Skip to main content
SpringerLink
Log in

Homotopy Methods

  • Chapter
Handbook of Global Optimization

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 2))

Abstract

This contribution discusses globally convergent methods for the solution of systems of nonlinear equations. The methods discussed are either based on piecewise linear approximations and pivoting steps or on differential topology and path following. Homotopy methods work by first solving a simple problem and then deforming this problem into the original complicated problem. During this deformation or homotopy we follow paths from the solutions of the simple problem to the solutions of the complicated problem. Concepts needed for understanding homotopies are explained and details of algorithms are given.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 469.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 599.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 599.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Abraham, R. (1967) Transversal mappings and flows, Benjamin, New York.

    MATH  Google Scholar 

  • Alexander, J.C. (1978a) ‘Matrices, eigenvalues, and complex projective space’, American Mathematical Monthly 85, 727-733.

    MathSciNet  MATH  Google Scholar 

  • Alexander, J.C. (1978b) ‘The additive inverse eigenvalue problem and topological degree’, Proceedings of the A.M.S. 70, 5-7.

    Google Scholar 

  • Alexander, J.C. (1979) ‘Numerical continuation methods and bifurcation’, in H.O. Peitgen, H.O. Walther, (eds.), Functional differential equations and approximations of fixed points, Springer, Berlin, pp. 1-15.

    Google Scholar 

  • Alexander, J.C., Yorke, J.A. (1978) ‘The homotopy continuation method: Numerically implementable topological procedures’, Trans. AMS 242, 271-284.

    MathSciNet  MATH  Google Scholar 

  • Allgower, E.L., Georg, K. (1980) ‘Simplicial and continuation methods for approximating fixed points and solutions to systems of equations’, SIAM Review 22, 28-85.

    MathSciNet  MATH  Google Scholar 

  • Allgower, E.L., Georg, K. (1990) Numerical continuation methods, Springer, Berlin.

    Google Scholar 

  • Allgower, E.L., Georg, K. (1992) ‘Continuation and path following’, Acta Numerica, 1-64.

    Google Scholar 

  • Allgower, E.L., Glashoff, K., Peitgen, H.O. (1981) Numerical solution of nonlinear equations, Springer, Berlin.

    MATH  Google Scholar 

  • Asmuth, R. , Eaves, B.C., Peterson, E. (1979), ‘Computing economic equilibria on affine networks with Lemke’s algorithm’, Mathematics of Operations Research 4, 209-214.

    MathSciNet  MATH  Google Scholar 

  • Bezout, £. (1779) ‘Theorie generale des equations algebriques’, Ph.D. Pierres, Paris.

    Google Scholar 

  • Bland, R.G. (1977) ‘New finite pivoting rules for the simplex method’, Mathematics of Operations Research 2, 103-107.

    MathSciNet  MATH  Google Scholar 

  • Borsuk, K. (1933) ‘Drei Satze liber die n-dimensionale euklidische Sphare’, Fundamenta Mathematicae 20, 177-190.

    Google Scholar 

  • Brooks, R.B.S., Brown, R.F., Pak, J., Taylor, D.H. (1975) ‘Nielsen number of maps of tori’, Proc. A.M.S. 52, 398-400.

    Google Scholar 

  • Brown, R.F. (1971) The Lefschetz fixed point theorem, Scott, Foresman and Company, Glenview.

    MATH  Google Scholar 

  • Brouwer, L.E.J. (1912) ‘Ober Abbildung von Mannigfaltigkeiten’, Mathematische Annalen 71, 97-115.

    MATH  Google Scholar 

  • Brouwer, L.E.J. (1926) ‘On transformations of projective spaces’, KNAW Proc. 29, 864-865 = L.E.J. Brouwer Collected Works, North-Holland, Amsterdam, 1976, pp. 475-476.

    Google Scholar 

  • Chakraborty, A., Allison, D.C.S., Ribbens, C.J., Watson, L.T. (1991) ‘Note on unit tangent vector computation for homotopy curve tracking on a hypercube’, Parallel Computing 17, 1385-1395.

    MathSciNet  MATH  Google Scholar 

  • Charnes, A. (1952) ‘Optimality and degeneracy in linear programming’, Econometrica 20, 160-170.

    MathSciNet  MATH  Google Scholar 

  • Charnes, A., Garcia, C.B., Lemke, C.E. (1977) ‘Constructive proofs of theorems relating to: F(x) = y , with applications’, Mathematical Programming 12, 328-343.

    MathSciNet  MATH  Google Scholar 

  • Chow, S. , Mallet-Paret, J., Yorke, J.A. (1978) ‘Finding zeroes of maps: Homotopy methods that are constructive with probability one’, Mathematics of Computation 32, 887-899.

    MathSciNet  MATH  Google Scholar 

  • Cohn, P.M. (1974) Agebra, Volume 1, Wiley, London.

    Google Scholar 

  • Colville, A.R. (1968) ‘A comparative study of nonlinear programming codes’, IBM New York Scientific Center Report No. 320-2949.

    Google Scholar 

  • Cottle, R.W. (1972) ‘Monotone solutions of the parametric linear complementarity problem’, Mathematical Programming 3, 210-224.

    MathSciNet  MATH  Google Scholar 

  • Cottle, R.W. (1977) ‘Computational experience with large-scale linear complementarity problems’, in S. Karamardian (ed.), Fixed points - algorithms and applications, Academic press, New York, pp. 281-313.

    Google Scholar 

  • Cottle, R.W., Pang, J.S. (1978) ‘A least-element theory of solving linear complementarity problems as linear programs’, Mathematics of Operations Research 3, 155-170.

    MathSciNet  MATH  Google Scholar 

  • Cottle, R.W., Pang, J.S., Stone, R.E. (1992) The linear complementarity problem, Academic Press, New York.

    MATH  Google Scholar 

  • Dahlquist, G. (1959) ‘Stability and error bounds in the numerical integration of ordinary differential equations’, Trans. Roy. Inst. Techno1., Nr. 130, Stockholm.

    Google Scholar 

  • Dang, C. (1991a) ‘The Dl-Triangulation of (Rn for simplicial algorithms for computing solutions of nonlinear equations’, Mathematics of Operations Research 16, 148-161.

    MathSciNet  MATH  Google Scholar 

  • Dang, C. (1991b) The Dl-Triangulation in simplicial algorithms, Ph.D. thesis, University of Tilburg, Netherlands.

    Google Scholar 

  • Dang, C. (1992) ‘ D3-Triangulation for simplicial deformation algorithms for computing solutions of nonlinear equations’, Journal of Optimization Theory and Applications 75, 51-67.

    MathSciNet  MATH  Google Scholar 

  • Dantzig, G.B., Orden, A., Wolfe, P. (1955) ‘The generalized simplex method for minimizing a linear form under linear inequality restraints’, Pacific J. Math. 5, 183-195.

    MathSciNet  MATH  Google Scholar 

  • Davenport, J.H., Siret, Y., et al. (1988) Computer algebra, Academic Press, London.

    MATH  Google Scholar 

  • Davidenko, D.F. (1953) ‘On the approximate solution of a system of nonlinear equations’, Dokl. Akad. Nauk. SSSR 88, 601-602.

    MathSciNet  MATH  Google Scholar 

  • Debreu, G. (1959) Theory of value, Yale University Press, New Haven.

    MATH  Google Scholar 

  • Dehn, M., Heegaard, P. (1907) ‘Analysis Situs’, Enzykl. Math. Wiss. Ill A B 3, Leipzig.

    Google Scholar 

  • Deschamps, P.J. (1977) ‘Pricing for congestion in telephone networks: A numerical example’ , in S. Karamardian (ed.), Fixed points - algorithms and applications, Academic press, New York, pp. 473-494.

    Google Scholar 

  • Deuflhard, P., Fiedler, B. , Kunkel, P. (1987) ‘Efficient numerical pathfollowing beyond critical points’, SIAM J. Numer. Anal. 24, 912-927.

    MathSciNet  MATH  Google Scholar 

  • Dunyak, J.P., Junkins, J.L., Watson, L.T. (1984) ‘Robust nonlinear least squares estimation using the Chow-Yorke homotopy method’, Journal of Guidance , Control, and Dynamics 7, 752-754.

    MATH  Google Scholar 

  • Eaves, B.C. (1971) ‘The linear complementarity problem’, Management Sciences 17, 612-634.

    MathSciNet  MATH  Google Scholar 

  • Eaves, B.C. (1979) ‘A view of complementary pivot theory (or solving equations with homotopies)’, in H.O. Peitgen, H.O. Walther, (eds.), Functional differential equations and approximations of fixed points, Springer, Berlin, pp. 89-111.

    Google Scholar 

  • Eaves, B.C. (1984) A course in triangulations for solving equations with deformations, Springer, Berlin.

    MATH  Google Scholar 

  • Eaves, B.C., Gould, F.J., Peitgen, H.O., Todd, M.J. (1983) Homotopy methods and global convergence, Plenum Press, New York.

    MATH  Google Scholar 

  • Eaves, B.C., Scarf, H. (1976) ‘The solution of systems of piecewise linear equations’, Mathematics of Operations Research 1, 1-27.

    MathSciNet  MATH  Google Scholar 

  • Engles, C.R. (1980) ‘Economic equilibrium under deformation of the economy’, in S.M. Robinson, (ed.), Analysis and computation of fixed points, Academic Press, New York, pp. 213-410.

    Google Scholar 

  • Fadell, E. , Fournier, G. (1981) Fixed point theory, Springer, Berlin.

    Google Scholar 

  • Fournier, G. (1981) ’A simplicial approach to the fixed point index, in E. Fadell, G. Fournier, (eds.), Fixed point theory, Springer, Berlin, pp. 73-102.

    Google Scholar 

  • Forster, W. (1978) ‘The structure of computational methods: O.D.E’s and the algebra C[a, b] ’, ZAMM 57, T 411-T 478.

    Google Scholar 

  • Forster, W. (1980) Numerical solution of highly nonlinear problems, North-Holland, Amsterdam.

    MATH  Google Scholar 

  • Forster, W. (1980) ‘Fixed point algorithms and array processors’, in W. Forster, (ed.), Numerical solution of highly nonlinear problems, North-Holland, Amsterdam, pp. 169-179.

    Google Scholar 

  • Forster, W. (1980) ’Fifty years of further development of a combinatorial lemma, Part C, in W. Forster, (ed.), Numerical solution of highly nonlinear problems, North-Holland, Amsterdam, pp. 215-217.

    Google Scholar 

  • Forster, W. (1987) ‘Computing all solutions of systems of polynomial equations by simplicial fixed point algorithms’, in A. Talman, G. van der Laan (eds.), The computation and modelling of economic equilibria, North-Holland, Amsterdam.

    Google Scholar 

  • Forster, W. (1992) ‘Some computational methods for systems of nonlinear equations and systems of polynomial equations’, Journal of Global Optimization 2, 317-356.

    MathSciNet  MATH  Google Scholar 

  • Freudenthal, H. (1942) ‘Simplizialzerlegungen von beschrankter Flachheit’, Annals of Mathematics 43, 580-582.

    MathSciNet  MATH  Google Scholar 

  • Freund, R.M., Todd, M.J. (1979) ‘A constructive proof of Tucker’s combinatorial lemma’, Technical Report No. 403, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York.

    Google Scholar 

  • Freund, R.M. (1986) ‘Combinatorial theorems on the simplotope that generalize results on the simplex and cube’, Math. Oper. Res. 11, 169-179.

    MathSciNet  MATH  Google Scholar 

  • Garcia, C.B. (1973) ‘Some classes of matrices in linear complementarity theory’, Mathematical Programming 5, 299-310.

    MathSciNet  MATH  Google Scholar 

  • Garcia, C.B., Gould, F.J., Turnbull, T.R. (1983) ‘Relations between pi maps, complementary cones, and degree in linear complementarity problems’, in B.C. Eaves, F.J. Gould, H.O. Peitgen, M.J. Todd (eds.), Homotopy methods and global convergence, Plenum Press, New York, pp. 91-144.

    Google Scholar 

  • Georg, K. (1979) ‘An application of simplicial algorithms to variational inequalities’, in H.O. Peitgen, H.O. Walther, (eds.), Functional differential equations and approximations of fixed points, Springer, Berlin, pp. 126-135.

    Google Scholar 

  • Gould, F.J. (1980) ‘Recent and past developments in the simplicial approximation approach to solving systems of nonlinear equations - a subjective view’, in A.V. Fiacco, K.O. Kortanek (eds.), Extremal methods in systems analysis, Springer, Berlin, pp. 466-480.

    Google Scholar 

  • Gould, F.J., Tolle, J.W. (1983) Complementary pivoting on a pseudomanifold structure with applications in the decision sciences, Heldermann, Berlin.

    Google Scholar 

  • Guillemin, V., Pollack, A. (1974) Differential topology, Prentice Hall, Englewood Cliffs, New Jersey.

    Google Scholar 

  • Hadley, G. (1964) Nonlinear and dynamic programming, Addison Wesley, Reading, Massachusetts.

    MATH  Google Scholar 

  • Henrici, P. (1962) Discrete variable methods in ordinary differential equations’, Wiley, New York.

    Google Scholar 

  • Heruska, M.W., Watson, L.T., Sankara, K.K. (1986) ‘Micropolar flow past a porous stretching sheet’, Computers & Fluids 14, 117-129.

    Google Scholar 

  • Hirsch, M. (1963) ‘A proof of the nonretractibility of a cell onto its boundary’, Proc. Amer. Math. Soc. 14, 364-365.

    MathSciNet  MATH  Google Scholar 

  • Hocking, J.G., Young, G.S. (1961) Topology, Addison-Wesley, Reading, Massachusetts.

    MATH  Google Scholar 

  • Holodniok, M. , Kubicek, M. (1984) ‘ DERPER - An algorithm for the continuation of periodic solutions in ordinary differential equations’, J. Comp. Phys. 55, 254-267.

    MathSciNet  MATH  Google Scholar 

  • Hopf, H. (1927) ‘Uber Mindestzahlen von Fixpunkten’, Mathematische Zeitschrift 26, 762-774.

    MathSciNet  MATH  Google Scholar 

  • Hopf, H. (1945) ‘Eine Verallgemeinerung bekannter Abbildungs und Oberdeckungssatze’, Portug. Math. 4, 129-139.

    MathSciNet  Google Scholar 

  • Horst, R., Tuy, H. (1990) Global Optimization, Springer, Berlin.

    Google Scholar 

  • Hurewicz, W. (1935a) ‘Beitrage zur Toplogie der Deformationen: I. Hoherdimensionale Homotopiegruppen’, Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Sciences 38, 112-119.

    Google Scholar 

  • Hurewicz, W. (1935b) ‘Beitrage zur Toplogie der Deformationen: II. Homotopie- und Homologiegruppen’, Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Sciences 38, 521-528.

    Google Scholar 

  • Hurewicz, W. (1936a) ‘Beitrage zur Toplogie der Deformationen: III. Klassen und Homologietypen von Abbildungen’, Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Sciences 39, 117-126.

    Google Scholar 

  • Hurewicz, W. (1936b) ‘Beitrage zur Toplogie der Deformationen: IV. Aspharische Raume’, Koninklijke Akademie van Wetenschappen te Amsterdam, Proceedings of the Section of Sciences 39, 215-224.

    MATH  Google Scholar 

  • Jiang, B. (1983) Lectures on Nielsen fixed point theory, A.M.S., Providence, Rhode Island.

    MATH  Google Scholar 

  • Jordan, C. (1866) ‘Des contours traces sur les surfaces’, J. Math, Pures Appl. 2, 110-130.

    Google Scholar 

  • Jurgens, H. , Peitgen, H.O., Saupe, D. (1980) ‘Topological perturbation in the numerical study of nonlinear eigenvalue and bifurcation problems’, in S.M. Robinson, (ed.), Analysis and computation of fixed points, Academic Press, New York, pp. 139-181.

    Google Scholar 

  • Kaneko, I. (1979) ‘The number of solutions of a class of linear complementarity problems’, Mathematical Programming 17, 104-105.

    MathSciNet  MATH  Google Scholar 

  • Kakutani, S. (1941) ‘A generalization of Brouwer’s fixed point theorem’, Duke Math. J. 8, 457-459.

    MathSciNet  Google Scholar 

  • Karamardian, S. (1977) Fixed points - algorithms and applications, Academic Press, New York.

    MATH  Google Scholar 

  • Kellogg, R.B., Li, T.Y., Yorke, J. (1976) ‘A constructive proof of the Brouwer fixed-point theorem and computational results’, SIAM J. Num. Anal. 13, 473-483.

    MathSciNet  MATH  Google Scholar 

  • Kiang, Tsai-Han (1989) The theory of fixed point classes, Springer, Berlin.

    MATH  Google Scholar 

  • Knaster, B. , Kuratowski, K. , Mazurkiewicz, S. (1929) ’Ein Beweis des Fixpunktsatzes fur n-dimensionale Simplexe, Fundamenta Math. 14, 132-137.

    MATH  Google Scholar 

  • Kojima, M. (1981) Simplicial fixed point algorithms, Tokyo, (in Japanese).

    Google Scholar 

  • Kojima, M. (1981) ‘An introduction to variable dimension algorithms for solving systems of equations’, in E.L. Allgower, K. Glashoff, H.O. Peitgen, (eds.),Numerical solution of nonlinear equations, Springer, Berlin, pp. 199-237.

    Google Scholar 

  • Kojima, M. , Mizuno, S. (1983) ‘Computation of all solutions to a system of polynomial equations’, Mathematical Programming 25, 131-157.

    MathSciNet  MATH  Google Scholar 

  • Kojima, M., Yamamoto, Y. (1984) ‘A unified approach to the implementation of several fixed point algorithms and a new variable dimension algorithm’, Mathematical Programming 28, 288-328.

    MathSciNet  MATH  Google Scholar 

  • Kostreva, M.M., Kinard, L.A. (1991) ‘A differentiable homotopy approach for solving polynomial optimization problems and noncooperative games’, Computers Math. Applic. 21, 135-143.

    MathSciNet  MATH  Google Scholar 

  • Kubicek, M. (1976) ‘Algorithm 502. Dependence of solutions of nonlinear systems on a parameter’, ACM Trans. Math. Software 2, 98-107.

    MATH  Google Scholar 

  • Kuhn, H.W. (1974) ‘A new proof of the fundamental theorem of algebra’, Mathematical Programming Study 1, 148-158.

    Google Scholar 

  • Kuhn, H.W. (1977) ‘Finding roots of polynomials by pivoting’, in S. Karamardian (ed.), Fixed points - Algorithms and applications, Academic Press, New York, pp. 156-163.

    Google Scholar 

  • Kuhn, H.W. (1980) ‘Pivotal exchange algorithms’, in L.C.W. Dixon, E. Spedicato, G.P. Szego (eds.), Nonlinear optimization, Birkhauser, Boston, pp. 319-386.

    Google Scholar 

  • Kuhn, H.W., Wang, Z. , Xu, S. (1984) ‘On the cost of computing roots of polynomials’, Mathematical Programming 32, 301-318.

    MathSciNet  Google Scholar 

  • Kumar, S.K., Thacker, W.I., Watson, L.T. (1986 ?) ’Magnetohydrodynamic flow and heat transfer about a rotating disk with suction and injection at the disk surface, Report, Department of Electrical Engineering and Computer Science, University of Michigan.

    Google Scholar 

  • Ky Fan (1952) ‘A generalization of Tucker’s combinatorial lemma with topological applications’, Ann. of Math. 56, 431-437.

    MathSciNet  MATH  Google Scholar 

  • Ky Fan (1960) ‘Combinatorial properties of certain simplicial and cubical vertex maps’, Arch. Math. 11, 368-377.

    MATH  Google Scholar 

  • Lemke, C.E. (1965) ‘Bimatrix equilibrium points and mathematical programming’, Management Sciences 11, 681-689.

    MathSciNet  MATH  Google Scholar 

  • Lemke, C.E. (1970) ‘Recent results on complementarity problems’, in J.B. Rosen, O.L. Mangasarian, K. Ritter (eds.), Nonlinear programming, Academic Press, New York, pp. 349-384.

    Google Scholar 

  • Lemke, C.E. (1980) ‘A survey of complementarity theory’, in R.W. Cottle, F. Giannessi, J.L. Lions (eds.), Variational inequalities and complementarity problems, Wiley, Chichester, pp. 213-239.

    Google Scholar 

  • Lemke, C.E., Howson, J.T. (1964) ‘Equilibrium points of bimatrix games’, J. Soc. Indust. Appl. Math. 12, 413-423.

    MathSciNet  MATH  Google Scholar 

  • Li, T.Y. (1983) ‘On locating all zeros of an analytic function within a bounded domain by a revised Delves/Lyness method’, SIAM J. Numer. Anal. 20, 865-871.

    MATH  Google Scholar 

  • Li, T.Y., Rhee, N.H. (1989) ‘Homotopy algorithms for symmetric eigenvalue problems’ , Numer. Math. 55, 265-280.

    MathSciNet  MATH  Google Scholar 

  • Li, T.Y., Zeng, Z., Cong, L. (1992) ‘Solving eigenvalue problems of real nonsymmetric matrices with real homotopies’, SIAM J. Numer. Anal. 29, 229-248.

    MathSciNet  MATH  Google Scholar 

  • Liithi, H.J. (1976) Komplementaritats- und Fixpunktalgorithmen in der mathematischen Programmierung, Spieltheorie und Okonomie, Springer Verlag, Berlin.

    Google Scholar 

  • MacKinnon, J. (1977) ‘Solving economic general equilibrium models by the sandwich method’, in S. Karamardian (ed.), Fixed points - algorithms and applications, Academic press, New York, pp. 367-402.

    Google Scholar 

  • MacKinnon, J. (1980) ‘Solving urban general equilibrium models by fixed point methods’, in S.M. Robinson, (ed.), Analysis and computation of fixed points, Academic Press, New York, pp. 197-212.

    Google Scholar 

  • Mangasarian, O.L. (1976) ‘Equivalence of the complementarity problem to a system of nonlinear equations’, SIAM J. Appl. Math. 31, 89-92.

    MathSciNet  MATH  Google Scholar 

  • Mathiesen, L., Hansen, T. (1980) ‘An equilibrium model for an open economy with institutional constraints on factor prices’, in W. Forster, (ed.), Numerical solution of highly nonlinear problems, North-Holland, Amsterdam, pp. 337-360.

    Google Scholar 

  • Merrill, O.H. (1972) Applications and extensions of an algorithm that computes fixed points of certain upper semi-continuous point to set mappings, Ph.D. thesis, University of Michigan.

    Google Scholar 

  • Meyerson, M.D., Wright, A.H. (1979) ‘A new constructive proof of the Borsuk-Ulam theorem’, Proceedings A.M.S. 73, 134-136.

    Google Scholar 

  • Miller, M.H., Spencer, J.E. (1977) ‘The static economic effects of the U.K. joining the E.E.C.: A general equilibrium approach’, The Review of Economic Studies 44, 71-93.

    Google Scholar 

  • Milnor, J.W. (1965) Topology from the differentiable viewpoint, The University Press of Virginia, Charlottesville.

    MATH  Google Scholar 

  • Morgan, A. (1983) ‘A method for computing all solutions to systems of polynomial equations’, ACM Transactions n Mathematical Software 9, 1-17.

    MATH  Google Scholar 

  • Morgan, A. (1986) ‘A homotopy for solving polynomial systems’, Applied Mathematics and Computation 18, 87-92.

    MathSciNet  MATH  Google Scholar 

  • Morgan, A. (1987) Solving polynomial systems using continuation for engineering and scientific problems, Prentice Hall, Englewood Cliffs, New Jersey.

    Google Scholar 

  • Morgan, A., Sommese, A.J., Wampler, C.W.(1991) ‘Computing singular solutions to nonlinear analytic systems’, Numer. Math. 58, 669-684.

    MathSciNet  MATH  Google Scholar 

  • Murty, K.G. (1972) ‘On the number of solutions to the complementarity problem and spanning properties of complementary cones’, L. of Linear Algebra and its Applications 5, 65-108.

    MATH  Google Scholar 

  • Nielsen, J. (1921) ‘Ober die Minimalzahl der Fixpunkte bei den Abbildungen der Ringflachen’, Mathematische Annalen 82, 83-93.

    Google Scholar 

  • Ortega, J.M., Rheinboldt, W.C. (1970) Iterative solution of nonlinear equations in several variables, Academic Press, New York.

    MATH  Google Scholar 

  • Pang, J.S. (1979) ‘On Q-matrices’, Mathematical Programming 17, 243-247.

    MathSciNet  MATH  Google Scholar 

  • Pang, J.S., Kaneko, I., Hallman, W.P. (1979) ’On the solution of some (parametric) linear complementarity problems with applications to portfolio selection, structural engineering, and actuarial graduation*, Mathematical Programming 16, 325-347.

    MathSciNet  MATH  Google Scholar 

  • Peitgen, H.O., Priifer, M. (1979) ‘The Leray-Schauder continuation method is a constructive element in the numerical study of nonlinear eigenvalue and bifurcation problems’, in H.O. Peitgen, H.O. Walther, (eds.), Functional differential equations and approximations of fixed points, Springer, Berlin, pp. 326-409.

    Google Scholar 

  • Peitgen, H.O., Siegberg, H.W. (1979) ‘On computational aspects of topological degree in IR ‘ , in H.O. Peitgen, H.O. Walther, (eds.), Functional differential equations and approximations of fixed points, Springer, Berlin, pp. 410-433.

    Google Scholar 

  • Peitgen, H.O., Siegberg, H.W.(1981) ‘An e-perturbation of Brouwer’s definition of degree’, in E. Fadell, G. Fournier, (eds.), Fixed point theory, Springer, Berlin, pp. 331-366.

    Google Scholar 

  • Peitgen, H.O., Walther, H.O. (1979) Functional differential equations and approximations of fixed points, Springer, Berlin.

    Google Scholar 

  • Priifer, M. (1978) ‘Calculating global bifurcation’, in H. Wacker (ed.), Continuation methods, Academic Press, New York, pp. 187-213.

    Google Scholar 

  • Reiser, P.M. (1981) ‘A modified integer labeling for complementarity algorithms’, Mathematics of Operations Research 6, 129-139.

    MathSciNet  MATH  Google Scholar 

  • Renegar, J. (1985) ‘On the cost of approximating all rots of a complex polynomial’, Mathematical Programming 32, 319-336.

    MathSciNet  MATH  Google Scholar 

  • Rheinboldt, W.C., Burkardt, J.V. (1983) ‘A locally-parametrized continuation process’, ACM Transactions on Mathematical Software 9, 215-235.

    MathSciNet  MATH  Google Scholar 

  • Robinson, S.M. (1980) Analysis and computation of fixed points, Academic Press, New York.

    MATH  Google Scholar 

  • Ruys, P.H.M., Van der Laan, G. (1987) ‘Computation of an industrial equilibrium’, in A. Talman and G. Van der Laan (eds.), The computation and modelling of economic equilibria, North-Holland, Amsterdam, pp. 205-229.

    Google Scholar 

  • Saigal, R. (1977) ‘On the convergence rate of algorithms for solving equations that are based on methods of complementary pivoting’, Mathematics of Operations Research 2, 108-124.

    MathSciNet  MATH  Google Scholar 

  • Saigal, R. , Shin, Y.S, (1979) ‘Perturbation in fixed point algorithms’, in H.O. Peitgen, H.O. Walther, (eds.), Functional differential equations and approximations of fixed points, Springer, Berlin, pp. 434-441.

    Google Scholar 

  • Saigal, R. , Todd, M.J. (1978) ‘Efficient acceleration techniques for fixed point algorithms’, SIAM J. Numer. Anal. 15, 997-1007.

    MathSciNet  MATH  Google Scholar 

  • Sankara, K.K., Watson, L.T. (1985) ‘Micropolar flow past a stretching sheet’, Journal of Applied Mathematics and Physics (ZAMP) 36, 845-853.

    MathSciNet  MATH  Google Scholar 

  • Scarf, H. (1967) ‘The approximation of fixed points of a continuous mapping’, SIAM J. Applied Math. 15, 1328-1343.

    MathSciNet  MATH  Google Scholar 

  • Scarf, H. (1973) The computation of economic equilibria, Yale University Press, New Haven.

    MATH  Google Scholar 

  • Shapley, L.S. (1974) ‘A note on the Lemke-Howson algorithm’, Mathematical Programming Study 1, 175-189.

    MathSciNet  Google Scholar 

  • Shoven, J.B. (1977) ‘Applying fixed point algorithms to the analysis of tax policies’, in S. Karamardian (ed.), Fixed points - algorithms and applications, Academic press, New York, pp.403-434.

    Google Scholar 

  • Shoven, J.B. (1983) ‘The application of fixed point methods to economics’, in B.C. Eaves et al. (eds.), Homotopy methods and global convergence, Plenum Press, New York, pp. 249-261.

    Google Scholar 

  • Sperner, E. (1928) ‘Neuer Beweis fur die Invarianz der Dimensionszahl und des Gebietes’, Abh. math. Sem. Hamburg 6, 265-272.

    MathSciNet  MATH  Google Scholar 

  • Sperner, E. (1980) ’Fifty years of further development of a combinatorial lemma, Part B, in Forster, W. (ed.), Numerical solution of highly nonlinear problems, North-Holland, Amsterdam, pp. 199-214.

    Google Scholar 

  • Stickney, S. , Watson, L.T. (1978) ‘Digraph models of bard-type algorithms for the linear complementarity problem’, Mathematics of Operations Research 3, 322-333.

    MathSciNet  MATH  Google Scholar 

  • Talman, A.J.J. (1980) Variable dimension fixed point algorithms and triangulations, Mathematisch Centrum, Amsterdam.

    MATH  Google Scholar 

  • Talman, A.J.J., Van der Heyden, L. (1983) ‘Algorithms for the linear complementarity problem which allow an arbitrary starting point’, in B.C. Eaves, F.J. Gould, H.0. Peitgen, M.J. Todd (eds.), Homotopy methods and global convergence, Plenum Press, New York, pp. 267-285.

    Google Scholar 

  • Talman, A.J.J., Van der Laan, G. (1987) The computation and modelling of economic equilibria, North-Holland, Amsterdam.

    Google Scholar 

  • Todd, M.J. (1974) ‘A generalized complementary pivoting algorithm’, Mathematical Programming 6, 243-263.

    MathSciNet  MATH  Google Scholar 

  • Todd, M.J. (1976) The computation of fixed points and applications, Springer, Berlin.

    MATH  Google Scholar 

  • Todd, M.J. (1976) ‘Union Jack triangulations’, in S. Karamardian (ed.), Fixed points - algorithms and applications, Academic press, New York, pp.315-336.

    Google Scholar 

  • Todd, M.J. (1980) ‘PLALGO: A FORTRAN implementation of a piecewise linear homotopy algorithm for solving systems of nonlinear equations’, Technical Report No. 452, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York.

    Google Scholar 

  • Todd, M.J. (1987) ‘Reformulations of economic equilibrium problems for solution by quasi-Newton and simplicial algorithms’, in A. Talman and G. Van der Laan (eds.), The computation and modelling of economic equilibria, North-Holland, Amsterdam, pp. 19-37.

    Google Scholar 

  • Tucker, A.W. (1945) ‘Some topological properties of disk and sphere’, in Proc. First Can. Math. Congress, Montreal, pp. 285-309.

    Google Scholar 

  • Van de Panne, C., (1974) ‘A complementary variant of Lemke’s method for the linear complementarity problem’, Mathematical Programming 7, 283-310.

    MathSciNet  MATH  Google Scholar 

  • Van der Heyden, L. (1979) ‘Refinement methods for computing fixed points using primitive sets’, Ph.D. thesis, Yale University.

    Google Scholar 

  • Van der Heyden, L. (1980) ‘A variable dimension algorithm for the linear complementarity problem’, Mathematical Programming 19, 328-346.

    MathSciNet  MATH  Google Scholar 

  • Van der Heyden, L. (1987) ‘On a theorem of Scarf’, in A. Talman and G. Van der Laan (eds.), The computation and modelling of economic equilibria, North-Holland, Amsterdam, pp. 177-192.

    Google Scholar 

  • Van der Laan, G. (1980), Simplicial fixed point algorithms, Mathematisch Centrum, Amsterdam.

    MATH  Google Scholar 

  • Van der Laan, G. (1987) ‘Adjustment processes for finding economic equilibria’, in A. Talman and G. Van der Laan (eds.), The computation and modelling of economic equilibria, North-Holland, Amsterdam, pp. 85-123.

    Google Scholar 

  • Van der Laan, G. , Talman, A.J.J. (1979) ‘A restart algorithm without an artificial level for computing fixed points on unbounded regions’ in H.0. Peitgen, H.0. Walther, (eds.),

    Google Scholar 

  • Functional differential equations and approximations of fixed points, Springer, Berlin, pp. 247-256.

    Google Scholar 

  • Van der Laan, G. , Talman, A.J.J. (1981) ‘Labelling rules and orientation: On Sperner’s lemma and Brouwer degree’, in E.L.

    Google Scholar 

  • Allgower, K. Glashoff, H.O. Peitgen, (eds.),Numerical solution of nonlinear equations, Springer, Berlin, pp. 238-257.

    Google Scholar 

  • Wang, Z. (1986) Simplicial fixed point algorithms, Peking, (in Chinese).

    Google Scholar 

  • Watson, L.T. (1979a) ‘Solving the nonlinear complementarity problem by a homotopy method’ , SIAM J. of Control and Optimization 17, 36-46.

    MATH  Google Scholar 

  • Watson, L.T. (1979b) ‘A globally convergent algorithm for computing 2 fixed points of C maps’, Applied Mathematics and Computation 5, 297-311.

    MathSciNet  MATH  Google Scholar 

  • Watson, L.T. (1981) ‘Engineering applications of the Chow-Yorke algorithm’, Applied Mathematics and Computation 9, 111-133.

    MathSciNet  MATH  Google Scholar 

  • Watson, L.T. (1986) ‘Numerical linear algebra aspects of globally convergent homotopy methods’, SIAM Review 28, 529-545.

    MathSciNet  MATH  Google Scholar 

  • Watson, L.T., Billups, S.C., Morgan, A.P. (1987) ‘Algorithm 652 Hopmpack: A suite of codes for globally convergent homotopy algorithms’, ACM Transactions on Mathematical Software 13, 281-310.

    MathSciNet  MATH  Google Scholar 

  • Watson, L.T., Fenner, D. (1980) ‘Algorithm 555. Chow-Yorke 2 algorithm for fixed points or zeros of C maps’, ACM Transactions on Mathematical Software 6, 252-259.

    MATH  Google Scholar 

  • Watson, L.T., Haftka, R.T. (1989) ‘Modern homotopy methods in optimization’ , Computer Methods in Applied Mechanics and Engineering 74, 289-305.

    MathSciNet  MATH  Google Scholar 

  • Watson, L.T., Holzer, S.M., Hansen, M.C. (1983) ‘Tracking nonlinear equilibrium paths by a homotopy method’, Nonlinear Analysis, Theory, Methods & Applications 7, 1271-1282.

    MathSciNet  MATH  Google Scholar 

  • Watson, L.T., Wang, C.Y. (1982) ‘Overhang of a heavy elastic sheet’, Journal of Applied Mathematics and Physics 33, 17-23.

    MATH  Google Scholar 

  • Watson, L., Yang, W.H. (1980) ‘Optimal design by a homotopy method’, Applicable Analysis 10, 275-284.

    MathSciNet  MATH  Google Scholar 

  • Whalley, J. (1977) ‘Fiscal harmonization in the EEC; Some preliminary findings of fixed point calculations’, in S.

    Google Scholar 

  • Karamardian (ed.), Fixed points - algorithms and applications, Academic press, New York, pp. 435-471.

    Google Scholar 

  • Whalley, J., Piggott, J. (1980) ‘General equilibrium analysis of taxation policy’, in S.M. Robinson, (ed.), Analysis and computation of fixed points, Academic Press, New York, pp. 183-195.

    Google Scholar 

  • Wille, F. (1981) ‘On a conjecture of Hopf for a-separating maps from manifolds into spheres’, in E. Fadell and G. Fournier (ems.), Fixed Point Theory, Springer, Berlin, pp. 435-446.

    Google Scholar 

  • Wilmuth, R. (1977) ’A computational comparison of fixed point algorithms which use complementary pivoting’, in S. Karamardian (ed.), Fixed points - algorithms and applications, Academic press, New York, pp. 249-280.

    Google Scholar 

  • Wilmuth, R. (1977) ‘A computational comparison of fixed point algorithms which use complementary pivoting’, in S. Karamardian (ed.), Fixed points - algorithms and applications, Academic press, New York, pp. 249-280.

    Google Scholar 

  • Wright, A.H. (1985) ‘Finding all solutions to a system of polynomial equations’, Mathematics of Computation 44, 125-133.

    MathSciNet  MATH  Google Scholar 

  • Yamamoto, Y. (1988) ‘Orientability of pseudomanifold and generalizations of Sperner’s lemma’, Journal of the Operations Research Society of Japan 31, 19-42.

    MathSciNet  MATH  Google Scholar 

  • Yoseloff, M. (1974) ‘Topological proofs of some combinatorial theorems’, J. Comb. Theory (A) 17, 95-111.

    MathSciNet  MATH  Google Scholar 

  • Zangwill, W.I., Garcia, C.B. (1981) Pathways to solutions, fixed points, and equilibria, Prentice Hall, Englewood Cliffs.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematical Studies, University of Southampton, Southampton, England

    W. Forster

Authors
  1. W. Forster
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Trier, Trier, Germany

    Reiner Horst

  2. Department of Industrial and Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Forster, W. (1995). Homotopy Methods. In: Horst, R., Pardalos, P.M. (eds) Handbook of Global Optimization. Nonconvex Optimization and Its Applications, vol 2. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2025-2_13

Download citation

  • DOI: https://doi.org/10.1007/978-1-4615-2025-2_13

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-5838-1

  • Online ISBN: 978-1-4615-2025-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation