Skip to main content
SpringerLink
Log in

Introduction

  • Chapter
  • First Online:
Optimization in Banach Spaces

Part of the book series: SpringerBriefs in Optimization ((BRIEFSOPTI))

  • 441 Accesses

Abstract

In this book, we study algorithms for constrained minimization problems in a general Banach space. Our goal is to obtain a good approximate solution of the problem in the presence of computational errors. It is shown that the algorithm generates a good approximate solution, if the sequence of computational errors is bounded from above by a small constant. In this section, we discuss several algorithms that are studied in the book. We also prove a convergence result for an unconstrained problem that is a prototype of our results for the constrained problem.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 19.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 29.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alber YI (1971) On minimization of smooth functional by gradient methods. USSR Comp Math Math Phys 11:752–758

    MathSciNet  Google Scholar 

  2. Alber YI, Iusem AN, Solodov MV (1997) Minimization of nonsmooth convex functionals in Banach spaces. J Convex Anal 4:235–255

    MathSciNet  MATH  Google Scholar 

  3. Alber YI, Iusem AN, Solodov MV (1998) On the projected subgradient method for nonsmooth convex optimization in a Hilbert space. Math Program 81:23–35

    Article  MathSciNet  MATH  Google Scholar 

  4. Barty K, Roy J-S, Strugarek C (2007) Hilbert-valued perturbed subgradient algorithms. Math Oper Res 32:551–562

    Article  MathSciNet  MATH  Google Scholar 

  5. Bauschke H, Wang C, Wang X, Xu J (2015) On subgradient projectors. SIAM J Optim 25:1064–1082

    Article  MathSciNet  MATH  Google Scholar 

  6. Beck A, Teboulle M (2003) Mirror descent and nonlinear projected subgradient methods for convex optimization. Oper Res Lett 31:167–175

    Article  MathSciNet  MATH  Google Scholar 

  7. Burachik RS, Grana Drummond LM, Iusem AN, Svaiter BF (1995) Full convergence of the steepest descent method with inexact line searches. Optimization 32:137–146

    Article  MathSciNet  MATH  Google Scholar 

  8. Ceng LC, Hadjisavvas N, Wong NC (2010) Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J Glob Optim 46:635–646

    Article  MathSciNet  MATH  Google Scholar 

  9. Ceng LC, Wong NC, Yao JC (2015) Hybrid extragradient methods for finding minimum norm solutions of split feasibility problems. J Nonlinear and Convex Analysis 16:1965–1983

    MathSciNet  MATH  Google Scholar 

  10. Censor Y, Gibali A, Reich S (2011) The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl 148:318–335

    Article  MathSciNet  MATH  Google Scholar 

  11. Censor Y, Gibali A, Reich S, Sabach S (2012) Common solutions to variational inequalities. Set-Valued Var Anal 20:229–247

    Article  MathSciNet  MATH  Google Scholar 

  12. Censor Y, Davidi R, Herman GT, Schulte RW, Tetruashvili L (2014) Projected subgradient minimization versus superiorization. J Optim Theory Appl 160:730–747

    Article  MathSciNet  MATH  Google Scholar 

  13. Chadli O, Konnov IV, Yao JC (2004) Descent methods for equilibrium problems in a Banach space. Comput Math Appl 48:609–616

    Article  MathSciNet  MATH  Google Scholar 

  14. Demyanov VF, Rubinov AM (1967) The minimization of a smooth convex functional on a convex set. SIAM J Control 5:280–294

    Article  MathSciNet  Google Scholar 

  15. Demyanov VF, Vasilyev LV (1985) Nondifferentiable optimization. Optimization Software, New York

    Book  Google Scholar 

  16. DeVore RA, Temlyakov VN (2016) Convex optimization on Banach spaces. Found Comput Math 16:369–394

    Article  MathSciNet  MATH  Google Scholar 

  17. Gao Z, Petrova G (2019) Rescaled pure greedy algorithm for convex optimization. Calcolo: 56:14 pp

    Google Scholar 

  18. Gibali A, Jadamba B, Khan AA, Raciti F, Winkler B (2016) Gradient and extragradient methods for the elasticity imaging inverse problem using an equation error formulation: a comparative numerical study. Nonlinear Analysis and Optimization Contemp Math 659:65–89

    Article  MathSciNet  MATH  Google Scholar 

  19. Griva I (2018) Convergence analysis of augmented Lagrangian-fast projected gradient method for convex quadratic problems. Pure Appl Funct Anal 3:417–428

    MathSciNet  MATH  Google Scholar 

  20. Hiriart-Urruty J-B, Lemarechal C (1993) Convex analysis and minimization algorithms. Springer, Berlin

    Book  MATH  Google Scholar 

  21. Konnov IV (2003) On convergence properties of a subgradient method. Optim Methods Softw 18:53–62

    Article  MathSciNet  MATH  Google Scholar 

  22. Konnov IV (2009) A descent method with inexact linear search for mixed variational inequalities. Russian Math (Iz VUZ) 53:29–35

    Article  MathSciNet  MATH  Google Scholar 

  23. Konnov IV (2018) Simplified versions of the conditional gradient method. Optimization 67:2275–2290

    Article  MathSciNet  MATH  Google Scholar 

  24. Korpelevich GM (1976) The extragradient method for finding saddle points and other problems. Ekon Matem Metody 12:747–756

    MathSciNet  MATH  Google Scholar 

  25. Liu L, Qin X, Yao J-C (2019) A hybrid descent method for solving a convex constrained optimization problem with applications. Math Methods Appl Sci 42:7367–7380

    Article  MathSciNet  MATH  Google Scholar 

  26. Mordukhovich BS (2006) Variational analysis and generalized differentiation, I: I: Basic Theory. Springer, Berlin

    Google Scholar 

  27. Mordukhovich BS, Nam NM (2014) An easy path to convex analysis and applications. Morgan and Clayton Publishers, San Rafael

    Book  MATH  Google Scholar 

  28. Nadezhkina N, Takahashi Wataru (2004) Modified extragradient method for solving variational inequalities in real Hilbert spaces. In: Nonlinear analysis and convex analysis, pp 359–366. Yokohama Publication, Yokohama

    Google Scholar 

  29. Nedic A, Ozdaglar A (2009) Subgradient methods for saddle-point problems. J Optim Theory Appl 142:205–228

    Article  MathSciNet  MATH  Google Scholar 

  30. ODHara JG, Pillay P, Xu HK (2006) Iterative approaches to convex feasibility problems in Banach spaces. Nonlinear Anal 64:2022–2042

    Google Scholar 

  31. Pallaschke D, Recht P (1985) On the steepest–descent method for a class of quasidifferentiable optimization problems. In: Nondifferentiable optimization: motivations and applications (Sopron, 1984), pp 252–263. Lecture Notes in Economics and Mathematical Systems, vol 255. Springer, Berlin

    Google Scholar 

  32. Polyak BT (1987) Introduction to optimization. Optimization Software, New York

    MATH  Google Scholar 

  33. Polyak RA (2015) Projected gradient method for non-negative least squares. Contemp Math 636:167–179

    Article  MathSciNet  MATH  Google Scholar 

  34. Qin X, Cho SY, Kang SM (2011) An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings. J Global Optim 49:679–693

    Article  MathSciNet  MATH  Google Scholar 

  35. Reich S, Zaslavski AJ (2014) Genericity in nonlinear analysis. Developments in Mathematics. Springer, New York

    Google Scholar 

  36. Shor NZ (1985) Minimization methods for non-differentiable functions. Springer, Berlin

    Book  MATH  Google Scholar 

  37. Solodov MV, Zavriev SK (1998) Error stability properties of generalized gradient-type algorithms. J Optim Theory Appl 98:663–680

    Article  MathSciNet  MATH  Google Scholar 

  38. Su M, Xu H-K (2010) Remarks on the gradient-projection algorithm. J Nonlinear Anal Optim 1:35–43

    MathSciNet  MATH  Google Scholar 

  39. Temlyakov VN (2015) Greedy approximation in convex optimization. Constr Approx 41:269–296

    Article  MathSciNet  MATH  Google Scholar 

  40. Thuy LQ, Wen C-F, Yao J-C, Hai TN (2018) An extragradient-like parallel method for pseudomonotone equilibrium problems and semigroup of nonexpansive mappings. Miskolc Math Notes 19:1185–1201

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang H, Xu H-K (2018) A note on the accelerated proximal gradient method for nonconvex optimization. Carpathian J Math 34:449–457

    Article  MathSciNet  MATH  Google Scholar 

  42. Xu H-K (2011) Averaged mappings and the gradient-projection algorithm. J Optim Theory Appl 150:360–378

    Article  MathSciNet  MATH  Google Scholar 

  43. Xu H-K (2017) Bounded perturbation resilience and superiorization techniques for the projected scaled gradient method. Inverse Prob 33:19 pp

    Google Scholar 

  44. Yao Y, Postolache M, Yao J-C (2019) Convergence of an extragradient algorithm for fixed point and variational inequality problems. J Nonlinear Convex Anal 20:2623–2631

    MathSciNet  MATH  Google Scholar 

  45. Zaslavski AJ (2020) The projected subgradient algorithm in convex optimization. Springer, Berlin. Briefs in Optimization (2020)

    Google Scholar 

  46. Zaslavski AJ (2010) The projected subgradient method for nonsmooth convex optimization in the presence of computational errors. Numer Funct Anal Optim 31:616–633

    Article  MathSciNet  MATH  Google Scholar 

  47. Zaslavski AJ (2012) The extragradient method for convex optimization in the presence of computational errors. Numer Funct Anal Optim 33:1399–1412

    Article  MathSciNet  MATH  Google Scholar 

  48. Zaslavski AJ (2012) The extragradient method for solving variational inequalities in the presence of computational errors. J Optim Theory Appl 153:602–618

    Article  MathSciNet  MATH  Google Scholar 

  49. Zaslavski AJ (2013) The extragradient method for finding a common solution of a finite family of variational inequalities and a finite family of fixed point problems in the presence of computational errors. J Math Anal Appl 400:651–663

    Article  MathSciNet  MATH  Google Scholar 

  50. Zaslavski AJ (2016) Numerical optimization with computational errors. Springer, Cham

    Book  MATH  Google Scholar 

  51. Zaslavski AJ (2020) Convex optimization with computational errors. In: Springer Optimization and Its Applications. Springer, Cham

    Google Scholar 

  52. Zeng LC, Yao JC (2006) Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwanese J Math 10:1293–1303

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Technion – Israel Institute of Technology, Haifa, Israel

    Alexander J. Zaslavski

Authors
  1. Alexander J. Zaslavski
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Zaslavski, A.J. (2022). Introduction. In: Optimization in Banach Spaces. SpringerBriefs in Optimization. Springer, Cham. https://doi.org/10.1007/978-3-031-12644-4_1

Download citation

Publish with us

Policies and ethics

Search

Navigation