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IMPROVED conic REFORMULATIONS FOR K-MEANS CLUSTERING

SIAM JOURNAL ON OPTIMIZATION 2018年第4期28卷 3105-3126页

作者： Prasad, Madhushini Narayana Hanasusanto, Grani A. Univ Texas Austin Grad Program Operat Res & Ind Engn Austin TX 78712 USA

In this paper, we show that the popular K-means clustering problem can equivalently be reformulated as a conic program of polynomial size. The arising convex optimization problem is NP-hard, but amenable to a tractable semidefinite programming (SDP) relaxation that is tighter than the current SDP relaxation schemes in the literature. In contrast to the existing schemes, our proposed SDP formulation gives rise to solutions that can be leveraged to identify the clusters. We devise a new approximation algorithm for K-means clustering that utilizes the improved formulation and empirically illustrate its superiority over the state-of-the-art solution schemes.

关键词： semidefinite programming K-means conic programming copositive programming

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Limit analysis, rammed earth material and Casagrande test

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COMPTES RENDUS MECANIQUE 2018年第2期346卷 98-109页

作者： El-Nabouch, Ranime Pastor, Joseph Quoc-Bao Bui Ple, Olivier Univ Savoie Mt Blanc CNRS LOCIE F-73000 Chambery France Ton Duc Thang Univ Fac Civil Engn Ho Chi Minh City Vietnam

The present paper is concerned with the simulation of the Casagrande test carried out on a rammed earth material for wall-type structures in the framework of Limit Analysis (LA). In a preliminary study, the material is considered as a homogeneous Coulomb material, and existing LA static and kinematic codes are used for the simulation of the test. In each loading case, static and kinematic bounds coincide;the corresponding exact solution is a two-rigid-block mechanism together with a quasi-constant stress vector and a velocity jump also constant along the interface, for the three loading cases. In a second study, to take into account the influence of compressive loadings related to the porosity of the material, an elliptic criterion (denoted Cohesive Cam-Clay, CCC) is defined based on recent homogenization results about the hollow sphere model for porous Coulomb materials. Finally, original finite element formulations of the static and mixed kinematic methods for the CCC material are developed and applied to the Casagrande test. The results are the same than above, except that this time the velocity jump depends on the compressive loading, which is more realistic but not satisfying fully the experimental observations. Therefore, the possible extensions of this work towards non-standard direct methods are analyzed in the conclusion section. (C) 2017 Academie des sciences. Published by Elsevier Masson SAS.

关键词： Limit analysis Lower and upper bound methods Cohesive Cam-Clay material Micromechanics identification Casagrande geotechnical test conic programming

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Primal interior-point decomposition algorithms for two-stage stochastic extended second-order cone programming

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OPTIMIZATION 2018年第12期67卷 2291-2323页

作者： Alzalg, Baha Univ Jordan Dept Math Amman 11942 Jordan

We study and solve the two-stage stochastic extended second-order cone programming problem. We show that the barrier recourse functions and the composite barrier functions for this optimization problem are self-concordant families with respect to barrier parameters. These results are used to develop primal decomposition-based interior-point algorithms. The worst case iteration complexity of the developed algorithms is shown to be the same as that for the short- and long-step primal interior algorithms applied to the extensive formulation of our problem.

关键词： Extended second order cone conic programming stochastic programming interior-point algorithms self-concordance

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The Equilibrium Cell-Based Smooth Finite Element Method for Shakedown Analysis of Structures

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INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS 2019年第5期16卷

作者： Ho, P. L. H. Le, C., V Chu, T. Q. Univ Tech Educ HCMC Fac Appl Mech & Civil Engn Ho Chi Minh City Vietnam Int Univ VNU HCM Dept Civil Engn Ho Chi Minh City Vietnam

This paper presents a novel equilibrium formulation, that uses the cell-based smoothed method and conic programming, for limit and shakedown analysis of structures. The virtual strains are computed using straining cell-based smoothing technique based on elements of discretizexl mesh. Fictitious elastic stresses are also determined within the framework of finite element method (CS-FEM)-based Galerkin procedure, and equilibrium equations for residual stresses are satisfied in an average sense at every cell-based smoothing cell. All constrains are imposed at only one point in the smoothing domains, instead of Gauss points as in a standard FEM-based procedure. The resulting optimization problem is then handled using the highly efficient solvers. Various numerical examples are investigated, and obtained solutions are compared with available results in the literature.

关键词： Limit analysis shakedown analysis smoothed finite element method CS-FEM conic programming

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Strong Stationarity for Optimization Problems with Complementarity Constraints in Absence of Polyhedricity With Applications to Optimization with Semidefinite and Second-Order-Cone Complementarity Constraints

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SET-VALUED AND VARIATIONAL ANALYSIS 2017年第1期25卷 133-175页

作者： Wachsmuth, Gerd Tech Univ Chemnitz Fac Math Professorship Numer Math Partial Differential Equ D-09107 Chemnitz Germany

We consider mathematical programs with complementarity constraints in Banach spaces. In particular, we focus on the situation that the complementarity constraint is defined by a non-polyhedric cone K. We demonstrate how strong stationarity conditions can be obtained in an abstract setting. These conditions and their verification can be made more precise in the case that Z is a Hilbert space and if the projection onto K is directionally differentiable with a derivative as given in Haraux (Journal of the Mathematical Society of Japan 29(4), 615-631, 1977, Theorem 1). Finally, we apply the theory to optimization problems with semidefinite and second-order-cone complementarity constraints. We obtain that local minimizers are strongly stationary under a variant of the linear-independence constraint qualification, and these are novel results.

关键词： Mathematical programs with complementarity constraints Strong stationarity conic programming Semidefinite cone Second-order cone Polyhedricity

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Mixed integer programming with a class of nonlinear convex constraints

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DISCRETE OPTIMIZATION 2017年 24卷 66-86页

作者： Vinel, Alexander Krokhmal, Pavlo A. Auburn Univ Dept Ind & Syst Engn Shelby Ctr 3301 Auburn AL 36849 USA Univ Arizona Dept Syst & Ind Engn 1127 E James E Rogers Way Tucson AZ 85721 USA

We study solution approaches to a class of mixed-integer nonlinear programming problems that arise from recent developments in risk-averse stochastic optimization and contain second- and p-order cone programming as special cases. We explore possible applications of some of the solution techniques that have been successfully used in mixed-integer conic programming and show how they can be generalized to the problems under consideration. Particularly, we consider a branch-and-bound method based on outer polyhedral approximations, lifted nonlinear cuts, and linear disjunctive cuts. Results of numerical experiments with discrete portfolio optimization models are presented. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Mixed-integer nonlinear programming Measures of risk Quasi-arithmetic average Valid inequalities conic programming

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A semi-smooth Newton method for projection equations and linear complementarity problems with respect to the second order cone

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LINEAR ALGEBRA AND ITS APPLICATIONS 2017年 513卷 160-181页

作者： Cruz, J. Y. Bello Ferreira, O. P. Nemeth, S. Z. Prudente, L. F. Northern Illinois Univ Dept Math Sci WH 366 De Kalb IL 60115 USA Univ Fed Goias IME Ave Esperanga S-NCampus Samambaia BR-74690900 Goiania Go Brazil Univ Birmingham Sch Math Watson Bldg Birmingham B15 2TT W Midlands England

In this paper a special semi-smooth equation associated to the second order cone is studied. It is shown that, under mild assumptions, the semi-smooth Newton method applied to this equation is well-defined and the generated sequence is globally and Q-linearly convergent to a solution. As an application, the obtained results are used to study the linear second order cone complementarity problem, with special emphasis on the particular case of positive definite matrices. Moreover, some computational experiments designed to investigate the practical viability of the method are presented. (C) 2016 Published by Elsevier Inc.

关键词： Semi-smooth system conic programming Second order cone Semi-smooth Newton method

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Real-time microgrid economic dispatch based on model predictive control strategy

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Journal of Modern Power Systems and Clean Energy 2017年第5期5卷 787-796页

作者： Yan DU Wei PEI Naishi CHEN Xianjun GE Hao XIAO Institute of Electrical Engineering Chinese Academy of SciencesBeijing 100190China University of Chinese Academy of Sciences Beijing 100049China China Electric Power Research Institute Beijing 100192China

To deal with uncertainties of renewable energy,demand and price signals in real-time microgrid operation,this paper proposes a model predictive control strategy for microgrid economic dispatch, where hourly schedule is constantly optimized according to the current system state and latest forecast information. Moreover, implicit network topology of the microgrid and corresponding power flow constraints are considered, which leads to a mixed integer nonlinear optimal power flow problem. Given the non-convexity feature of the original problem, the technique of conic programming is applied to efficiently crack the nut. Simulation results from a reconstructed IEEE-33 bus system and comparisons with the routine day-ahead microgrid schedule sufficiently substantiate the effectiveness of the proposed MPC strategy and the conic programming method.

关键词： conic programming Economic dispatch(ED) Microgrid Mixed-integer nonlinear programming(MINLP) Model predictive control(MPC) Optimal power flow(OPF)

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The Lyapunov rank of an improper cone

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OPTIMIZATION METHODS & SOFTWARE 2017年第1期32卷 109-125页

作者： Orlitzky, Michael Univ Maryland Baltimore Cty Dept Math & Stat Baltimore MD 21228 USA

Let K be a closed convex cone with dual We say that a linear transformation The dimension of the space of all such transformations is called the Lyapunov rank ofK. This number was introduced and studied by Rudolf etal.[Bilinear optimality constraints for the cone of positive polynomials, Math. Program., Ser. B 129 (2011), pp. 5-31] for proper cones because of its connection to conic programming and complementarity problems. The assumption that K is proper turns out to be nonessential. We first develop the basic theory for cones that are merely closed and convex. We then devise a way to compute the Lyapunov rank of any closed convex cone and show that the Lyapunov-like transformations on a closed convex cone are related to the Lie algebra of its automorphism group. Next, we extend some results for proper polyhedral cones. Finally, we devise algorithms to compute both the space of all Lyapunov-like transformations and the Lyapunov rank of a polyhedral closed convex cone.

关键词： Lyapunov rank Lyapunov-like transformation conic programming Lie algebra automorphism group

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Hollow sphere models, conic programming and third stress invariant

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EUROPEAN JOURNAL OF MECHANICS A-SOLIDS 2011年第2期30卷 63-71页

作者： Thore, Ph. Pastor, F. Pastor, J. Univ Savoie POLYTECHSavoie Lab LOCIE F-73376 Le Bourget Du Lac France Univ Lille 1 Ctr Hyperfrequences & Semicond CNRS LMLUMR 8107 F-59655 Villeneuve Dascq France

In this paper the hollow sphere model is investigated within the framework of limit analysis (LA), using the classical two-part velocity field, i.e., the exact solution for hydrostatic loading plus a linear solution, in both cases of von Mises and Drucker-Prager matrices. We use the kinematic LA approach in a quasi-analytical approach by imposing the plastic admissibility (PA) condition (Drucker-Prager matrix) or upper bounding the dissipated power (von Mises matrix) in a sufficiently high number of distributed points, thanks to conic programming formulations. Then we analyze the "porous Drucker-Prager" case to confirm that the so-called UBM (Upper Bound Model) approach of Guo et al. (2008) is only an estimate, although a good one in fact. Moreover, it is shown that the mean stress axis should not be a strict axis of symmetry for the macroscopic criterion. Then, considering the "Porous von Mises" case, we obtain that the real criterion is not only lower than the Gurson criterion, but probably non-symmetric with respect to the mean stress axis, more than in the Drucker-Prager case. We finally use ad hoc updated 3D-FEM LA codes to confirm the previous results and to evaluate the entire influence of the third stress invariant: the classical (Sigma(m), Sigma(eqv)) formulation of the Gurson criterion clearly overestimates the real, non-symmetric solution of the hollow sphere model, at least for porosities of the same order of magnitude as the value used in this work. (C) 2010 Elsevier Masson SAS. All rights reserved.

关键词： Gurson models "Porous Drucker-Prager" "Porous von Mises" Limit analysis Third stress invariant conic programming

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