Smoothing Newton and Quasi-Newton Methods for Mixed Complementarity Problems
Computational Optimization and Applications
Applications of smoothing methods in numerical analysis and optimization
Focus on computational neurobiology
Applied Numerical Mathematics - 6th IMACS International symposium on iterative methods in scientific computing
Applied Numerical Mathematics - 6th IMACS International symposium on iterative methods in scientific computing
Computation of generalized differentials in nonlinear complementarity problems
Computational Optimization and Applications
A minimal norm corrected underdetermined Gauß-Newton procedure
Applied Numerical Mathematics
Reconstructing a matrix from a partial sampling of Pareto eigenvalues
Computational Optimization and Applications
Computing Isolated Singular Solutions of Polynomial Systems: Case of Breadth One
SIAM Journal on Numerical Analysis
Verified error bounds for isolated singular solutions of polynomial systems: Case of breadth one
Theoretical Computer Science
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We present a local convergence analysis of generalized Newton methods for singular smooth and nonsmooth operator equations using adaptive constructs of outer inverses. We prove that for a solution $x^*$ of $F(x)=0$, there exists a ball $S=S(x^*,r)$, $r0$ such that for any starting point $x_0\in S$ the method converges to a solution $\bar{x}^*\in S$ of $\Gamma F(x)=0$, where $\Gamma$ is a bounded linear operator that depends on the Fréchet derivative of $F$ at $x_0$ or on a generalized Jacobian of $F$ at $x_0$. Point $\bar{x}^*$ may be different from $x^*$ when $x^*$ is not an isolated solution. Moreover, we prove that the convergence is quadratic if the operator is smooth and superlinear if the operator is locally Lipschitz. These results are sharp in the sense that they reduce in the case of an invertible derivative or generalized derivative to earlier theorems with no additional assumptions. The results are illustrated by a system of smooth equations and a system of nonsmooth equations, each of which is equivalent to a nonlinear complementarity problem.