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A smooth path-following algorithm for market equilibrium under a class of piecewise-smooth concave utilities

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  • Yang Zhan

    (City University of Hong Kong)

  • Chuangyin Dang

    (City University of Hong Kong)

Abstract

This paper presents a smooth path-following algorithm for computing market equilibrium in a pure exchange economy under a class of piecewise-smooth concave utilities, which can be expressed as $$u(x)=\min _\ell \{f_\ell (x)\}$$ u ( x ) = min ℓ { f ℓ ( x ) } with $$f_\ell (x)$$ f ℓ ( x ) being a smooth concave function for all $$\ell $$ ℓ . As a result of a smooth technique for minimax problems, a smooth homotopy mapping is derived from the introduction of logarithmic barrier terms and an extra variable. With this mapping, it is proved that there always exists a smooth path leading to a market equilibrium as the extra variable approaches zero. A predictor–corrector method is adapted for numerically following this path. Numerical results are given to further demonstrate the effectiveness and efficiency of the algorithm.

Suggested Citation

  • Yang Zhan & Chuangyin Dang, 2018. "A smooth path-following algorithm for market equilibrium under a class of piecewise-smooth concave utilities," Computational Optimization and Applications, Springer, vol. 71(2), pages 381-402, November.
    • Handle: RePEc:spr:coopap:v:71:y:2018:i:2:d:10.1007_s10589-018-0009-z
    DOI: 10.1007/s10589-018-0009-z
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    Cited by:

    1. Yang Zhan & Peixuan Li & Chuangyin Dang, 2020. "A differentiable path-following algorithm for computing perfect stationary points," Computational Optimization and Applications, Springer, vol. 76(2), pages 571-588, June.
    2. Dang, Chuangyin & Herings, P. Jean-Jacques & Li, Peixuan, 2020. "An Interior-Point Path-Following Method to Compute Stationary Equilibria in Stochastic Games," Research Memorandum 001, Maastricht University, Graduate School of Business and Economics (GSBE).
    3. Chuangyin Dang & P. Jean-Jacques Herings & Peixuan Li, 2022. "An Interior-Point Differentiable Path-Following Method to Compute Stationary Equilibria in Stochastic Games," INFORMS Journal on Computing, INFORMS, vol. 34(3), pages 1403-1418, May.
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    5. Yang Zhan & Chuangyin Dang, 2021. "Computing equilibria for markets with constant returns production technologies," Annals of Operations Research, Springer, vol. 301(1), pages 269-284, June.

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