Foundations of mathematical optimization: convex analysis without linearity.

*(English)*Zbl 0887.49001
Mathematics and its Applications (Dordrecht). 388. Dordrecht: Kluwer Academic Publishers. xii, 582 p. (1997).

Written by two well-known mathematicians this book gives an interesting overview on mathematical foundations of optimization theory in a general and modern setting with full proofs containing a lot of own results and ideas of the authors.

In Chapter 1 they develop convex analysis over a set \(X\), which not necessarily has a linear or topological structure, but often an order structure (a relation) is given. Nevertheless, so-called \(\Phi\)-convexity, duality, polarity, different constraints to optimization problems, marginal functions, optimality conditions and so on can be treated successfully.

In Chapter 2 optimization in metric spaces is considered. So, lower semicontinuity, peaking property, Ekeland’s variational principle (and related theorems), (weak) well-posedness and duality (with special results in connection with Lipschitzian functions, especially with respect to single valuedness of subdifferentials), globalization property, denseness results for subgradients, the smooth variational principle (Borwein and Preiss), approximation of sets, subgradient calculus (also in normed spaces) including interesting examples find their place.

Chapter 3 is devoted to multifunctions and marginal functions (semicontinuity and stability theorems under different conditions) in metric spaces.

In Chapter 4 well-posedness in Banach spaces is considered. We find especially theorems around Rolewicz’s drop property and streams. The known theorem, that a Banach space is reflexive if and only if its norm has the weak drop property follows as corollary. Finally, in this chapter uniform well-posedness properties are discussed.

Chapter 5 deals especially with regularization methods in infinite-dimensional spaces using inf-convolution technique in connection with Fenchel-dual and paraconvex functions. The results of Lasry and Lions find their place followed by Asplund’s results on denseness of points of Gâteaux differentiability.

Chapter 6 gives necessary optimality conditions using some tangent cones (with interesting examples as Sierpiński’s “carpet”) and a lot of important theorems on properties of cones. Following, Doleckij limits of multifunctions are discussed between spaces, which are not necessarily metrizable (filter, grills, convergence theory). Then, different optimality conditions follow (Dubowitzkij-Miljutin, Kuhn-Tucker, Clarke, applications of derivatives of set-valued mappings) accompanied by theorems such as Ljusternik, open mapping, Robinson-Ursescu and so on and by discussions about constraint qualifications.

Chapter 7 contains optimality conditions of higher-order, especially Rolewicz’s method of reduction of constraints.

Chapter 8 on nondifferentiable optimization, beginning with an overview on developments in quasidifferentiable analysis, contains the big calculus of DC-functions (class of real-valued functions which can be represented as a difference of two convex functions) and similar classes of functions, followed by considerations to construct and characterize (universal) derivatives, which fulfil some basic properties. Then, corresponding optimality conditions follow (results of Pallaschke and coworkers and of Demyanov and Rubinov).

In Chapter 9 some numerical aspects can be found (bundle method, subgradient method, connections to nonlinear complementarity problems, set covering problem).

Finally, Chapter 10 considers multicriteria problems and at first the authors discuss Pareto minimum and order Pareto infimum (not only in the case, where the order is given by a pointed convex cone), then scalarization methods and efficiency theorems, optimality conditions, and (at first in vector lattices then for general orders) duality theory are developed. Following, results of Doleckij and Malivert strong duality can be characterized by a suitable subdifferentiability of the value multifunction (similar as in the scalar case).

20 pages with references, a subject and an author index and a list of symbols round off the book.

As was shown, the monography is a rich collection of theorems which, together with their proofs, belong to the modern foundations of mathematical optimization theory. Furthermore, methods and definitions are motivated in an impressive way and the necessity of conditions or assumptions in the theorems are tested carefully with help of suitable examples. It is recommended for every mathematician, who works in this field, to take not only one look at this beautiful book.

In Chapter 1 they develop convex analysis over a set \(X\), which not necessarily has a linear or topological structure, but often an order structure (a relation) is given. Nevertheless, so-called \(\Phi\)-convexity, duality, polarity, different constraints to optimization problems, marginal functions, optimality conditions and so on can be treated successfully.

In Chapter 2 optimization in metric spaces is considered. So, lower semicontinuity, peaking property, Ekeland’s variational principle (and related theorems), (weak) well-posedness and duality (with special results in connection with Lipschitzian functions, especially with respect to single valuedness of subdifferentials), globalization property, denseness results for subgradients, the smooth variational principle (Borwein and Preiss), approximation of sets, subgradient calculus (also in normed spaces) including interesting examples find their place.

Chapter 3 is devoted to multifunctions and marginal functions (semicontinuity and stability theorems under different conditions) in metric spaces.

In Chapter 4 well-posedness in Banach spaces is considered. We find especially theorems around Rolewicz’s drop property and streams. The known theorem, that a Banach space is reflexive if and only if its norm has the weak drop property follows as corollary. Finally, in this chapter uniform well-posedness properties are discussed.

Chapter 5 deals especially with regularization methods in infinite-dimensional spaces using inf-convolution technique in connection with Fenchel-dual and paraconvex functions. The results of Lasry and Lions find their place followed by Asplund’s results on denseness of points of Gâteaux differentiability.

Chapter 6 gives necessary optimality conditions using some tangent cones (with interesting examples as Sierpiński’s “carpet”) and a lot of important theorems on properties of cones. Following, Doleckij limits of multifunctions are discussed between spaces, which are not necessarily metrizable (filter, grills, convergence theory). Then, different optimality conditions follow (Dubowitzkij-Miljutin, Kuhn-Tucker, Clarke, applications of derivatives of set-valued mappings) accompanied by theorems such as Ljusternik, open mapping, Robinson-Ursescu and so on and by discussions about constraint qualifications.

Chapter 7 contains optimality conditions of higher-order, especially Rolewicz’s method of reduction of constraints.

Chapter 8 on nondifferentiable optimization, beginning with an overview on developments in quasidifferentiable analysis, contains the big calculus of DC-functions (class of real-valued functions which can be represented as a difference of two convex functions) and similar classes of functions, followed by considerations to construct and characterize (universal) derivatives, which fulfil some basic properties. Then, corresponding optimality conditions follow (results of Pallaschke and coworkers and of Demyanov and Rubinov).

In Chapter 9 some numerical aspects can be found (bundle method, subgradient method, connections to nonlinear complementarity problems, set covering problem).

Finally, Chapter 10 considers multicriteria problems and at first the authors discuss Pareto minimum and order Pareto infimum (not only in the case, where the order is given by a pointed convex cone), then scalarization methods and efficiency theorems, optimality conditions, and (at first in vector lattices then for general orders) duality theory are developed. Following, results of Doleckij and Malivert strong duality can be characterized by a suitable subdifferentiability of the value multifunction (similar as in the scalar case).

20 pages with references, a subject and an author index and a list of symbols round off the book.

As was shown, the monography is a rich collection of theorems which, together with their proofs, belong to the modern foundations of mathematical optimization theory. Furthermore, methods and definitions are motivated in an impressive way and the necessity of conditions or assumptions in the theorems are tested carefully with help of suitable examples. It is recommended for every mathematician, who works in this field, to take not only one look at this beautiful book.

Reviewer: A.Göpfert (Halle)

##### MSC:

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

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |

90C48 | Programming in abstract spaces |

49K27 | Optimality conditions for problems in abstract spaces |

49J52 | Nonsmooth analysis |