The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Jeanlouis Goffin - One of the best experts on this subject based on the ideXlab platform.
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an interior point cutting Plane Method for the convex feasibility problem with second order cone inequalities
Mathematics of Operations Research, 2005Co-Authors: Mohammad R Oskoorouchi, Jeanlouis GoffinAbstract:The convex feasibility problem in general is a problem of finding a point in a convex set that contains a full dimensional ball and is contained in a compact convex set. We assume that the outer set is described by second-order cone inequalities and propose an analytic center cutting Plane technique to solve this problem. We discuss primal and dual settings simultaneously. Two complexity results are reported: the complexity of restoration procedure and complexity of the overall algorithm. We prove that an approximate analytic center is updated after adding a second-order cone cut (SOCC) inO(1) Newton step, and that the analytic center cutting Plane Method (ACCPM) with SOCC is a fully polynomial algorithm.
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the integration of an interior point cutting Plane Method within a branch and price algorithm
Mathematical Programming, 2004Co-Authors: Samir Elhedhli, Jeanlouis GoffinAbstract:This paper presents a novel integration of interior point cutting Plane Methods within branch-and-price algorithms. Unlike the classical Method, columns are generated at a ‘‘central’’ dual solution by applying the analytic centre cutting Plane Method (ACCPM) on the dual of the full master problem. First, we introduce some modifications to ACCPM. We propose a new procedure to recover primal feasibility after adding cuts and use, for the first time, a dual Newton’s Method to calculate the new analytic centre after branching. Second, we discuss the integration of ACCPM within the branch-and-price algorithm. We detail the use of ACCPM as the search goes deep in the branch and bound tree, making full utilization of past information as a warm start. We exploit dual information from ACCPM to generate incumbent feasible solutions and to guide branching. Finally, the overall approach is implemented and tested for the bin-packing problem and the capacitated facility location problem with single sourcing. We compare against Cplex-MIP 7.5 as well as a classical branch-and-price algorithm.
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solving variational inequalities with a quadratic cut Method a primal dual jacobian free approach
Les Cahiers du GERAD, 2002Co-Authors: Michel Denault, Jeanlouis GoffinAbstract:We extend in two directions the Analytic Center, Cutting Plane Method for Variational Inequalities with quadratic cuts, ACCPM-VI(quadratic cuts), introduced by Denault and Goffin in 1998. First, we define a primal-dual Method to find the analytic center at each iteration. Second, the Broyden-Fletcher-Goldfarb-Shanno Jacobian approximation, of quasi-Newton fame, is used in the definition of the cuts, making the algorithm applicable to problems without tractable Jacobians. The algorithm is tested on a variety of variational inequality problems, including one challenging problem of pricing the pollution permits put forward in the Kyoto Protocol.
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convex nondifferentiable optimization a survey focused on the analytic center cutting Plane Method
Optimization Methods & Software, 2002Co-Authors: Jeanlouis Goffin, Jeanphilippe VialAbstract:We present a survey of nondifferentiable optimization problems and Methods with special focus on the analytic center cutting Plane Method. We propose a self-contained convergence analysis that uses the formalism of the theory of self-concordant functions, but for the main results, we give direct proofs based on the properties of the logarithmic function. We also provide an in-depth analysis of two extensions that are very relevant to practical problems: the case of multiple cuts and the case of deep cuts. We further examine extensions to problems including feasible sets partially described by an explicit barrier function, and to the case of nonlinear cuts. Finally, we review several implementation issues and discuss some applications.
Daniel D Lee - One of the best experts on this subject based on the ideXlab platform.
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learning data manifolds with a cutting Plane Method
Neural Computation, 2018Co-Authors: Sueyeon Chung, Uri Cohen, Haim Sompolinsky, Daniel D LeeAbstract:We consider the problem of classifying data manifolds where each manifold represents invariances that are parameterized by continuous degrees of freedom. Conventional data augmentation Methods rely...
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learning data manifolds with a cutting Plane Method
arXiv: Learning, 2017Co-Authors: Sueyeon Chung, Uri Cohen, Haim Sompolinsky, Daniel D LeeAbstract:We consider the problem of classifying data manifolds where each manifold represents invariances that are parameterized by continuous degrees of freedom. Conventional data augmentation Methods rely upon sampling large numbers of training examples from these manifolds; instead, we propose an iterative algorithm called M_{CP} based upon a cutting-Plane approach that efficiently solves a quadratic semi-infinite programming problem to find the maximum margin solution. We provide a proof of convergence as well as a polynomial bound on the number of iterations required for a desired tolerance in the objective function. The efficiency and performance of M_{CP} are demonstrated in high-dimensional simulations and on image manifolds generated from the ImageNet dataset. Our results indicate that M_{CP} is able to rapidly learn good classifiers and shows superior generalization performance compared with conventional maximum margin Methods using data augmentation Methods.
Jeanphilippe Vial - One of the best experts on this subject based on the ideXlab platform.
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accpm with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems
Mathematical Programming, 2009Co-Authors: Frederic Louis Francois Babonneau, Jeanphilippe VialAbstract:This paper proposes an implementation of a constrained analytic center cutting Plane Method to solve nonlinear multicommodity flow problems. The new approach exploits the property that the objective of the Lagrangian dual problem has a smooth component with second order derivatives readily available in closed form. The cutting Planes issued from the nonsmooth component and the epigraph set of the smooth component form a localization set that is endowed with a self-concordant augmented barrier. Our implementation uses an approximate analytic center associated with that barrier to query the oracle of the nonsmooth component. The paper also proposes an approximation scheme for the original objective. An active set strategy can be applied to the transformed problem: it reduces the dimension of the dual space and accelerates computations. The new approach solves huge instances with high accuracy. The Method is compared to alternative approaches proposed in the literature.
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an efficient Method to compute traffic assignment problems with elastic demands
Transportation Science, 2008Co-Authors: Frederic Louis Francois Babonneau, Jeanphilippe VialAbstract:The traffic assignment problem (TAP) with elastic demands can be formulated as an optimization problem, the objective of which is the sum of a congestion function and a disutility function. We propose to use a variant of the analytic center cutting Plane Method to solve this problem. We test the Method on TAP instances with the Bureau of Public Roads congestion function and different demand functions (constant elasticity and linear). The results of the numerical experiments show that it is possible to solve large instances with high accuracy.
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convex nondifferentiable optimization a survey focused on the analytic center cutting Plane Method
Optimization Methods & Software, 2002Co-Authors: Jeanlouis Goffin, Jeanphilippe VialAbstract:We present a survey of nondifferentiable optimization problems and Methods with special focus on the analytic center cutting Plane Method. We propose a self-contained convergence analysis that uses the formalism of the theory of self-concordant functions, but for the main results, we give direct proofs based on the properties of the logarithmic function. We also provide an in-depth analysis of two extensions that are very relevant to practical problems: the case of multiple cuts and the case of deep cuts. We further examine extensions to problems including feasible sets partially described by an explicit barrier function, and to the case of nonlinear cuts. Finally, we review several implementation issues and discuss some applications.
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proximal accpm a cutting Plane Method for column generation and lagrangian relaxation application to the p median problem
2002Co-Authors: Olivier Du Merle, Jeanphilippe VialAbstract:Proximal ACCPM is a variant of the analytic center cutting Plane Method, in which a proximal term is added to the barrier function that defines the center. The present paper gives a detailed presentation of the Method and of its implementation. Proximal ACCPM is used to solve the Lagrangian relaxation of the p-median problem on two sets of problem instances. Problems of the same collection are tentatively solved with the classical column generation scheme.
Sam Chiuwai Wong - One of the best experts on this subject based on the ideXlab platform.
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an improved cutting Plane Method for convex optimization convex concave games and its applications
arXiv: Data Structures and Algorithms, 2020Co-Authors: Haotia Jiang, Yin Ta Lee, Zhao Song, Sam Chiuwai WongAbstract:Given a separation oracle for a convex set $K \subset \mathbb{R}^n$ that is contained in a box of radius $R$, the goal is to either compute a point in $K$ or prove that $K$ does not contain a ball of radius $\epsilon$. We propose a new cutting Plane algorithm that uses an optimal $O(n \log (\kappa))$ evaluations of the oracle and an additional $O(n^2)$ time per evaluation, where $\kappa = nR/\epsilon$. $\bullet$ This improves upon Vaidya's $O( \text{SO} \cdot n \log (\kappa) + n^{\omega+1} \log (\kappa))$ time algorithm [Vaidya, FOCS 1989a] in terms of polynomial dependence on $n$, where $\omega < 2.373$ is the exponent of matrix multiplication and $\text{SO}$ is the time for oracle evaluation. $\bullet$ This improves upon Lee-Sidford-Wong's $O( \text{SO} \cdot n \log (\kappa) + n^3 \log^{O(1)} (\kappa))$ time algorithm [Lee, Sidford and Wong, FOCS 2015] in terms of dependence on $\kappa$. For many important applications in economics, $\kappa = \Omega(\exp(n))$ and this leads to a significant difference between $\log(\kappa)$ and $\mathrm{poly}(\log (\kappa))$. We also provide evidence that the $n^2$ time per evaluation cannot be improved and thus our running time is optimal. A bottleneck of previous cutting Plane Methods is to compute leverage scores, a measure of the relative importance of past constraints. Our result is achieved by a novel multi-layered data structure for leverage score maintenance, which is a sophisticated combination of diverse techniques such as random projection, batched low-rank update, inverse maintenance, polynomial interpolation, and fast rectangular matrix multiplication. Interestingly, our Method requires a combination of different fast rectangular matrix multiplication algorithms.
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a faster cutting Plane Method and its implications for combinatorial and convex optimization
Foundations of Computer Science, 2015Co-Authors: Yin Ta Lee, Aaro Sidford, Sam Chiuwai WongAbstract:In this paper we improve upon the running time for finding a point in a convex set given a separation oracle. In particular, given a separation oracle for a convex set K a#x2282; Rn that is contained in a box of radius R we show how to either compute a point in K or prove that K does not contain a ball of radius a#x03B5; using an expected O(n log(nR=a#x03B5;)) evaluations of the oracle and additional time O(n3 logO(1)(nR=a#x03B5;)). This matches the oracle complexity and improves upon the O(na#x03C9;+1 log(nR=a#x03B5;)) additional time of the previous fastest algorithm achieved over 25 years ago by Vaidya [91] for the current value of the matrix multiplication constant a#x03C9; < 2:373 [98], [36] when R=a#x03B5; = O(poly(n)). Using a mix of standard reductions and new techniques we show how our algorithm can be used to improve the running time for solving classic problems in continuous and combinatorial optimization. In particular we provide the following running time improvements: a#x03B5; Sub modular Function Minimization: n is the size of the ground set, M is the maximum absolute value of function values and EO is the time for function evaluation. Our weakly and strongly polynomial time algorithms have a running time of O(n2 lognM EO+n3 logO(1) nM) and O(n3 log2 n EO+n4 logO(1) n), improving upon the previous best of O((n4 · EO+n5) logM) and O(n5 · EO + n6) respectively. a#x03B5; Sub modular Flow: n = |V|, m = |E|, C is the maximum edge cost in absolute value and U is maximum edge capacity in absolute value. We obtain a faster weakly polynomial running time of O(n2 log nCU · EO + n3 logO(1) nCU), improving upon the previous best of O(mn5 log nU · EO) and O(n4h min {log C, log U}) from 15 years ago by a factor of O(n4). We also achieve faster strongly polynomial time algorithms as a consequence of our result on sub modular minimization. a#x03B5; Matroid Intersection: n is the size of the ground set, r is the maximum size of independent sets, M is the maximum absolute value of element weight, Trank and Tind are the time for each rank and independence oracle query. We obtain a running time of O((nr log2 nTrank+n3 logO(1) n) lognM) and O((n2 log nTind+n3 logO(1) n) lognM), achieving the first quadratic bound on the query complexity for the independence and rank oracles. In the unweighted case, this is the first improvement since 1986 for independence oracle. a#x03B5; Semi definite Programming: n is the number of constraints, m is the number of dimensions and S is the total number of non-zeros in the constraint matrices. We obtain a running time of O(n(n2 +ma#x03C9; +S)), improving upon the previous best of O(n(na#x03C9; +ma#x03C9; +S)) for the regime S is small.
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a faster cutting Plane Method and its implications for combinatorial and convex optimization
arXiv: Data Structures and Algorithms, 2015Co-Authors: Yin Ta Lee, Aaro Sidford, Sam Chiuwai WongAbstract:We improve upon the running time for finding a point in a convex set given a separation oracle. In particular, given a separation oracle for a convex set $K\subset \mathbb{R}^n$ contained in a box of radius $R$, we show how to either find a point in $K$ or prove that $K$ does not contain a ball of radius $\epsilon$ using an expected $O(n\log(nR/\epsilon))$ oracle evaluations and additional time $O(n^3\log^{O(1)}(nR/\epsilon))$. This matches the oracle complexity and improves upon the $O(n^{\omega+1}\log(nR/\epsilon))$ additional time of the previous fastest algorithm achieved over 25 years ago by Vaidya for the current matrix multiplication constant $\omega<2.373$ when $R/\epsilon=n^{O(1)}$. Using a mix of standard reductions and new techniques, our algorithm yields improved runtimes for solving classic problems in continuous and combinatorial optimization: Submodular Minimization: Our weakly and strongly polynomial time algorithms have runtimes of $O(n^2\log nM\cdot\text{EO}+n^3\log^{O(1)}nM)$ and $O(n^3\log^2 n\cdot\text{EO}+n^4\log^{O(1)}n)$, improving upon the previous best of $O((n^4\text{EO}+n^5)\log M)$ and $O(n^5\text{EO}+n^6)$. Matroid Intersection: Our runtimes are $O(nrT_{\text{rank}}\log n\log (nM) +n^3\log^{O(1)}(nM))$ and $O(n^2\log (nM) T_{\text{ind}}+n^3 \log^{O(1)} (nM))$, achieving the first quadratic bound on the query complexity for the independence and rank oracles. In the unweighted case, this is the first improvement since 1986 for independence oracle. Submodular Flow: Our runtime is $O(n^2\log nCU\cdot\text{EO}+n^3\log^{O(1)}nCU)$, improving upon the previous bests from 15 years ago roughly by a factor of $O(n^4)$. Semidefinite Programming: Our runtime is $\tilde{O}(n(n^2+m^{\omega}+S))$, improving upon the previous best of $\tilde{O}(n(n^{\omega}+m^{\omega}+S))$ for the regime where the number of nonzeros $S$ is small.
Sueyeon Chung - One of the best experts on this subject based on the ideXlab platform.
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learning data manifolds with a cutting Plane Method
Neural Computation, 2018Co-Authors: Sueyeon Chung, Uri Cohen, Haim Sompolinsky, Daniel D LeeAbstract:We consider the problem of classifying data manifolds where each manifold represents invariances that are parameterized by continuous degrees of freedom. Conventional data augmentation Methods rely...
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learning data manifolds with a cutting Plane Method
arXiv: Learning, 2017Co-Authors: Sueyeon Chung, Uri Cohen, Haim Sompolinsky, Daniel D LeeAbstract:We consider the problem of classifying data manifolds where each manifold represents invariances that are parameterized by continuous degrees of freedom. Conventional data augmentation Methods rely upon sampling large numbers of training examples from these manifolds; instead, we propose an iterative algorithm called M_{CP} based upon a cutting-Plane approach that efficiently solves a quadratic semi-infinite programming problem to find the maximum margin solution. We provide a proof of convergence as well as a polynomial bound on the number of iterations required for a desired tolerance in the objective function. The efficiency and performance of M_{CP} are demonstrated in high-dimensional simulations and on image manifolds generated from the ImageNet dataset. Our results indicate that M_{CP} is able to rapidly learn good classifiers and shows superior generalization performance compared with conventional maximum margin Methods using data augmentation Methods.