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OFFO minimization algorithms for second-order optimality and their complexity

S. Gratton () and Ph. L. Toint ()
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S. Gratton: Université de Toulouse
Ph. L. Toint: University of Namur

Computational Optimization and Applications, 2023, vol. 84, issue 2, No 10, 573-607

Abstract: 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.

Keywords: Second-order optimality; Objective-function-free optimization (OFFO); Adagrad; Global rate of convergence; Evaluation complexity (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10589-022-00435-2

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