A regularized Newton method without line search for unconstrained optimization
In this paper, we propose a regularized Newton method without line search. The proposed method controls a regularization parameter instead of a step size in order to guarantee the global convergence. We show that the proposed algorithm has the following convergence properties. (a) The proposed algorithm has global convergence under appropriate conditions. (b) It has superlinear rate of convergence under the local error bound condition. (c) An upper bound of the number of iterations required to obtain an approximate solution <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$x$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>x</mi> </math> </EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\Vert \nabla f(x) \Vert \le \varepsilon $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mo stretchy="false">‖</mo> <mi mathvariant="normal">∇</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">‖</mo> <mo>≤</mo> <mi mathvariant="italic">ε</mi> </mrow> </math> </EquationSource> </InlineEquation> is <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$O(\varepsilon ^{-2})$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi mathvariant="italic">ε</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$f$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>f</mi> </math> </EquationSource> </InlineEquation> is the objective function and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\varepsilon $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">ε</mi> </math> </EquationSource> </InlineEquation> is a given positive constant. Copyright Springer Science+Business Media New York 2014
