Abstract
To permit the stable solution of ill-posed problems, the Proximal Point Algorithm (PPA) was introduced by Martinet (RIRO 4:154–159, 1970) and further developed by Rockafellar (SIAM J Control Optim 14:877–898, 1976). Later on, the usual proximal distance function was replaced by the more general class of Bregman(-like) functions and related distances; see e.g. Chen and Teboulle (SIAM J Optim 3:538–543, 1993), Eckstein (Math Program 83:113–123, 1998), Kaplan and Tichatschke (Optimization 56(1–2):253–265, 2007), and Solodov and Svaiter (Math Oper Res 25:214–230, 2000). An adequate use of such generalized non-quadratic distance kernels admits to obtain an interior-point-effect, that is, the auxiliary problems may be treated as unconstrained ones. In the above mentioned works and nearly all other works related with this topic it was assumed that the operator of the considered variational inequality is a maximal monotone and paramonotone operator. The approaches of El-Farouq (JOTA 109:311–326, 2001), and Schaible et al. (Taiwan J Math 10(2):497–513, 2006) only need pseudomonotonicity (in the sense of Karamardian in JOTA 18:445–454, 1976); however, they make use of other restrictive assumptions which on the one hand contradict the desired interior-point-effect and on the other hand imply uniqueness of the solution of the problem. The present work points to the discussion of the Bregman algorithm under significantly weaker assumptions, namely pseudomonotonicity [and an additional assumption much less restrictive than the ones used by El-Farouq and Schaible et al. We will be able to show that convergence results known from the monotone case still hold true; some of them will be sharpened or are even new. An interior-point-effect is obtained, and for the generated subproblems we allow inexact solutions by means of a unified use of a summable-error-criterion and an error criterion of fixed-relative-error-type (this combination is also new in the literature).
Similar content being viewed by others
References
Brézis H.: Équations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier 18, 115–175 (1968)
Burachik R., Iusem A.: A generalized Proximal Point algorithm for the variational inequality problem in Hilbert space. SIAM J. Optim. 8, 197–216 (1998)
Burachik R., Svaiter B.: \({\varepsilon}\) -enlargement of maximal monotone operators in Banach spaces. Set-Valued Anal. 7, 117–132 (1999)
Censor Y., Iusem A., Zenios S.A.: An interior point method with Bregman functions for the variational inequality problem with paramonotone operators. Math. Program. 81, 373–400 (1998)
Chen G., Teboulle M.: Convergence analysis of a proximal-like minimization algorithm using Bregman functions. SIAM J. Optim. 3, 538–543 (1993)
Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Crouzeix J.-P.: Pseudo-monotone variational inequality problems: existence of solutions. Math. Program. 78, 305–314 (1997)
Crouzeix J.-P., Zhu D.L., Marcotte P.: Conditions ensuring the applicability of cutting-plane methods for solving variational inequalities. JOTA 87, 457–471 (1995)
Eckstein J.: Approximate iterations in Bregman-function-based proximal algorithms. Math. Program. 83, 113–123 (1998)
El-Farouq N.: Pseudomonotone variational inequalities: convergence of proximal methods. JOTA 109, 311–326 (2001)
Facchinei F., Pang J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
Hadjisavvas N.: Continuity and maximality properties of pseudomonotone operators. J. Convex Anal. 10, 459–469 (2003)
Hadjisavvas N., Schaible S.: On a generalization of paramonotone maps and its application to solving the Stampacchia variational inequality. Optimization 55, 593–604 (2006)
Iusem A.: On some properties of paramonotone operators. J. Conv. Anal. 5, 269–278 (1998)
Kaplan, A., Tichatschke, R.: On the convergence of interior proximal methods for variational inequalities on non-polyhedral sets. Discuss. Math. (2009, to appear)
Kaplan A., Tichatschke R.: On inexact generalized proximal methods with a weakened error tolerance criterion. Optimization 53, 3–17 (2004)
Kaplan A., Tichatschke R.: Bregman functions and proximal methods for variational problems with nonlinear constraints. Optimization 56(1–2), 253–265 (2007)
Kaplan A., Tichatschke R.: Interior proximal method for variational inequalities on non-polyhedral sets. Discuss. Math. 27(1), 71–93 (2007)
Karamardian S.: Complementarity problems over cones with monotone and pseudomonotone maps. JOTA 18, 445–454 (1976)
Kien B., Yao J.-C., Yen N.: On the solution existence of pseudomonotone variational inequalities. J. Glob. Optim. 41, 135–145 (2008)
Langenberg, N.: Convergence analysis of an extended auxiliary problem principle with various stopping criteria. Optim. Methods Softw. (2009, accepted)
Martinet B.: Régularisation d’inéquations variationelles par approximations successives. RIRO 4, 154–159 (1970)
Polyak B.T.: Introduction to Optimization. Optimization Software, Inc. Publ. Division, New York (1987)
Rockafellar R.T.: Monotone operators and the Proximal Point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Schaible S., Yao J., Zeng L.: A proximal method for pseudomonotone type variational-like inequalities. Taiwan. J. Math. 10(2), 497–513 (2006)
Solodov M., Svaiter B.: An inexact hybrid generalized proximal point algorithm and some new results on the theory of Bregman functions. Math. Oper. Res. 25, 214–230 (2000)
Tam N., Yao J., Yen N.: Solution methods for pseudomonotone variational inequalities. JOTA 138, 253–273 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research is supported by the Landesgraduiertenförderung Rheinland-Pfalz.
Rights and permissions
About this article
Cite this article
Langenberg, N. Pseudomonotone operators and the Bregman Proximal Point Algorithm. J Glob Optim 47, 537–555 (2010). https://doi.org/10.1007/s10898-009-9470-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-009-9470-7
Keywords
- Pseudomonotone operators
- Variational inequalities
- Bregman distances
- Proximal Point algorithm
- Interior-point-effect