zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Proximal-point algorithm using a linear proximal term. (English) Zbl 1170.49008
Summary: Proximal-Point Algorithms (PPAs) are classical solvers for convex optimization problems and monotone Variational Inequalities (VIs). The proximal term in existing PPAs usually is the gradient of a certain function. This paper presents a class of PPA-based methods for monotone VIs. For a given current point, a proximal point is obtained via solving a PPA-like subproblem whose proximal term is linear but may not be the gradient of any functions. The new iterate is updated via an additional slight calculation. Global convergence of the method is proved under the same mild assumptions as the original PPA. Finally, profiting from the less restrictions on the linear proximal terms, we propose some parallel splitting augmented Lagrangian methods for structured variational inequalities with separable operators.
MSC:
49J40Variational methods including variational inequalities
49M30Other numerical methods in calculus of variations
90C25Convex programming
References:
[1]Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39(4), 669–713 (1997) · Zbl 0891.90158 · doi:10.1137/S0036144595285963
[2]He, B.S.: Inexact implicit methods for monotone general variational inequalities. Math. Program. Ser. A 86(1), 199–217 (1999) · Zbl 0979.49006 · doi:10.1007/s101070050086
[3]Martinet, B.: Régularization d’inéquations variationelles par approximations sucessives. Rev. Fr. Inf. Rech. Opér. Ser. R-3 4, 154–158 (1970)
[4]Auslender, A., Teboulle, M.: Lagrangian duality and related multiplier methods for variational inequality problems. SIAM J. Optim. 10(4), 1097–1115 (2000) · Zbl 0996.49005 · doi:10.1137/S1052623499352656
[5]Burke, J., Qian, M.J.: On the superlinear convergence of the variable metric proximal point algorithm using Broyden and BFGS matrix secant updating. Math. Program. 88(1), 157–181 (2000) · doi:10.1007/PL00011373
[6]Eckstein, J.: Approximate iterations in Bregman-function-based proximal algorithms. Math. Program. Ser. A 83(1), 113–123 (1998)
[7]Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976) · Zbl 0358.90053 · doi:10.1137/0314056
[8]Teboulle, M.: Convergence of proximal-like algorithms. SIAM J. Optim. 7, 1069–1083 (1997) · Zbl 0890.90151 · doi:10.1137/S1052623495292130
[9]Burachik, R.S., Iusem, A.N.: A generalized proximal point algorithm for the variational inequality problem in a Hilbert space. SIAM J. Optim. 8(1), 197–216 (1998) · Zbl 0911.90273 · doi:10.1137/S1052623495286302
[10]Auslender, A., Teboulle, M., Ben-Tiba, S.: A logarithmic-quadratic proximal method for variational inequalities. Comput. Optim. Appl. 12, 31–40 (1999) · Zbl 1039.90529 · doi:10.1023/A:1008607511915
[11]Kincaid, D., Cheney, W.: Numerical Analysis: Mathematics of Scientific Computing, 3rd edn. Thomson Learning Inc. (2002)
[12]Blum, E., Oettli, W.: Mathematische Optimierung. Econometrics and Operations Research XX. Springer, Berlin (1975)
[13]Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM Studies in Applied Mathematics. SIAM, Philadelphia (1989)
[14]Fukushima, M.: Application of the alternating direction method of multipliers to separable convex programming problems. Comput. Optim. Appl. 1(1), 93–111 (1992) · Zbl 0763.90071 · doi:10.1007/BF00247655
[15]Tseng, P.: Alternating projection-proximal methods for convex programming and variational inequalities. SIAM J. Optim. 7, 951–965 (1997) · Zbl 0914.90218 · doi:10.1137/S1052623495279797
[16]Xue, G.L., Ye, Y.Y.: An efficient algorithm for minimizing a sum of Euclidean norms with applications. SIAM J. Optim. 7, 1017–1036 (1997) · Zbl 0885.68074 · doi:10.1137/S1052623495288362