Goemans, M.X., Tardos, É., Williamson, D.P.: Approximating the smallest \\(k\\)-edge connected spanning subgraph by LP-rounding. Networks 53(4), 345–357 (2009)",{"id":20,"text":1028,"url":20,"identifiers":20},"Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM J. Comput. 24(2), 296–317 (1995)",{"id":20,"text":1030,"url":20,"identifiers":20},"Grandoni, F., Kalaitzis, C., Zenklusen, R.: Improved approximation for tree augmentation: saving by rewiring. 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Cambridge University Press, Cambridge (2011)",{"id":20,"text":1040,"url":20,"identifiers":20},"Lovàsz, L., Plummer, M.D.: Matching Theory, volume 367 of AMS\u002FChelsea Publishing. American Mathematical Society, Providence (2009)",{"id":20,"text":1042,"url":20,"identifiers":20},"Nagamochi, H.: An approximation for finding a smallest 2-edge connected subgraph containing a specified spanning tree. Discrete Appl. Math. 126, 83–113 (2003)",{"id":20,"text":1044,"url":20,"identifiers":20},"Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)",{"id":20,"text":1046,"url":20,"identifiers":20},"Sebö, A., Vygen, J.: Shorter tours by nicer ears: 7\u002F5-approximation for the graph-TSP, 3\u002F2 for the path version, and 4\u002F3 for two-edge-connected subgraphs. Combinatorica 34(5), 597–629 (2014)",{"id":20,"text":1048,"url":20,"identifiers":20},"Vempala, S., Vetta, A.: Factor 4\u002F3 approximations for minimum 2-connected subgraphs. In: Jansen, K., Khuller, S. (eds.) Approximation Algorithms for Combinatorial Optimization, Third International Workshop, APPROX 2000, Proceedings, LNCS 1913, pp. 262–273. Springer, Berlin (2000)",{"id":20,"text":1050,"url":20,"identifiers":20},"Whitney, H.: Non-separable and planar graphs. Trans. Am. Math. Soc. 34, 339–362 (1932)",{"id":20,"text":1052,"url":20,"identifiers":20},"Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011)",{"id":1054,"createTime":1055,"updateTime":1056,"relativeEntities":1057,"slug":1058,"properties":1059,"entityType":273,"verifyStatus":274,"verifyTime":1070,"verifyNote":276,"languages":20,"translateLanguages":20,"viewCount":21,"primaryUrl":1071,"fullTextUrl":20,"authors":1072,"publicationType":295,"publisherRelationship":1088,"citationCount":21,"citationInfo":1140,"publishDate":1143,"publishYear":1141,"citationAnalyzeStatus":19,"lastCitationAnalyze":1144,"indexDatabases":1145,"openAccess":20,"references":20,"isForceReanalyzing":353},"2a89fa4e-2ab5-4567-80c8-2945b21f0f3e","2024-01-11T17:58:10.897+00:00","2026-07-27T15:16:48.633+00:00",[],"A-quadratically-convergent-O-kappa-1-sqrt-n-L-iteration-algorithm-for-theP-%CE%BA-matrix-linear-complementarity-problem",{"abstract":1060,"title":1062,"gsPaper":1064,"references":1066,"doi":1068},{"EN":1061},"An interior-point predictor-corrector algorithm for theP\n*(κ)-matrix linear complementarity problem is proposed. The algorithm is an extension of Mizuno—Todd—Ye's predictor—corrector algorithm for linear programming problem. The extended algorithm is quadratically convergent with iteration complexity\n                  \n                    \n                  \n                  \n$$O((\\kappa  + 1)\\sqrt n L)$$\n\n                . It is the first polynomially and quadratically convergent algorithm for a class of LCPs that are not necessarily monotone.",{"EN":1063},"A quadratically convergent $$O((\\kappa + 1)\\sqrt n L)$$ -iteration algorithm for theP *(κ)-matrix linear complementarity problem",{"VOID":1065},"[\"10754318314789659554\"]",{"VOID":1067},"R.W. Cottle, J.S. Pang and R.E. Stone,The Linear Complementarity Problem (Academic Press, Boston, MA, 1992).\nJ. Ding and T.Y. Li, “A polynomial-time predictor—corrector algorithm for a class of linear complementarity problems,”SIAM Journal on Optimization 1 (1991) 83–92.\nA.J. Hoffman, “On approximate solutions of systems of linear inequalities,”Journal of Research of the National Bureau of Standards 49 (1952) 263–265.\nJ. Ji, F. Potra and S. Huang, “A predictor—corrector method for linear complementarity problems with polynomial complexity and superlinear convergence,” Technical Report 18, Department of Mathematics, University of Iowa, Iowa City, IA, 1991.\nM. Kojima, N. Megiddo, T. Noma and A. Yoshise,A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Lecture Notes in Computer Science, Vol. 538 (Springer, New York, 1991).\nM. Kojima, N. Megiddo and Y. Ye “An interior point potential reduction algorithm for the linear complementarity problems,”Mathematical Programming 54 (1992) 267–279.\nM. Kojima, S. Mizuno and A. Yoshise, “A primal—dual algorithm for a class of linear complementarity problems,”Mathematical Programming 44 (1989) 1–26.\nM. Kojima, S. Mizuno and A. Yoshise, “An\\(O(\\sqrt n L)\\) iteration potential reduction algorithm for linear complementarity problems,”Mathematical Programming 50 (1991) 331–342.\nS. Mizuno, “A new polynomial time method for a linear complementarity problem,”Mathematical Programming 56 (1992) 31–43.\nS. Mizuno, M. Todd and Y. Ye, “On adaptive-step primal-dual interior-point algorithms for linear programming,”Mathematics of Operations Research 18 (1992) 964–981.\nR.C. Monteiro and I. Adler, “Interior path following primal-dual algorithms. Part II: Convex quadratic programming,”Mathematical Programming 44 (1989) 43–66.\nR.C. Monteiro and S. Wright, “Local convergence of interior-point algorithms for degenerate monotone LCP,” Technical Report MCS-P357, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 1993.\nH. Samelson, R.M. Thrall and O. Wesler, “A partition theorem for Euclideann-space,”Proceedings of the American Mathematical Society 9 (1958) 805–807.\nY. Ye, “A further result on the potential reduction algorithm for theP-matrix linear complementarity problem,”Advances in Optimization and Parallel Computing (1992) 310–316.\nY. Ye and K. Anstreicher, “On quadratic and\\(O(\\sqrt n L)\\) convergence of a predictor—corrector algorithm for LCP,”Mathematical Programming 62 (1993) 537–551.\nY. Ye, O. Güler, R.A. Tapia and Y. Zhang, “A quadratically convergent\\(O(\\sqrt n L)\\)-iteration algorithm for linear programming,”Mathematical Programming 59 (1993) 151–162.\nY. Ye and P.M. Pardalos, “A class of linear complementarity problems solvable in polynomial time,”Linear Algebra and its Applications 152 (1991) 3–17.\nY. Ye, R. Tapia and Y. Zhang, “A superlinearly convergent\\(O(\\sqrt n L)\\)-iteration algorithm for linear programming,” Technical Report TR91-22, Department of Mathematical Sciences, Rice University, Houston, TX, 1991.",{"VOID":1069},"10.1007\u002FBF01585565","2024-05-11T22:43:28.046+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF01585565",[1073],{"id":1074,"sortIndex":21,"researcher":20,"roles":1075,"affiliations":1076,"properties":1085,"displayName":1087,"givenName":20,"familyName":20},"9e10e05e-ba15-45c3-adeb-e0bf73cbef1e",[282],[1077],{"id":1078,"sortIndex":21,"affiliation":1079,"properties":20},"7c9c2586-5be5-4d38-8e7c-5394db0a0eae",{"id":1078,"createTime":20,"updateTime":20,"relativeEntities":1080,"slug":20,"properties":1081,"entityType":20,"verifyStatus":20,"verifyTime":20,"verifyNote":20,"languages":20,"translateLanguages":20,"viewCount":20,"url":20,"parentIds":1084,"statistic":20},[],{"title":1082},{"VI":1083},"RUTCOR — Rutgers Center for Operations Research, Rutgers 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graph partition problem is the problem of partitioning the vertex set of a graph into a fixed number of sets of given sizes such that the sum of weights of edges joining different sets is optimized. In this paper we simplify a known matrix-lifting semidefinite programming relaxation of the graph partition problem for several classes of graphs and also show how to aggregate additional triangle and independent set constraints for graphs with symmetry. We present an eigenvalue bound for the graph partition problem of a strongly regular graph, extending a similar result for the equipartition problem. We also derive a linear programming bound of the graph partition problem for certain Johnson and Kneser graphs. Using what we call the Laplacian algebra of a graph, we derive an eigenvalue bound for the graph partition problem that is the first known closed form bound that is applicable to any graph, thereby extending a well-known result in spectral graph theory. Finally, we strengthen a known semidefinite programming relaxation of a specific quadratic assignment problem and the above-mentioned matrix-lifting semidefinite programming relaxation by adding two constraints that correspond to assigning two vertices of the graph to different parts of the partition. This strengthening performs well on highly symmetric graphs when other relaxations provide weak or trivial bounds.",{"EN":1156},"Semidefinite programming and eigenvalue bounds for the graph partition problem",{"VOID":1158},"[\"16290272782914265046\"]",{"VOID":1160},"Alizadeh, F.: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim. 5, 13–51 (1995)\nArmbruster, M., Helmberg, C., Fügenschuh, M., Martin, A.: LP and SDP branch-and-cut algorithms for the minimum graph bisection problem: a computational comparison. Math. Program. Comput. 4(3), 275–306 (2012)\nBiswas, R., Hendrickson, B., Karypis, G.: Graph partitioning and parallel computing. Parallel Comput. 26(12), 1515–1517 (2000)\nBrouwer, A.E.: Chang graphs. http:\u002F\u002Fwww.win.tue.nl\u002F~aeb\u002Fgraphs\u002FChang.html\nBrouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Berlin (1989)\nBrouwer, A.E., Haemers, W.H.: Spectra of Graphs, Springer, New York. http:\u002F\u002Fhomepages.cwi.nl\u002F~aeb\u002Fmath\u002Fipm\u002F (2012)\nBuluç, A., Meyerhenke, H., Safro, I., Sanders, P., Schulz, C.: Recent advances in graph partitioning. Preprint 2013. arXiv:1311.3144\nChang, L.C.: The uniqueness and nonuniqueness of triangular association schemes. Sci. Rec. 3, 604–613 (1959)\nDai, W., Kuh, E.: Simultaneous floor planning and global routing for hierarchical building-block layout. IEEE Trans. Comput.-Aided Des. Integr. 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In: Anjos, M.F., Lasserre, J.B. (eds.) Handbook of Semidefinite, Cone and Polynomial Optimization: Theory, Algorithms, Software and Applications, pp. 171–200. Springer, New York (2012)\nDe Klerk, E., Nagy, M., Sotirov, R., Truetsch, U.: Symmetry in RLT-type relaxations for the quadratic assignment and standard quadratic optimization problems. EJOR 233(3), 488–499 (2014)\nDelsarte, P.: An agebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10 (1973)\nDonath, W.E., Hoffman, A.J.: Lower bounds for the partitioning of graphs. IBM J. Res. Dev. 17, 420–425 (1973)\nFalkner, J., Rendl, F., Wolkowicz, H.: A computational study of graph partitioning. Math. Program. 66, 211–239 (1994)\nFiduccia, C.M., Mattheyses, R.M.: A linear-time heuristic for improving network partitions. In: Proceedings of the 19th Design Automation Conference, pp. 175–181 (1982)\nGarey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theor. Comput. 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Comput. 12, 177–191 (2000)\nKarisch, S.E., Rendl, F.: Semidefinite programming and graph equipartition. In: Pardalos, P.M., Wolkowicz, H. (eds.) Topics in Semidefinite and Interior-Point Methods, vol. 18, pp. 77–96. American Mathematical Society, Providence (1998)\nKarloff, H.: How good is the Goemans–Williamson max cut algorithm? SIAM J. Comput. 29(1), 336–350 (1999)\nLengauer, T.: Combinatorial Algorithms for Integrated Circuit Layout. Wiley, Chicester (1990)\nLöfberg, J.: YALMIP: a toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan, pp. 284–289. http:\u002F\u002Fusers.isy.liu.se\u002Fjohanl\u002Fyalmip\u002F (2004)\nMohar, B., Poljak, S.: Eigenvalues in combinatorial optimization. In: Brualdi, R.A., Friedland, S., Klee, V. (eds.) Combinatorial and Graph-Theoretical Problems in Linear Algebra. IMA Volumes in Mathematics and Its Applications, vol. 50, pp. 107–151. Springer, Berlin (1993)\nPong, T.K., Sun, H., Wang, N., Wolkowicz, H.: Eigenvalue, quadratic programming, and semidefinite programming bounds for vertex separators. Technical report, University of Waterloo, Canada (2014)\nPovh, J., Rendl, F.: Copositive and semidefinite relaxations of the quadratic assignment problem. Discrete Optim. 6(3), 231–241 (2009)\nRendl, F., Sotirov, R.: Bounds for the quadratic assignment problem using the bundle method. Math. Program. Ser. B 109(2–3), 505–524 (2007)\nRendl, F., Wolkowicz, H.: A projection technique for partitioning nodes of a graph. Ann. Oper. Res. 58, 155–179 (1995)\nRendl, F., Lisser, A., Piacentini, M.: Bandwidth, vertex separators and eigenvalue optimization. In: Bezdek, K., et al. (eds.) Discrete Geometry and Optimization, volume 69 of Fields Institute for Research in Mathematical Sciences, Communication Series, pp. 249–263. 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Version 4.5.6. http:\u002F\u002Fwww.gap-system.org (2012)\nvan Dam, E.R., Sotirov, R.: On bounding the bandwidth of graphs with symmetry. INFORMS J. Comput. (to appear)\nWedderburn, J.H.M.: On hypercomplex numbers. Proc. Lond. Math. Soc. 6(2), 77–118 (1907)\nZhao, Q., Karisch, S.E., Rendl, F., Wolkowicz, H.: Semidefinite programming relaxations for the quadratic assignment problem. J. Comb. Optim. 2, 71–109 (1998)\nWolkowicz, H., Zhao, Q.: Semidefinite programming relaxations for the graph partitioning problem. Discrete Appl. Math. 96\u002F97, 461–479 (1999)",{"VOID":1162},"10.1007\u002Fs10107-014-0817-6","2024-06-27T00:06:24.586+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs10107-014-0817-6",[1166,1183],{"id":1167,"sortIndex":21,"researcher":20,"roles":1168,"affiliations":1169,"properties":1178,"displayName":1180,"givenName":20,"familyName":20},"676e587a-93e0-4cb7-b0be-aaa3ce9016f2",[282],[1170],{"id":1171,"sortIndex":21,"affiliation":1172,"properties":20},"835847e6-1079-4ad4-bb52-6046465a38c3",{"id":1171,"createTime":20,"updateTime":20,"relativeEntities":1173,"slug":20,"properties":1174,"entityType":20,"verifyStatus":20,"verifyTime":20,"verifyNote":20,"languages":20,"translateLanguages":20,"viewCount":20,"url":20,"parentIds":1177,"statistic":20},[],{"title":1175},{"VI":1176},"Department of Econometrics and OR, Tilburg University, Tilburg, The Netherlands",[],{"title":1179,"gsAuthor":1181},{"VI":1180},"Edwin R. van Dam",{"VOID":1182},"[\"xflyEPEAAAAJ\"]",{"id":1184,"sortIndex":390,"researcher":20,"roles":1185,"affiliations":1186,"properties":1193,"displayName":1195,"givenName":20,"familyName":20},"70f04a5f-906b-48d1-adef-e9db7ea49da5",[282],[1187],{"id":1171,"sortIndex":21,"affiliation":1188,"properties":20},{"id":1171,"createTime":20,"updateTime":20,"relativeEntities":1189,"slug":20,"properties":1190,"entityType":20,"verifyStatus":20,"verifyTime":20,"verifyNote":20,"languages":20,"translateLanguages":20,"viewCount":20,"url":20,"parentIds":1192,"statistic":20},[],{"title":1191},{"VI":1176},[],{"title":1194,"gsAuthor":1196},{"VI":1195},"Renata 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describe a new potential function and a sequence of ellipsoids in the path-following algorithm for convex quadratic programming. Each ellipsoid in the sequence contains all of the optimal primal and dual slack vectors. Furthermore, the volumes of the ellipsoids shrink at the ratio\n                  \n                    \n                  \n                  \n$$2^{ - \\Omega (\\sqrt n )} $$\n\n                , in comparison to 2−Ω(1) in Karmarkar's algorithm and 2−Ω(1\u002Fn) in the ellipsoid method. We also show how to use these ellipsoids to identify the optimal basis in the course of the algorithm for linear programming.",{"EN":1267},"Containing and shrinking ellipsoids in the path-following algorithm",{"VOID":1269},"[\"17796420973546771557\"]",{"VOID":1271},"D. Bayer and J.C. Lagarias, “The non-linear geometry of linear programming, I. Affine and projective scaling trajectories, II. Legendre transform coordinates and central trajectories,”Transactions of the American Mathematical Society 314 (1989) 499–581.\nB.P. Burrell and M.J. Todd, “The ellipsoid method generates dual variables,”Mathematics of Operations Research 10 (1985) 688–700.\nR.W. Cottle and G.B. Dantzig, “Complementary pivot theory of mathematical programming“,Linear Algebra and its Applications 1 (1968) 103–125.\nG.B. Dantzig,Linear Programming and Extensions (Princeton University Press, Princeton, NJ, 1963).\nC.C. Gonzaga, “An algorithm for solving linear programming problems in O(n 3 L) operations,” in: N. Megiddo, ed.,Progress in Mathematical Programming (Springer, New York, 1988) pp. 1–28.\nC.C. Gonzaga, “Conical projection algorithms for linear programming,”Mathematical Programming 43 (1989) 151–173.\nN. Karmarkar, “A new polynomial-time algorithm for linear programming,”Combinatorica 4 (1984) 373–395.\nL.G. Khachiyan, “A polynomial algorithm for linear programming,”Doklady Akademii Nauk SSSR 244 (1979) 1093–96. [Translated in:Soviet Mathematics Doklady 20 (1979) 191–94.]\nM. Kojima, S. Mizuno and A. Yoshise, “A polynomial-time algorithm for a class of linear complementarity problems,”Mathematical Programming 44 (1989) 1–26.\nM.K. Kozlov, S.P. Tarasov and L.G. Khachiyan, “Polynomial solvability of convex quadratic programming,”Doklady Akademii Nauk SSSR 5 (1979) 1051–1053.\nN. Megiddo, “Pathways to the optimal set in linear programming,”Proceedings of the 6th Mathematical Programming Symposium of Japan (Nagoya, Japan, 1986) 1–35.\nR.C. Monteiro and I. Adler, “Interior path following primal-dual algorithms. Part II: Convex quadratic programming,”Mathematical Programming 44 (1989) 43–66.\nC.H. Papadimitriou and K. Steiglitz,Combinatorial Optimization: Algorithms and Complexity (Prentice-Hall, Englewood Cliffs, NJ, 1982).\nJ. Renegar, “A polynomial-time algorithm, based on Newton's method, for linear programming,”Mathematical Programming 40 (1988) 59–93.\nG. Sonnevend, “An ‘analytic center’ for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming,”Proceedings of the 12th IFIP Conference on System Modeling and Optimization (Budapest, 1985).\nM.J. Todd, “Improved bounds and containing ellipsoids in Karmarkar's linear programming algorithm,”Mathematics of Operations Research 13 (1988) 650–659.\nP.M. Vaidya, “An algorithm for linear programming which requires O((m+n)n 2+(m+n 1.5 n)L) arithmetic operations,” to appear in:Mathematical Programming 47 (1990) 175–201, next issue.\nY. Ye, “Interior algorithms for linear, quadratic, and linearly constrained convex programming,” Ph.D. Thesis, Department of Engineering-Economic Systems, Stanford University (Stanford, CA, 1987).\nY. Ye, “Recovering optimal basis in Karmarkar's polynomial algorithm for linear programming,” to appear in:Mathematics of Operations Research (1990).\nY. 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