Nonsmooth optimization. Analysis and algorithms with applications to optimal control.

*(English)*Zbl 0757.49017
Singapore: World Scientific. xii, 254 p. (1992).

This book is an extended version of a thesis which I have reviewed in 1991 [M. M. Mäkelä, “Nonsmooth optimization. Theory and algorithms with applications to optimal control”, Univ. Jyväskylä/Finland, Dep. Math., Report No. 47, (1990; Zbl 0731.49002)].

Part I (Nonsmooth Analysis) is devoted to the finite-dimensional theory of subdifferentials for convex functions, Clarke subdifferentials and \(\varepsilon\)-subdifferentials for locally Lipschitzian functions as well as optimality conditions for nonsmooth programming problems.

Part II (Nonsmooth Optimization) contains a survey of several existing numerical methods for nonsmooth optimization, a detailed presentation of the proximal bundle method (developed by the first author) and the results of some numerical experiments performed with this method.

In Part III (Nonsmooth Optimal Control) applications of the proximal bundle method to optimal control problems are presented. The authors consider the following kinds of problems: distributed parameter control problems, optimal shape designs, design of optimal covering and boundary control for Stefan type problems. For each kind of problem, the following questions are discussed: setting of the problem, discretization of the problem, sensitivity analysis and numerical results. The authors compare the nonsmooth optimization approach with the regularization technique and smooth optimization approaches.

In Parts I and II only some small changes are made in comparison with the previous version. The most important one is that some drawings are added which may help the reader to develop his geometric intuition (although, in my opinion, Figure 3.1 on p. 50 does not help very much to understand the notion of Clarke’s subdifferential since the formula for the plotted function is not included). Unfortunately, some errors pointed out in my previous review have not been corrected in this new edition. Part III, however, has been substantially enlarged. In particular, more information concerning the theoretical aspects of the discussed problems is given.

Finally, the book contains an extensive and up-to-date bibliography on nonsmooth optimization.

Part I (Nonsmooth Analysis) is devoted to the finite-dimensional theory of subdifferentials for convex functions, Clarke subdifferentials and \(\varepsilon\)-subdifferentials for locally Lipschitzian functions as well as optimality conditions for nonsmooth programming problems.

Part II (Nonsmooth Optimization) contains a survey of several existing numerical methods for nonsmooth optimization, a detailed presentation of the proximal bundle method (developed by the first author) and the results of some numerical experiments performed with this method.

In Part III (Nonsmooth Optimal Control) applications of the proximal bundle method to optimal control problems are presented. The authors consider the following kinds of problems: distributed parameter control problems, optimal shape designs, design of optimal covering and boundary control for Stefan type problems. For each kind of problem, the following questions are discussed: setting of the problem, discretization of the problem, sensitivity analysis and numerical results. The authors compare the nonsmooth optimization approach with the regularization technique and smooth optimization approaches.

In Parts I and II only some small changes are made in comparison with the previous version. The most important one is that some drawings are added which may help the reader to develop his geometric intuition (although, in my opinion, Figure 3.1 on p. 50 does not help very much to understand the notion of Clarke’s subdifferential since the formula for the plotted function is not included). Unfortunately, some errors pointed out in my previous review have not been corrected in this new edition. Part III, however, has been substantially enlarged. In particular, more information concerning the theoretical aspects of the discussed problems is given.

Finally, the book contains an extensive and up-to-date bibliography on nonsmooth optimization.

Reviewer: M.Studniarski (Łódź)

##### MSC:

49J52 | Nonsmooth analysis |

90C30 | Nonlinear programming |

90C25 | Convex programming |

49J20 | Existence theories for optimal control problems involving partial differential equations |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

65K05 | Numerical mathematical programming methods |