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A simplicial algorithm for stationary point problems on polytopes

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  • Talman, A.J.J.

    (Tilburg University, School of Economics and Management)

  • Yamamoto, Y.

Abstract

A simplicial variable dimension restart algorithm for the stationary point problem or variational inequality problem on a polytope is proposed. Given a polytope C in (R-openface) n and a continuous function f : C (rightarrow) (R-openface) n , find a point (bigcirc) in C such that f ( (bigcirc) ) · (bigcirc) (ge) f ( (bigcirc) ) · x for any point x in C . Starting from an arbitrary point v in C , the algorithm generates a piecewise linear path of points in C . This path is followed by alternating linear programming pivot steps to follow a linear piece of the path and replacement steps in a simplicial subdivision of C . Within a finite number of function evaluations and linear programming pivot steps the algorithm finds an approximate stationary point. The algorithm leaves the starting point v along a ray pointing to one of the vertices w of C . The vertex w is obtained from the optimum solution of the linear programming problem maximize f ( v ) · x subject to x (in) C .
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Suggested Citation

  • Talman, A.J.J. & Yamamoto, Y., 1989. "A simplicial algorithm for stationary point problems on polytopes," Other publications TiSEM 0d6b2de0-17c0-4d5e-963f-5, Tilburg University, School of Economics and Management.
    • Handle: RePEc:tiu:tiutis:0d6b2de0-17c0-4d5e-963f-5498eeaa104a
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    Cited by:

    1. Dolf Talman & Zaifu Yang, 2012. "On a Parameterized System of Nonlinear Equations with Economic Applications," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 644-671, August.
    2. Talman, A.J.J. & Yamamoto, M., 2001. "Contiuum of Zero Points of a Mapping on a Compact Convex Set," Discussion Paper 2001-56, Tilburg University, Center for Economic Research.
    3. Talman, A.J.J., 1990. "General equilibrium programming," Other publications TiSEM b2dccd7c-f4e9-4e3c-9f67-f, Tilburg University, School of Economics and Management.
    4. Talman, A.J.J. & Yang, Z.F., 2003. "On the Connectedness of Coincidences and Zero Points of Mappings," Other publications TiSEM b061dd09-2b7f-4fe3-af6e-d, Tilburg University, School of Economics and Management.
    5. Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 1999. "Variational Inequality Problems With a Continuum of Solutions : Existence and Computation," Other publications TiSEM 73e2f01b-ad4d-4447-95ba-a, Tilburg University, School of Economics and Management.
    6. Jean-Jacques Herings & Gerard van der Laan & Dolf Talman & Zaifu Yang, 2004. "A Fixed Point Theorem for Discontinuous Functions," Tinbergen Institute Discussion Papers 05-004/1, Tinbergen Institute.
    7. Talman, A.J.J. & Thijssen, J.J.J., 2006. "Existence of equilibrium and price adjustments in a finance economy with incomplete markets," Journal of Mathematical Economics, Elsevier, vol. 42(3), pages 255-268, June.
    8. Dai, Y. & Talman, D., 1990. "Linear Stationary Point Problems On Unbounded Polyhedra," Papers 9067, Tilburg - Center for Economic Research.
    9. van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2002. "Perfection and Stability of Stationary Points with Applications in Noncooperative Games," Other publications TiSEM fc1f47c6-314f-4932-80c6-1, Tilburg University, School of Economics and Management.
    10. Thijssen, J.J.J., 2003. "Investment under uncertainty, market evolution and coalition spillovers in a game theoretic perspective," Other publications TiSEM 672073a6-492e-4621-8d4a-0, Tilburg University, School of Economics and Management.
    11. Clemens J M Kool & Tom Van Veen & Bertrand Chandelon & Katharina Raabe, 2005. "The Feasibility of a Fixed Exchange Rate Regime for New EU-members Evidence from Real Exchange Rates," Money Macro and Finance (MMF) Research Group Conference 2005 20, Money Macro and Finance Research Group.
    12. Dai, Y., 1989. "A simplicial algorithm for the nonlinear stationary point problem of an unbounded polyhedron," Serie Research Memoranda 0068, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.

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