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81
A Spectral Bundle Method for Semidefinite Programming
 SIAM Journal on Optimization
, 1997
"... . A central drawback of primaldual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices. This restricts applicability to problems of small dimension. Typically semidefinite relaxations arising in combinatorial applic ..."
Abstract

Cited by 141 (6 self)
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. A central drawback of primaldual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices. This restricts applicability to problems of small dimension. Typically semidefinite relaxations arising in combinatorial applications have sparse and well structured cost and coefficient matrices of huge order. We present a method that allows to compute acceptable approximations to the optimal solution of large problems within reasonable time. Semidefinite programming problems with constant trace on the primal feasible set are equivalent to eigenvalue optimization problems. These are convex nonsmooth programming problems and can be solved by bundle methods. We propose replacing the traditional polyhedral cutting plane model constructed from subgradient information by a semidefinite model that is tailored for eigenvalue problems. Convergence follows from the traditional approach but a proof is included for completene...
Learning Structural SVMs with Latent Variables
"... It is well known in statistics and machine learning that the combination of latent (or hidden) variables and observed variables offer more expressive power than models with observed variables alone. Latent variables ..."
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Cited by 114 (2 self)
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It is well known in statistics and machine learning that the combination of latent (or hidden) variables and observed variables offer more expressive power than models with observed variables alone. Latent variables
Dual Decomposition in Stochastic Integer Programming
 Operations Research Letters
, 1997
"... We present an algorithm for solving stochastic integer programming problems with recourse, based on a dual decomposition scheme and Lagrangian relaxation. The approach can be applied to multistage problems with mixedinteger variables in each time stage. Numerical experience is presented for so ..."
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Cited by 61 (5 self)
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We present an algorithm for solving stochastic integer programming problems with recourse, based on a dual decomposition scheme and Lagrangian relaxation. The approach can be applied to multistage problems with mixedinteger variables in each time stage. Numerical experience is presented for some twostage test problems.
Decomposition Algorithms for Stochastic Programming on a Computational Grid
 Computational Optimization and Applications
, 2001
"... . We describe algorithms for twostage stochastic linear programming with recourse and their implementation on a grid computing platform. In particular, we examine serial and asynchronous versions of the Lshaped method and a trustregion method. The parallel platform of choice is the dynamic, heter ..."
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Cited by 56 (9 self)
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. We describe algorithms for twostage stochastic linear programming with recourse and their implementation on a grid computing platform. In particular, we examine serial and asynchronous versions of the Lshaped method and a trustregion method. The parallel platform of choice is the dynamic, heterogeneous, opportunistic platform provided by the Condor system. The algorithms are of masterworker type (with the workers being used to solve secondstage problems), and the MW runtime support library (which supports masterworker computations) is key to the implementation. Computational results are presented on large sample average approximations of problems from the literature. 1.
Practical Aspects of the MoreauYosida Regularization I: Theoretical Properties
, 1994
"... When computing the infimal convolution of a convex function f with the squared norm, one obtains the socalled MoreauYosida regularization of f . Among other things, this function has a Lipschitzian gradient. We investigate some more of its properties, relevant for optimization. Our main result co ..."
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Cited by 49 (2 self)
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When computing the infimal convolution of a convex function f with the squared norm, one obtains the socalled MoreauYosida regularization of f . Among other things, this function has a Lipschitzian gradient. We investigate some more of its properties, relevant for optimization. Our main result concerns secondorder differentiability and is as follows. Under assumptions that are quite reasonable in optimization, the MoreauYosida is twice diffferentiable if and only if f is twice differentiable as well. In the course of our development, we give some results of general interest in convex analysis. In particular, we establish primaldual relationship between the remainder terms in the firstorder development of a convex function and its conjugate.
Variable Metric Bundle Methods: from Conceptual to Implementable Forms
, 1996
"... To minimize a convex function, we combine MoreauYosida regularizations, quasiNewton matrices and bundling mechanisms. First we develop conceptual forms using "reversal " quasiNewton formulae and we state their global and local convergence. Then, to produce implementable versions, we inco ..."
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Cited by 40 (8 self)
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To minimize a convex function, we combine MoreauYosida regularizations, quasiNewton matrices and bundling mechanisms. First we develop conceptual forms using "reversal " quasiNewton formulae and we state their global and local convergence. Then, to produce implementable versions, we incorporate a bundle strategy together with a "curvesearch". No convergence results are given for the implementable versions; however some numerical illustrations show their good behaviour even for largescale problems.
Bundle Methods to Minimize the Maximum Eigenvalue Function
, 1999
"... this paper. 1.9.1 The spectral bundle method ..."
A Spectral Bundle Method with Bounds
 MATHEMATICAL PROGRAMMING
, 1999
"... Semidefinite relaxations of quadratic 01 programming or graph partitioning problems are well known to be of high quality. However, solving them by primaldual interior point methods can take much time even for problems of moderate size. The recent spectral bundle method of Helmberg and Rendl can sol ..."
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Cited by 32 (2 self)
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Semidefinite relaxations of quadratic 01 programming or graph partitioning problems are well known to be of high quality. However, solving them by primaldual interior point methods can take much time even for problems of moderate size. The recent spectral bundle method of Helmberg and Rendl can solve quite efficiently large structured equalityconstrained semidefinite programs if the trace of the primal matrix variable is fixed, as happens in many applications. We extend the method so that it can handle inequality constraints without seriously increasing computation time. Encouraging preliminary computational results are reported.
Solving Nonlinear Multicommodity Flow Problems By The Analytic Center Cutting Plane Method
, 1995
"... The paper deals with nonlinear multicommodity flow problems with convex costs. A decomposition method is proposed to solve them. The approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear prog ..."
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Cited by 29 (14 self)
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The paper deals with nonlinear multicommodity flow problems with convex costs. A decomposition method is proposed to solve them. The approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems. Each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network. It is thus formulated as a shortest path problem and solved with the Dijkstra's dheap algorithm. An implementation is described that that takes full advantage of the supersparsity of the network in the linear algebra operations. Computational results show the efficiency of this approach on wellknown nondifferentiable problems and also large scale randomly generated problems (up to 1000 arcs and 5000 commodities). This research has been supported by the Fonds National de la Recherche Scientifique Suisse, grant #12 \Gamma 34002:92, NSERCCanada and ...