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Existence of solutions and star-shapedness in Minty variational inequalities

Author

Listed:
  • Crespi Giovanni

    () (University of Bocconi, Italy)

  • Ginchev Ivan

    () (Department of Mathematics, Technical University of Varna, Bulgaria)

  • Rocca Matteo

    () (Department of Economics, University of Insubria, Italy)

Abstract

Minty variational inequalities have proven to define a stronger notion of equilibrium than Stampacchia variational inequalities. This conclusion leads to argue that some regularity, e.g. convexity or generalized convexity, has to be implicit for any function that admits a solution of the corresponding integrable Minty variational inequality. Quasi-convexity arises almost naturally when functions of one variable are involved. However some differences appear when considering functions of several variables. In this case we show that existence of a solution does not necessarily imply quasi-convexity of the function and instead we prove that the level sets of the function must be star-shaped at a point which is a solution of the Minty variational inequality.

Suggested Citation

  • Crespi Giovanni & Ginchev Ivan & Rocca Matteo, 2002. "Existence of solutions and star-shapedness in Minty variational inequalities," Economics and Quantitative Methods qf0211, Department of Economics, University of Insubria.
  • Handle: RePEc:ins:quaeco:qf0211
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    File URL: http://eco.uninsubria.it/dipeco/Quaderni/files/QF2002_21.pdf
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    Keywords

    Minty variational inequality; generalized convexity; star-shaped sets; existence of solutions;

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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