Foundations of nonsmooth analysis and quasidifferential calculus. (Osnovy negladkogo analiza i kvazidifferentsial’noe ischislenie.) (Russian) Zbl 0728.49001
The monograph provides a rather complete picture of the present state and basic concepts of nonsmooth analysis. The initial question of the book is how to describe the behavior of \(f(x+\alpha g)\), \(x,g\in R^ n\); \(\alpha\downarrow 0\) for a nondifferentiable function (multivalued mapping) f by the first order approximating construction \(f(x)+\alpha \phi (x,g)\) with the function (mapping) \(\phi\) (x,\(\cdot)\) simple in some sense. The first part of the book (chapters I, II) contains a detailed investigation of an approximation technique with a positively homogeneous \(\phi\) (x,\(\cdot)\). Special attention is paid to the approximation by upper \((\phi (x,g)=f^{\uparrow}_{Cl}(x,g))\) and lower \((\phi (x,g)=f^{\downarrow}_{Cl}(x,g))\) Clarke’s derivatives where \(f^{\uparrow}_{Cl}(x,g)\) is a sublinear and \(f^{\downarrow}_{Cl}(x,g)\) is a superlinear function.
In the second part (chapters III, IV) the approximating problem is treated by means of the directional derivative \(\phi (x,g)=f'(x,g).\) The authors show that for a large family of functions it is useful to take as an approximating tool a sum of sublinear and superlinear functions p(x,\(\cdot)\), q(x,\(\cdot)\) so that \(f'(x,g)=p(x,g)+q(x,g).\) Here (p(x,\(\cdot)\), q(x,\(\cdot))\) corresponds to a pair of convex compact subsets of \(R^ n\), \(R^{n+1}\) (the quasidifferential, codifferential). A detailed account of the calculus of quasidifferentials and codifferentials is presented.
In the third part (chapters V-VIII) the quasidifferential and codifferential calculus is applied to a number of problems of optimization, game theory, optimal control and analysis.
In the second part (chapters III, IV) the approximating problem is treated by means of the directional derivative \(\phi (x,g)=f'(x,g).\) The authors show that for a large family of functions it is useful to take as an approximating tool a sum of sublinear and superlinear functions p(x,\(\cdot)\), q(x,\(\cdot)\) so that \(f'(x,g)=p(x,g)+q(x,g).\) Here (p(x,\(\cdot)\), q(x,\(\cdot))\) corresponds to a pair of convex compact subsets of \(R^ n\), \(R^{n+1}\) (the quasidifferential, codifferential). A detailed account of the calculus of quasidifferentials and codifferentials is presented.
In the third part (chapters V-VIII) the quasidifferential and codifferential calculus is applied to a number of problems of optimization, game theory, optimal control and analysis.
Reviewer: M.Yu.Kokurin (Yoshkar-Ola)
MSC:
49J52 | Nonsmooth analysis |
49J50 | Fréchet and Gateaux differentiability in optimization |
90-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming |
90C30 | Nonlinear programming |
91A05 | 2-person games |
26B05 | Continuity and differentiation questions |