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OFFO minimization algorithms for second-order optimality and their complexity

Author

Listed:
  • S. Gratton

    (Université de Toulouse)

  • Ph. L. Toint

    (University of Namur)

Abstract

An Adagrad-inspired class of algorithms for smooth unconstrained optimization is presented in which the objective function is never evaluated and yet the gradient norms decrease at least as fast as $$\mathcal{O}(1/\sqrt{k+1})$$ O ( 1 / k + 1 ) while second-order optimality measures converge to zero at least as fast as $$\mathcal{O}(1/(k+1)^{1/3})$$ O ( 1 / ( k + 1 ) 1 / 3 ) . This latter rate of convergence is shown to be essentially sharp and is identical to that known for more standard algorithms (like trust-region or adaptive-regularization methods) using both function and derivatives’ evaluations. A related “divergent stepsize” method is also described, whose essentially sharp rate of convergence is slighly inferior. It is finally discussed how to obtain weaker second-order optimality guarantees at a (much) reduced computational cost.

Suggested Citation

  • S. Gratton & Ph. L. Toint, 2023. "OFFO minimization algorithms for second-order optimality and their complexity," Computational Optimization and Applications, Springer, vol. 84(2), pages 573-607, March.
    • Handle: RePEc:spr:coopap:v:84:y:2023:i:2:d:10.1007_s10589-022-00435-2
    DOI: 10.1007/s10589-022-00435-2
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