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We consider the method of alternating projections for finding a point in the intersection of two possibly nonconvex closed sets. We present a local linear convergence result that makes no regularity assumptions on either set (unlike previous results), while at the same time weakening standard transver-sal intersection assumptions. The proof grows out of a… (More)

- Alexander D. Ioffe, Alexander J. Zaslavski
- SIAM J. Control and Optimization
- 2000

Necessary conditions are developed for a general problem in the calculus of variations in which the Lagrangian function, although finite, need not be Lipschitz continuous or convex in the velocity argument. For the first time in such a broadly nonsmooth, nonconvex setting, a full subgradient version of Euler's equation is derived for an arc that furnishes a… (More)

- Alexander D. Ioffe
- Math. Program.
- 2013

The paper studies regularity properties of set-valued mappings between metric spaces. In the context of metric regularity, nonlinear models correspond to nonlinear dependencies of estimates of error bounds in terms of residuals. Among the questions addressed in the paper are equivalence of the corresponding concepts of openness and " pseudo-Hölder "… (More)

- Dmitriy Drusvyatskiy, Alexander D. Ioffe, Adrian S. Lewis
- SIAM J. Control and Optimization
- 2015

Steepest descent is central in variational mathematics. We present a new transparent existence proof for curves of near-maximal slope — an influential notion of steepest descent in a nonsmooth setting. We moreover show that for semi-algebraic functions — prototypical nonpatho-logical functions in nonsmooth optimization — such curves are precisely the… (More)

We consider optimization algorithms that successively minimize simple Taylor-like models of the objective function. Methods of Gauss-Newton type for minimizing the composition of a convex function and a smooth map are common examples. Our main result is an explicit relationship between the step-size of any such algorithm and the slope of the function at a… (More)

- Alexander D. Ioffe, R. E. Lucchetti
- Math. Program.
- 2005

The paper contains calculus rules for coderivatives of compositions, sums and intersections of set-valued mappings. The types of coderivatives considered correspond to Dini-Hadamard and limiting Dini-Hadamard subdifferentials in Gâteaux differentiable spaces, Fréchet and limiting Fréchet subdifferentials in Asplund spaces and approximate subdifferentials in… (More)

- Alexander D. Ioffe, R. E. Lucchetti, Julian P. Revalski
- Math. Oper. Res.
- 2004

- Dmitriy Drusvyatskiy, Alexander D. Ioffe
- Math. Program.
- 2015

We show that quadratic growth of a semi-algebraic function is equivalent to strong metric subregularity of the subdifferential — a kind of stability of generalized critical points. In contrast, this equivalence can easily fail outside of the semi-algebraic setting.