J.-L. Goffin and J.-P. Vial. Convex nondifferentiable optimization: A survey focussed on the analytic center cutting plane mathod. Technical Report 99.02, McGill University, Canada, February 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....value V , and is a (componen[ wise) upper bound for p. The ACCPM can [hen be oufiined as given p(0) and a required [olerance 0 for [he duali W gap. k: 0. repeat 1. Compute the analytic center z of ( At the same time, obtain a lower bound (by duality of computing the analytic center, see [21 8] for details) 2. Let p( be the vector of the first L components of z( and compute V(p ) and a subgradient h (see section 6.2.1) The new localization information 20 (a) 7 (k) and its analytical center (b) The new localization set 7 (k ) Figure 6: One iteration step of the ACCPM. is that ....

....v(p ) T(p p( 3. Form the polyhedron 7 ( 7 ( 7 ) and update the upper bound V : min V, V(p ) 4. If the duality gap v , quit; else, let k : k 1. One iteration of the ACCPM is illustrated in figure 6. For computational details and convergence analysis of ACCPM, see [21, 22, 8] and references therein. We applied ACCPM to the example in section 5. Figure 5 also shows the dual objective function and the lower bound obtained by ACCPM versus number of iterations. Unlike the subgradient methods in section 6.3.1 where only the current subgradient is used at each step, in ....

J.-L. Goffin and J.-P. Vial. Convex nondifferentiable optimization: A survey focussed on the analytic center cutting plane mathod. Technical Report 99.02, McGill University, Canada, February 1999.

....V(p ) and a supergradient h . Then (p , V ) must lie in the halfspace t (p, v) I v v(p ) h( T(p 3. Form the polyhedron 7 ( 7 (k) A 7 , and update the upper bound : min 5, V(p( 4. IfS v ,quit;else, let k: k l. For computational details and convergence analysis of ACCPM, see [15, 9] and references therein. We applied ACCPM to solve the dual of the SRRA problem in section 5. The dual objective function and the lower bound are also ploted in figure 3. Compared to the subgradient methods, ACCPM usually converges faster. However, in ACCPM, all previous computed subgradients are ....

J.-L. Goffin and J.-P. Vial. Convex nondifferen- tiable optimization: A survey focussed on the analytic center cutting plane mathod. Technical Report 99.02, McGill University, Canada, February 1999.

....semidefinite programming. The website of semidefinite programming (http: www.zib.de helmberg semidef.html) contains a nice categorized list of papers in this area. Recently, there has been also an increasing interest in cutting plane methods based on analytic centers. The paper of Goffin and Vial [5] and the references therein provide a convenient overview on the related work. Our paper attempts to make a connection between the semidefinite programming techniques and the analytic center cutting plane methods. We hope that the analysis in this paper can stimulate further research in both ....

J.-L. Goffin and J.-P. Vial, "Convex nondifferentiable optimization: a survey focussed on the analytic center cutting plane method", Logilab Technical Report, Department of Management Studies, University of Geneva, Switzerland, February 1999.

....of Geneva, 102, Bd Carl Vogt, CH 1211 Gen eve 4, Switzerland. E mail: jpvial uni2a.unige.ch. 1 E 896 inferior to primal dual method that have become nowadays standard. On the other hand, the method, or a variant of it, is used with great success in the computation of analytic centers. See [7] for a survey on the analytic center cutting plane method. We feel it worth to deliver an improved presentation of the original method, since it is used in practice to solve, in a decomposition framework, huge and challenging nondifferentiable optimization problems [7] The approach in [3] treats ....

....of analytic centers. See [7] for a survey on the analytic center cutting plane method. We feel it worth to deliver an improved presentation of the original method, since it is used in practice to solve, in a decomposition framework, huge and challenging nondifferentiable optimization problems [7]. The approach in [3] treats the objective as the goal constraint of achieving a given target value. The target value is progressively triggered to its optimum level. The overall scheme can be interpreted as a sequence of subproblems; each one is an attempt at solving a pure feasibility problem, ....

J.-L. Goffin and J.-Ph. Vial (1999), "Convex nondifferentiable optimization: a survey on the analytic center cutting plane method", Tech. Report 99.02, Logilab/Management Studies, University of Geneva, Switzerland. 10

....methods have been used to find inexact solutions of the large master problems, or to approximately solve analytic center subproblems to generate test points. Approaches such as these are described in [Gondzio and Sarkissian, 1996] Gondzio and Kouwenberg, 1999] and the survey paper [Goffin and Vial, 1999]. Additionally, application of interior point methodology to nonconvex nonlinear programming has occupied many researchers for some time now. The methods that have been proposed to date contain many ingredients, including primal dual steps, barrier and merit functions, and scaled trust regions. ....

Goffin, J. and Vial, J. (1999). Convex nondifferentiable optimization: A survey based on the analytic center cutting plane method. Technical Report 99.02, Logilab, HEC, Section of Management Studies, University of Geneva.

....programming (http: www.zib.de helmberg semidef.html) contains a nice categorized list of papers in this area. Recently, there has been also an increasing interest in cutting plane methods based on analytic centers which was first proposed by Sonnevend [9] The paper of Goffin and Vial [5] and the references therein provide a convenient overview on the related work. Our paper attempts to make a connection between the semidefinite programming techniques and the analytic center cutting plane methods. We hope that the analysis in this paper can stimulate further research in both ....

....used in [4] In addition, a number of important estimates in [4] has to be re built using matrix analysis. Compared to [6] although both papers allow nonlinear cuts, we use moderately deep cuts rather than shallow cuts used in [6] for the meanings of shallow and moderately deep cuts, see [5]) Moreover, we include the barrier term of S m in our potential function rather than treat it as a set of convex constraints. Therefore the method proposed in this paper guarantees that all iterates are positive definite, a favorite property for many applications. 1.2 Notations and ....

J-L. Goffin and J-P. Vial, "Convex nondifferentiable optimization: a survey focused on the analytic center cutting plane method", HEC/Logilab Tech. Report 99-02, University of Geneva (1999).

....methods have been used to find inexact solutions of the large master problems, or to approximately solve analytic center subproblems to generate test points. Approaches such as these are described in [Gondzio and Sarkissian, 1996] Gondzio and Kouwenberg, 1999] and the survey paper [Goffin and Vial, 1999]. Additionally, application of interior point methodology to nonconvex nonlinear programming has occupied many researchers for some time now. The methods that have been proposed to date contain many ingredients, including primal dual steps, barrier and merit functions, and scaled trust regions. ....

Goffin, J. and Vial, J. (1999). Convex nondifferentiable optimization: A survey based on the analytic center cutting plane method. Technical Report 99.02, Logilab, HEC, Section of Management Studies, University of Geneva.

....way they select the query point in the localization set. Other differences pertain to the update of the localization set. For instance, the subgradient method discards all previous cuts and concatenates the subgradients inequalities (4) into a single one. For a survey of cutting plane methods, see [22]. In the literature, the terminology cutting plane method often refers to the specific implementation which selects as query point a minimizer of i in the localization set [2, 13, 31] This is easily done by solving a linear program. This approach in general performs well, but it is known to be ....

J.-L. Goffin and J.-P. Vial, Convex nondifferentiable optimization: a survey on the analytic center cutting plane method, tech. report, HEC/Logilab, University of Geneva, 1204 Geneva, Switzerland, November 1998.

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