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.