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Convex Optimization with Computational Errors de Alexander J. Zaslavski

Convex Optimization with Computational Errors: Springer Optimization and Its Applications, cartea 155

Autor Alexander J. Zaslavski
en Limba Engleză Hardback – feb 2020

În cadrul programelor de studii avansate în matematică aplicată și inginerie, studiul algoritmilor de optimizare ignoră adesea realitatea erorilor de calcul inerente procesării numerice. Convex Optimization with Computational Errors vine să umple această lacună, poziționându-se ca o resursă fundamentală pentru cercetătorii care operează în spații Hilbert. Apreciem în mod deosebit rigoarea cu care Alexander J. Zaslavski extinde cercetările sale anterioare din Numerical Optimization with Computational Errors, aducând o perspectivă mult mai nuanțată asupra practicii computaționale.

Observăm că elementul distinctiv al acestui volum, față de ediția din 2016, este recunoașterea faptului că o singură iterație a unui algoritm cuprinde pași multipli, fiecare generând erori de magnitudini diferite. De exemplu, în algoritmul proiecției subgradientului, calculul subgradientului unei funcții obiectiv complexe poate introduce erori semnificativ mai mari decât proiecția pe o mulțime fezabilă simplă. Această abordare granulară oferă o imagine mult mai fidelă a comportamentului de convergență în condiții reale. Cartea reprezintă o alternativă tehnică solidă la Convex Analysis and Minimization Algorithms I de Jean-Baptiste Hiriart-Urruty pentru cursurile de optimizare numerică, având avantajul integrării explicite a erorilor de calcul în analiza stabilității algoritmilor.

Structura celor 12 capitole indică o progresie logică, de la bazele algoritmului subgradient și mirror descent, până la aplicații complexe în procesarea semnalelor (Capitolul 5) și teoria jocurilor (Capitolul 8). Remarcăm includerea unor metode recente, precum algoritmul PDA-based pentru optimizarea convexă cu constrângeri, demonstrând relevanța actuală a lucrării pentru inginerie și economie.

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Specificații

ISBN-13: 9783030378219
ISBN-10: 3030378217
Pagini: 360
Ilustrații: XI, 360 p. 150 illus.
Dimensiuni: 155 x 235 mm
Greutate: 0.7 kg
Ediția:1st ed. 2020
Editura: Springer International Publishing
Colecția Springer
Seria Springer Optimization and Its Applications

Locul publicării:Cham, Switzerland

De ce să citești această carte

Această monografie este esențială pentru cercetătorii și inginerii care doresc să înțeleagă cum influențează erorile de calcul precizia soluțiilor în optimizarea convexă. Cititorul câștigă o metodologie clară pentru a determina numărul de iterații necesare obținerii unei soluții aproximative viabile. Este un instrument indispensabil pentru cei care dezvoltă software de optimizare unde stabilitatea numerică este critică.

Despre autor

Alexander J. Zaslavski este un cercetător recunoscut în domeniul analizei neliniare și teoriei optimizării, cu o activitate prolifică publicată sub egida Springer. Opera sa se concentrează pe comportamentul algoritmic în prezența erorilor și pe fenomenele de tip „turnpike” în controlul optimal, teme explorate în lucrări precum Turnpike Conditions in Infinite Dimensional Optimal Control. Prin volumele sale, Zaslavski a fundamentat studiul soluțiilor aproximative în spații Banach și Hilbert, fiind o autoritate în aplicarea matematicii riguroase la probleme practice de inginerie și economie.

Descriere scurtă

The book is devoted to the study of approximate solutions of optimization problems in the presence of computational errors. It contains a number of results on the convergence behavior of algorithms in a Hilbert space, which are known as important tools for solving optimization problems. The research presented in the book is the continuation and the further development of the author's (c) 2016 book Numerical Optimization with Computational Errors, Springer 2016. Both books study the algorithms taking into account computational errors which are always present in practice. The main goal is, for a known computational error, to find out what an approximate solution can be obtained and how many iterates one needs for this. 

The main difference between this new book and the 2016 book is that in this present book the discussion takes into consideration the fact that for every algorithm, its iteration consists of several steps and that computational errors for different steps are generally, different. This fact, which was not taken into account in the previous book, is indeed important in practice. For example, the subgradient projection algorithm consists of two steps. The first step is a calculation of a subgradient of the objective function while in the second one we calculate a projection on the feasible set. In each of these two steps there is a computational error and these two computational errors are different in general. 

It may happen that the feasible set is simple and the objective function is complicated. As a result, the computational error, made when one calculates the projection, is essentially smaller than the computational error of the calculation of the subgradient. Clearly, an opposite case is possible too. Another feature of this book is a study of a number of important algorithms which appeared recently in the literature and which are not discussed in the previous book. 

This monograph contains 12 chapters. Chapter 1 is an introduction. In Chapter 2 we study the subgradient projection algorithm for minimization of convex and nonsmooth functions. We generalize the results of [NOCE] and establish results which has no prototype in [NOCE]. In Chapter 3 we analyze the mirror descent algorithm for minimization of convex and nonsmooth functions, under the presence of computational errors.  For this algorithm each iteration consists of two steps. The first step is a calculation of a subgradient of the objective function while in the second one we solve an auxiliary minimization problem on the set of feasible points. In each of these two steps there is a computational error. We generalize the results of [NOCE] and establish results which has no prototype in [NOCE].  In Chapter 4 we analyze the projected gradient algorithm with a smooth objective function under the presence of computational errors.  In Chapter 5 we consider an algorithm, which is an extension of the projection gradient algorithm used for solving linear inverse problems arising in signal/image processing. In Chapter 6 we study continuous subgradient method and continuous subgradient projection algorithm for minimization of convex nonsmooth functions and for computing the saddle points of convex-concave functions, under the presence of computational errors.  All the results of this chapter has no prototype in [NOCE]. In Chapters 7-12 we analyze several algorithms under the presence of computational errors which were not considered in [NOCE]. Again, each step of an iteration has a computational errors and we take into account that these errors are, in general, different. An optimization problems with a composite objective function is studied in Chapter 7. A zero-sum game with two-players is considered in Chapter 8. A predicted decrease approximation-based method is used in Chapter 9 for constrained convex optimization. Chapter 10 is devoted tominimization of quasiconvex functions. Minimization of sharp weakly convex functions is discussed in Chapter 11. Chapter 12 is devoted to a generalized projected subgradient method for minimization of a convex function over a set which is not necessarily convex.

The book is of interest for researchers and engineers working in optimization. It also can be useful in preparation courses for graduate students.  The main feature of the book which appeals specifically to this audience is the study of the influence of computational errors for several important optimization algorithms. The book is of interest for experts in applications of optimization  to engineering and economics.

Cuprins

Preface.- 1. Introduction.- 2. Subgradient Projection Algorithm.- 3. The Mirror Descent Algorithm.- 4. Gradient Algorithm with a Smooth Objective Function.- 5. An Extension of the Gradient Algorithm.- 6. Continuous Subgradient Method.- 7. An optimization problems with a composite objective function.- 8. A zero-sum game with two-players.- 9. PDA-based method for convex optimization.- 10 Minimization of quasiconvex functions.-11. Minimization of sharp weakly convex functions.-12. A Projected Subgradient Method for Nonsmooth Problems.- References. -Index.  

Notă biografică

​Alexander J. Zaslavski is professor in the Department of Mathematics, Technion-Israel Institute of Technology, Haifa, Israel.

Caracteristici

Studies the influence of computational errors in numerical optimization, for minimization problems on unbounded sets, and time zero-sum games with two players Explains that for every algorithm its iteration consists of several steps and that computational errors for different steps are different Provides modern and interesting developments in the field