The gradient descent method minimizes an unconstrained nonlinear optimization problem with O(1/K)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {O}}(1/\sqrt{K})$$\end{document}, where K is the number of iterations performed by the gradient method. Traditionally, this analysis is obtained for smooth objective functions having Lipschitz continuous gradients. This paper aims to consider a more general class of nonlinear programming problems in which functions have Hölder continuous gradients. More precisely, for any function f in this class, denoted by CL1,ν\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {C}}}^{1,\nu }_L$$\end{document}, there is a ν∈(0,1]\documentclass[12pt]{minimal}
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\begin{document}$$\nu \in (0,1]$$\end{document} and L>0\documentclass[12pt]{minimal}
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\begin{document}$$L>0$$\end{document} such that for all x,y∈Rn\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{x,y}\in {{\mathbb {R}}}^n$$\end{document} the relation ‖∇f(x)-∇f(y)‖≤L‖x-y‖ν\documentclass[12pt]{minimal}
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\begin{document}$$\Vert \nabla f(\mathbf{x})-\nabla f(\mathbf{y})\Vert \le L \Vert \mathbf{x}-\mathbf{y}\Vert ^{\nu }$$\end{document} holds. We prove that the gradient descent method converges globally to a stationary point and exhibits a convergence rate of O(1/Kνν+1)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {O}}(1/K^{\frac{\nu }{\nu +1}})$$\end{document} when the step-size is chosen properly, i.e., less than [ν+1L]1ν‖∇f(xk)‖1ν-1\documentclass[12pt]{minimal}
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\begin{document}$$[\frac{\nu +1}{L}]^{\frac{1}{\nu }}\Vert \nabla f(\mathbf{x}_k)\Vert ^{\frac{1}{\nu }-1}$$\end{document}. Moreover, the algorithm employs O(1/ϵ1ν+1)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {O}}(1/\epsilon ^{\frac{1}{\nu }+1})$$\end{document} number of calls to an oracle to find x¯\documentclass[12pt]{minimal}
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\begin{document}$${\bar{\mathbf{x}}}$$\end{document} such that ‖∇f(x¯)‖<ϵ\documentclass[12pt]{minimal}
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\begin{document}$$\Vert \nabla f({{\bar{\mathbf{x}}}})\Vert <\epsilon $$\end{document}.
