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Convergence of the steepest descent method for minimizing quasiconvex functions

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Abstract

To minimize a continuously differentiable quasiconvex functionf: ℝn→ℝ, Armijo's steepest descent method generates a sequencex k+1 =x kt k f(x k), wheret k >0. We establish strong convergence properties of this classic method: either\(x^k \to \bar x,\), s.t.\(\nabla f(\bar x) = 0\); or arg minf = ∅, ∥x k∥ ↓ ∞ andf(x k)↓ inff. We also discuss extensions to other line searches.

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Communicated by O. L. Mangasarian

The research of the first author was supported by the Polish Academy of Sciences. The second author acknowledges the support of the Department of Industrial Engineering, Hong Kong University of Science and Technology.

We wish to thank two anonymous referees for their valuable comments. In particular, one referee has suggested the use of quasiconvexity instead of convexity off.

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Kiwiel, K.C., Murty, K. Convergence of the steepest descent method for minimizing quasiconvex functions. J Optim Theory Appl 89, 221–226 (1996). https://doi.org/10.1007/BF02192649

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