## Approximate Iterations in Bregman-Function-Based Proximal Algorithms (1998)

Citations: | 24 - 2 self |

### BibTeX

@MISC{Eckstein98approximateiterations,

author = {Jonathan Eckstein},

title = {Approximate Iterations in Bregman-Function-Based Proximal Algorithms},

year = {1998}

}

### OpenURL

### Abstract

This paper establishes convergence of generalized Bregman-function-based proximal point algorithms when the iterates are computed only approximately. The problem being solved is modeled as a general maximal monotone operator, and need not reduce to minimization of a function. The accuracy conditions on the iterates resemble those required for the classical "linear" proximal point algorithm, but are slightly stronger; they should be easier to verify or enforce in practice than conditions conditions given in earlier analyses of approximate generalized proximal methods. Subject to these practically enforceable accuracy restrictions, convergence is obtained under the same conditions currently established for exact Bregman-function-based proximal methods.

### Citations

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(Show Context)
Citation Context ... partial level sets L'(cc, y) -- {x ] Dh (x,y) ~ a} be bounded. However, this condition follows automatically from the observation that L'(O,y) = {y} for all y E S, the convexity Of Dh(X,y) in x, and =-=[23]-=-, Corollary 8.7.1; see [1] for a deeper analysis. Note that [18] proposes a weaker set of conditions than B1-B7, but assumes that T is the subdifferential of a convex function. 3. Analysis We now prov... |

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275 |
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(Show Context)
Citation Context ... guaranteed to exist even if T is maximal, but their existence is assured by some simple additional assumptions; see e.g. [22]. A classical method for this problem is the proximalpoint algorithm; see =-=[25]-=-, which draws on a large volume of prior work by various authors. Given a sequence of 1 E-mail: jeckstei@rutcor.rutgers.edu. 0025-5610/98/$19.00 9 1998 The Mathematical Programming Society, Inc. Publi... |

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(Show Context)
Citation Context ...e Q}, (13) whence (T + Q) (x) is equal to the Minkowski sum T(x) + Q(x). If T and Q are monotone, so is T + Q. If T and Q are both maximal and (ri dom T) rl (ri dom Q) # 0, then T + Q is also maximal =-=[24]-=-. Here, "ri X" denotes the relative interior of the set X, and "dom T" the domain of the operator T, dora T= {xl(x,y ) 9 T} = {xl r(x ) # 0}. Given any set C C_ N", its normal cone operator Nc C_ C x ... |

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(Show Context)
Citation Context ...g only Z # 0. These assumptions are: k=l [Idll < oc, (18) S(ek,x k) exists and is finite. (19) k=l We now proceed with the main convergence result, first stating a necessary technical lemma. Lemma 4 (=-=[21]-=-, Section 2.2). Suppose {ak}k~ 0, {Tk}~=0 C R are sequences such that {ak} is bounded below, Y~=0 7k exists and is finite, and the recursion ak+x ~ ak q- 7k holds for all k. Then {ak} is convergent. T... |

49 |
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(Show Context)
Citation Context ...med to be a Bregman function with zone S, defined similarly to B l-B7 below. Further analysis and applications appeared in [26], and the case of a general maximal monotone operator T was addressed in =-=[12]-=-. Subsequent papers, including for example [5,8,9,15,16,18], have dealt with many other variations, special cases, and related algorithms. Exactly computing solutions x k+x to Eq. (4) is fraught with ... |

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(Show Context)
Citation Context ...y) -- {x ] Dh (x,y) ~ a} be bounded. However, this condition follows automatically from the observation that L'(O,y) = {y} for all y E S, the convexity Of Dh(X,y) in x, and [23], Corollary 8.7.1; see =-=[1]-=- for a deeper analysis. Note that [18] proposes a weaker set of conditions than B1-B7, but assumes that T is the subdifferential of a convex function. 3. Analysis We now prove the convergence of the i... |

39 |
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(Show Context)
Citation Context ...ed similarly to B l-B7 below. Further analysis and applications appeared in [26], and the case of a general maximal monotone operator T was addressed in [12]. Subsequent papers, including for example =-=[5,8,9,15,16,18]-=-, have dealt with many other variations, special cases, and related algorithms. Exactly computing solutions x k+x to Eq. (4) is fraught with the same practical difficulties as apply to Eq. (3). This p... |

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(Show Context)
Citation Context ...ed similarly to B l-B7 below. Further analysis and applications appeared in [26], and the case of a general maximal monotone operator T was addressed in [12]. Subsequent papers, including for example =-=[5,8,9,15,16,18]-=-, have dealt with many other variations, special cases, and related algorithms. Exactly computing solutions x k+x to Eq. (4) is fraught with the same practical difficulties as apply to Eq. (3). This p... |

39 |
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(Show Context)
Citation Context ...tial mapping of a convex function, and, as in much subsequent work, h is assumed to be a Bregman function with zone S, defined similarly to B l-B7 below. Further analysis and applications appeared in =-=[26]-=-, and the case of a general maximal monotone operator T was addressed in [12]. Subsequent papers, including for example [5,8,9,15,16,18], have dealt with many other variations, special cases, and rela... |

34 |
Proximal minimization algorithm with D-functions
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(Show Context)
Citation Context ...nd therefore Eq. (4) reduces to Eq. (3). In this case, one also has Dh(x,.r ~) = 89 IIx -xql 2, and Eq. (6) is the classical proximal minimization formula. Early analyses of the iteration (4) include =-=[7,10,11]-=-. Refs. [10,11] assume that h is strongly convex, and make various assumptions on T. In [7], T is the subdifferential mapping of a convex function, and, as in much subsequent work, h is assumed to be ... |

31 |
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(Show Context)
Citation Context ...ts z E R n such that 0 E T(z), or equivalently (z, 0) E T. Such points are not guaranteed to exist even if T is maximal, but their existence is assured by some simple additional assumptions; see e.g. =-=[22]-=-. A classical method for this problem is the proximalpoint algorithm; see [25], which draws on a large volume of prior work by various authors. Given a sequence of 1 E-mail: jeckstei@rutcor.rutgers.ed... |

30 |
Auxiliary problem principle and decomposition of optimization problems
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(Show Context)
Citation Context ...nd therefore Eq. (4) reduces to Eq. (3). In this case, one also has Dh(x,.r ~) = 89 IIx -xql 2, and Eq. (6) is the classical proximal minimization formula. Early analyses of the iteration (4) include =-=[7,10,11]-=-. Refs. [10,11] assume that h is strongly convex, and make various assumptions on T. In [7], T is the subdifferential mapping of a convex function, and, as in much subsequent work, h is assumed to be ... |

26 | Enlargement of monotone operators with applications to variational inequalities
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(Show Context)
Citation Context ...n established. A1 and A2 appear in [12], and their treatment here is generally similar, except for the introduction of a non-zero {ek}. Condition A3 is a simplified version of some hypotheses used in =-=[5,6]-=-; similar assumptions have appeared in [3,4] for the analysis of generalized gradient projection algorithms. Note that if A1 holds but z = 0, an analysis virtually identical to that in [12] Shows that... |

21 |
A generalized proximal point algorithm for the variational inequality problem in a Hilbert space
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- 1998
(Show Context)
Citation Context ...ed similarly to B l-B7 below. Further analysis and applications appeared in [26], and the case of a general maximal monotone operator T was addressed in [12]. Subsequent papers, including for example =-=[5,8,9,15,16,18]-=-, have dealt with many other variations, special cases, and related algorithms. Exactly computing solutions x k+x to Eq. (4) is fraught with the same practical difficulties as apply to Eq. (3). This p... |

21 | Convergence of proximal-like algorithms
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- 1997
(Show Context)
Citation Context ...closed proper convex function. Here,3". Eckstein I Mathematical Programming 83 (1998) 113-123 115 the approximation protocol is not exactly as in Eq. (2), but uses the theory of e-subgradients. Ref. =-=[27]-=- gives related e-subgradient-based results in the context of iterations based on (p-divergence distances. Ref. [14] addresses some related topics in a different problem setting, again using e-subgradi... |

20 |
Entropy-Like proximal methods in convex programming
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(Show Context)
Citation Context |

16 |
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(Show Context)
Citation Context ...n protocol is not exactly as in Eq. (2), but uses the theory of e-subgradients. Ref. [27] gives related e-subgradient-based results in the context of iterations based on (p-divergence distances. Ref. =-=[14]-=- addresses some related topics in a different problem setting, again using e-subgradients. Ref. [6] introduced the notion of an c-enlargement T ~ of a general monotone operator T; this notion is somew... |

14 |
An interior point method with Bregman functions for the variational inequality problem with paramonotone operators
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- 1998
(Show Context)
Citation Context |

13 |
An iterative solution of a variational inequality for certain monotone operators in Hilbert space
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(Show Context)
Citation Context ...onvex function f: [~" ~ (-o% +oo]. A3. T has following two properties (see, e.g. [5,6,8]): (i) If {(xk,yk)} C T, {x k} C S, and {x k} is convergent, then {yk} has a limit point; (ii) T isparamonotone =-=[3,8]-=-, that is, (x,y), (x',y') E T and (x - x',y - y') = 0 collectively imply that (x,y') E T. The three alternatives A1-A3 represent essentially the same cases for which convergence of the exact Bregman p... |

12 |
Auxiliary problem principle extended to variational inequalities
- Cohen
- 1988
(Show Context)
Citation Context ...nd therefore Eq. (4) reduces to Eq. (3). In this case, one also has Dh(x,.r ~) = 89 IIx -xql 2, and Eq. (6) is the classical proximal minimization formula. Early analyses of the iteration (4) include =-=[7,10,11]-=-. Refs. [10,11] assume that h is strongly convex, and make various assumptions on T. In [7], T is the subdifferential mapping of a convex function, and, as in much subsequent work, h is assumed to be ... |

11 |
Convergence rate analysis of nonquadratic proximal method for convex and linear programming
- Iusem, Teboulle
- 1995
(Show Context)
Citation Context |

9 |
Proximal minimizations with D-functions and the massively parallel solution of linear stochastic network programs
- Nielsen, Zenios
(Show Context)
Citation Context ... e k+l . (10) This approximation condition is easier to check than c-enlargement- and e-subgradient-based protocols, and more closely captures the spirit of existing computational experiments such as =-=[20]-=-. Condition (10), as with the condition (2) of [25], can always be checked by the simple examination of a gradient, subgradient, or residual, whereas Eq. (7) or the related e-subgradient conditions wi... |

3 |
On the twice differentiable cubic augmented Lagrangian
- Kiwiel
- 1996
(Show Context)
Citation Context ... h(x) = }-~=1 x~ log xi - xi, but still admits many useful applications, such as to the smoothing of augmented Lagrangians for complementarity problems [13] and inequality constrained convex programs =-=[12,19,26]-=-. Of course, if A2 or A3 holds, then A1 is not needed. Assumption A3(ii) dates back at least to [3,4], where it appeared in connection with a different, related algorithm. The name "paramonotone" was ... |

2 | Methodes Proximales Entropiques. Doctoral thesis. Université de Montpellier - Kabbadj - 1994 |

2 |
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(Show Context)
Citation Context ...l] = 0, j~or j~zr Ck(j ) where we use x kUl, x *~/)+l ~ x ~, the continuity of ~Th at x ~, e k ~ 0, and {c~} bounded away from zero. Since T is maximal, it is a closed set in ~q" • W; see for example =-=[2]-=-. Therefore lim (x~(J)+',y k0)+t) = (x~,0) E T. j~oo As x ~176 E S, we also have (x~, 0) E 7", establishing the claim. Now suppose that A2 holds. Then y~(,)+l is a subgradient off f(z) >~ f(x <*)+') +... |

1 |
Corrigendum to [3
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(Show Context)
Citation Context ...nted Lagrangians for complementarity problems [13] and inequality constrained convex programs [12,19,26]. Of course, if A2 or A3 holds, then A1 is not needed. Assumption A3(ii) dates back at least to =-=[3,4]-=-, where it appeared in connection with a different, related algorithm. The name "paramonotone" was later applied in [8]. As for the error sequence {ek}, we will use a pair of assumptions that are slig... |

1 | Smooth methods of multipliers for monotone complementarity problems - Eckstein, Ferris - 1996 |