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Proximal Minimization Methods with Generalized Bregman Functions

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  • K. Kiwiel

Abstract

We consider methods for minimizing a convex function $f$ that generate a sequence $\{x^k\}$ by taking $x^{k+1}$ to be an approximate minimizer of $f(x)+D_h(x,x^k)/c_k$, where $c_k>0$ and $D_h$ is the $D$-function of a Bregman function $h$. Extensions are made to $B$-functions that generalize Bregman functions and cover more applications. Convergence is established under criteria amenable to implementation. Applications are made to nonquadratic multiplier methods for nonlinear programs.

Suggested Citation

  • K. Kiwiel, 1995. "Proximal Minimization Methods with Generalized Bregman Functions," Working Papers wp95024, International Institute for Applied Systems Analysis.
    • Handle: RePEc:wop:iasawp:wp95024
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    References listed on IDEAS

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    1. Marc Teboulle, 1992. "Entropic Proximal Mappings with Applications to Nonlinear Programming," Mathematics of Operations Research, INFORMS, vol. 17(3), pages 670-690, August.
    2. Jonathan Eckstein, 1993. "Nonlinear Proximal Point Algorithms Using Bregman Functions, with Applications to Convex Programming," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 202-226, February.
    3. Jong-Il Kim & Lawrence J. Lau, 1996. "The sources of Asian Pacific economic growth," Canadian Journal of Economics, Canadian Economics Association, vol. 29(s1), pages 448-454, April.
    4. Alfredo N. Iusem & B. F. Svaiter & Marc Teboulle, 1994. "Entropy-Like Proximal Methods in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 19(4), pages 790-814, November.
    5. Paul Tseng & Dimitri P. Bertsekas, 1991. "Relaxation Methods for Problems with Strictly Convex Costs and Linear Constraints," Mathematics of Operations Research, INFORMS, vol. 16(3), pages 462-481, August.
    6. K. Kiwiel, 1994. "Free-Steering Relaxation Methods for Problems with Strictly Convex Costs and Linear Constraints," Working Papers wp94089, International Institute for Applied Systems Analysis.
    7. Soren S. Nielsen & Stavros A. Zenios, 1992. "Massively Parallel Algorithms for Singly Constrained Convex Programs," INFORMS Journal on Computing, INFORMS, vol. 4(2), pages 166-181, May.
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