K.C. Kiwiel, 1983. An aggregate subgradient method for nonsmooth convex minimization, Mathematical Programming 27, 320-341.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....( is a nonsmooth concave function (the concavity follows from the variational characterization) In fact this function is nonsmooth precisely at those points where the minimum eigenvalue has a multiplicity greater than one. A general scheme to minimize such functions is the bundle scheme (Kiwiel [29, 30, 31], Lemarechal [34] and Schramm et al. [40] and the books by Urruty and Lemarechal [26, 27] especially Chapter XV (3) An excellent survey on eigenvalue optimization appears in Lewis and Overton [35] To motivate the bundle approach, we begin with some preliminaries for the minimum eigenvalue ....

K.C. Kiwiel, An aggregate subgradient method for nonsmooth convex minimization, Mathematical Programming 27(1983), pp. 320-341.

....fixed, we set y k 1 = y c and k is increased by 1; in some improved versions (see the considerations in x6 below) an additional test is made before increasing k. iii) The choices of the norming fMng given so far in the literature are: an abstract sequence, as in [10] Mn j I, as in [8], Mn = n I, with heuristic rules for computing n ; see [9] 15] RR n2128 4 Claude Lemar echal , Claudia Sagastiz abal An essential feature of our present development consists of a quasi Newton update of Mn [4] using the so called Moreau Yosida regularization ( 12] 19] we also pay some ....

K.C. Kiwiel. An aggregate subgradient method for nonsmooth convex minimization. Mathematical Programming, 27:320--341, 1983.

....proximal bundle method considers only polyhedral approximations of W . In practice it is usually impossible to include the subgradients of all previous iterates in the model. Without endangering convergence the number of subgradients can be controlled by introducing an aggregate subgradient [18], 15, x XV.3] Here, the aggregate subgradient of iteration k corresponds to W k of Theorem 1.3.1. The decisive two conditions for the convergence analysis of the proximal bundle method are W k 2 W k 1 and p k 1 p T k 1 2 W k 1 with p k 1 2 f(y k 1 ) 1.4.22) These conditions ensure that ....

Krzysztof C. Kiwiel. An aggregate subgradient method for nonsmooth convex minimization. Math. Programming, 27:320-341, 1983.

....(Procedure 1) terminates in a nite number of steps. Proof: Due to Lemma 16 and Assumption 3(i) the function f( k (x k ; is weakly upper semismooth. Therefore, the proof of Theorem 4.1 (a) in [19] can be easily adapted to our linesearch procedure. The following lemma due to Kiwiel [9] will be used in the proof of Theorem 3. In particular, given 0, this lemma implies existence of a nite number of successive null steps without reset such that jd k j . Lemma 18 [9, 10] Let w k = 1 2 jd k j 2 k and assume t k L = 0 and r k a = 0. Then, there exists a ....

....4.1 (a) in [19] can be easily adapted to our linesearch procedure. The following lemma due to Kiwiel [9] will be used in the proof of Theorem 3. In particular, given 0, this lemma implies existence of a nite number of successive null steps without reset such that jd k j . Lemma 18 [9, 10] Let w k = 1 2 jd k j 2 k and assume t k L = 0 and r k a = 0. Then, there exists a constant C independent of k such that 0 w k 1 w k (1 mR ) 2 (w k ) 2 =8C 2 : 53) We are now in a position to establish convergence of the generalized proximal bundle method. We ....

K. C. Kiwiel. An aggregate subgradient method for nonsmooth convex minimization. Mathematical Programming, 27:320-341, 1983.

....oe k is to one, the better p a (x k ) approximates the proximal point p(x k ) Remark. Procedure (2.2) 2.3) is the pure cutting plane algorithm without any strategy for dropping useless cutting planes. This is only for simplicity of presentation. Any improved cutting plane type procedure [1, 7, 15, 17] can be used as well to generate fy j g converging to the proximal point p(x k ) Actually we used a modified version of the cutting plane method with resetting and deleting rules [1, 7, 17] in our numerical experiments reported in Section 4. Now we state the algorithm. Algorithm 2.1 Choose an ....

K.C. Kiwiel, An aggregate subgradient method for nonsmooth convex minimization, Mathematical Programming 27 (1983) 320-341.

....oe k is to one, the better p a (x k ) approximates the proximal point p(x k ) Remark. Procedure (2.2) 2.3) is the pure cutting plane algorithm without any strategy for dropping useless cutting planes. This is only for simplicity of presentation. Any improved cutting plane type procedure [1, 7, 15, 17] can be used as well to generate fy j g converging to the proximal point p(x k ) Actually we used a modified version of the cutting plane method with resetting and deleting rules [1, 7, 17] in our numerical experiments reported in Section 4. Now we state the algorithm. Algorithm 2.1 Choose an ....

K. C. Kiwiel, "An aggregate subgradient method for nonsmooth convex minimization ", Mathematical Programming 27 (1983) 320-341.

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K.C. Kiwiel, 1983. An aggregate subgradient method for nonsmooth convex minimization, Mathematical Programming 27, 320-341.

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K.C. KIWIEL. An aggregate subgradient method for nonsmooth convex minimization. Math. Programming, 27:320--341, 1983.

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