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Nonsmooth optimization

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Nonlinear Differential Problems with Smooth and Nonsmooth Constraints systematically evaluates how to solve boundary value problems with smooth and nonsmooth constraints. Primarily covering nonlinear elliptic eigenvalue problems and quasilinear elliptic problems using techniques amalgamated from a range of sophisticated nonlinear analysis domains, the work is suitable for PhD and other early career researchers seeking solutions to nonlinear differential equations. Although an advanced work, the book is self-contained, requiring only graduate-level knowledge of functional analysis and topology. Whenever suitable, open problems are stated and partial solutions proposed. The work is accompanied by end-of-chapter problems and carefully curated references.

Nonlinear difference equations. --- Nonsmooth optimization.

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Nonsmooth optimization. --- Variational inequalities (Mathematics). --- Equilibrium.

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The authors consider operators of the form L=sum_{i=1}^{n}X_{i}^{2}+X_{0} in a bounded domain of mathbb{R}^{p} where X_{0},X_{1},ldots,X_{n} are nonsmooth Hörmander's vector fields of step r such that the highest order commutators are only Hölder continuous. Applying Levi's parametrix method the authors construct a local fundamental solution gamma for L and provide growth estimates for gamma and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients the authors prove that gamma also possesses second derivatives, and they deduce the local solvability of L, constructing, by means of gamma, a solution to Lu=f with Hölder continuous f. The authors also prove C_{X,loc}^{2,alpha} estimates on this solution.

Differential operators. --- Nonsmooth optimization. --- Vector fields.

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nonsmooth analysis --- nonsmooth optimization --- variational analysis --- equilibrium problems --- stochastic optimization --- bilevel optimization --- Nonsmooth optimization --- Nonsmooth analysis --- Optimization, Nonsmooth --- Mathematical optimization --- Nonsmooth optimization.