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Representing quadratically constrained quadratic programs as generalized copositive programs

OPERATIONS RESEARCH LETTERS 2012年第3期40卷 203-206页

作者： Burer, Samuel Dong, Hongbo Univ Wisconsin Wisconsin Inst Discovery Madison WI 53715 USA Univ Iowa Dept Management Sci Iowa City IA 52242 USA

We show that any (nonconvex) quadratically constrained quadratic program (QCQP) can be represented as a generalized copositive program. In fact, we provide two representations: one based on the concept of completely positive (CP) matrices over second-order cones, and one based on CP matrices over the positive semidefinite cone. (C) 2012 Elsevier B.V. All rights reserved.

关键词： conic programming Copositive programming Quadratically constrained quadratic programs

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Robust and low-order design of flexible AC transmission systems and power system stabilisers for oscillation damping

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IET GENERATION TRANSMISSION & DISTRIBUTION 2012年第5期6卷 445-452页

作者： Simfukwe, D. D. Pal, B. C. Jabr, R. A. Martins, N. Univ London Imperial Coll Sci Technol & Med Dept Elect & Elect Engn London SW7 2AZ England Amer Univ Beirut Dept Elect & Comp Engn Beirut 11072020 Lebanon Electrobras CEPEL BR-21941911 Rio De Janeiro RJ Brazil

The study presents a method for the coordinated design of low-order robust controllers for stabilising power system oscillations. The design uses conic programming to shift under-damped or unstable modes into a region of sufficient damping of the complex plane and involves two stages. The first stage is a phase compensation design that accounts for multiple operating conditions with flexible AC transmission systems (FACTS) and power system stabilisers (PSS), unlike our earlier approach involving PSS only. The second stage is gain tuning. This is done effectively in a coordinated way using conic programming. An example demonstrates the method's ability to design coordinated FACTS and PSS controllers resulting in damping oscillations over all given operating conditions of the power system with very simple and low-order control structure.

关键词： FACTS conic programming flexible AC transmission systems power system stabilisers gain tuning Robust and low-order design of low-order robust controllers a.c. transmission Power system control Control of electric power systems Power supply quality and harmonics power system stability power system harmonics Power transmission lines and cables oscillation damping

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POLYNOMIAL APPROXIMATIONS FOR CONTINUOUS LINEAR PROGRAMS

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SIAM JOURNAL ON OPTIMIZATION 2012年第2期22卷 628-648页

作者： Bampou, Dimitra Kuhn, Daniel Univ London Imperial Coll Sci Technol & Med Dept Comp London SW7 2BZ England

Continuous linear programs have attracted considerable interest due to their potential for modeling manufacturing, scheduling, and routing problems. While efficient simplex-type algorithms have been developed for separated continuous linear programs, crude time discretization remains the method of choice for solving general (nonseparated) problem instances. In this paper we propose a more generic approximation scheme for nonseparated continuous linear programs, where we approximate the functional decision variables (policies) by polynomial and piecewise polynomial decision rules. This restriction results in an upper bound on the original problem, which can be computed efficiently by solving a tractable semidefinite program. To estimate the approximation error, we also compute a lower bound by solving a dual continuous linear program in (piecewise) polynomial decision rules. We establish the convergence of the primal and dual approximations under Slater-type constraint qualifications. We also highlight the potential of our method for optimizing large-scale multiclass queueing systems and dynamic Leontief models.

关键词： continuous linear programming polynomial decision rules conic programming

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Full Nesterov-Todd step infeasible interior-point method for symmetric optimization

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2011年第3期214卷 473-484页

作者： Gu, G. Zangiabadi, M. Roos, C. Nanjing Univ Dept Math Nanjing Peoples R China Shahrekord Univ Dept Math Sci Shahrekord Iran Delft Univ Technol Fac Elect Engn Math & Comp Sci NL-2600 AA Delft Netherlands

Euclidean Jordan algebras were proved more than a decade ago to be an indispensable tool in the unified study of interior-point methods. By using it, we generalize the full-Newton step infeasible interior-point method for linear optimization of Roos [Roos, C., 2006. A full-Newton step O(n) infeasible interior-point algorithm for linear optimization. SIAM Journal on Optimization. 16(4), 1110-1136 (electronic)] to symmetric optimization. This unifies the analysis for linear, second-order cone and semidefinite optimizations. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Interior point methods conic programming Nesterov-Todd step

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FAST METHODS FOR COMPUTING THE p-RADIUS OF MATRICES

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2011年第3期33卷 1246-1266页

作者： Jungers, Raphael M. Protasov, Vladimir Y. MIT Informat & Decis Syst Lab Cambridge MA 02139 USA Moscow MV Lomonosov State Univ Dept Mech & Math Moscow 119992 Russia

The p-radius characterizes the average rate of growth of norms of matrices in a multiplicative semigroup. This quantity has found several applications in recent years. We raise the question of its computability. We prove that the complexity of its approximation increases exponentially with p. We then describe a series of approximations that converge to the p-radius with a priori computable accuracy. For nonnegative matrices, this gives efficient approximation schemes for the p-radius computation.

关键词： joint spectral radius p-radius approximation algorithm conic programming smoothness in L-p spaces

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SELF-CONCORDANCE AND DECOMPOSITION-BASED INTERIOR POINT METHODS FOR THE TWO-STAGE STOCHASTIC CONVEX OPTIMIZATION PROBLEM

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SIAM JOURNAL ON OPTIMIZATION 2011年第4期21卷 1667-1687页

作者： Chen, Michael Mehrotra, Sanjay York Univ Dept Math & Stat Toronto ON M3J 1P3 Canada York Univ Dept Math & Stat Toronto ON M3J 1P3 Canada Northwestern Univ Dept IE MS Evanston IL 60208 USA

We study the two-stage stochastic convex optimization problem whose first-and second-stage feasible regions admit a self-concordant barrier. We show that the barrier recourse functions and the composite barrier functions for this problem form self-concordant families. These results are used to develop prototype primal interior point decomposition algorithms that are more suitable for a heterogeneous distributed computing environment. We show that the worst case iteration complexity of the proposed algorithms is the same as that for the short-and long-step primal interior algorithms applied to the extensive formulation of this problem. The generality of our results allows the possibility of using barriers other than the standard log-barrier in an algorithmic framework.

关键词： stochastic programming convex programming conic programming Benders' decomposition interior point methods

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Numerical limit analysis bounds for ductile porous media with oblate voids

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MECHANICS RESEARCH COMMUNICATIONS 2011年第5期38卷 350-354页

作者： Pastor, F. Kondo, D. Pastor, J. UPMC CNRS UMR 7190 Inst DAlembert F-75252 Paris France CNRS UMR 8107 LML F-59655 Villeneuve Dascq France Univ Savoie POLYTECHAnnecy Chambery Lab LOCIE F-73376 Le Bourget Du Lac France

The paper is devoted to a numerical limit analysis of a hollow spheroidal model with a von Mises solid matrix. To this purpose, existing kinematic and static 3D-FEM codes for the case of spherical cavities have been modified and improved to account for the model of a spheroidal cavity confocal with the external spheroidal boundary. The optimized conic programming formulations and the resulting codes appear to be very efficient. This framework is then applied to the derivation of numerical upper and lower anisotropic bounds in the case of an oblate void. The numerical results obtained from a series of tests are presented and allow to assess the accuracy of closed-form expressions of the macroscopic criteria proposed by Gologanu et al. (1994, 1997) for porous media with oblate voids. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Gurson-type models Spheroidal voids Micromechanics Limit analysis Upper and lower bounds conic programming

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Constraint qualifications for optimality conditions and total Lagrange dualities in convex infinite programming

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2010年第5期73卷 1143-1159页

作者： Fang, D. H. Li, C. Ng, K. F. Jishou Univ Coll Math & Comp Sci Jishou 416000 Peoples R China Zhejiang Univ Dept Math Hangzhou 310027 Zhejiang Peoples R China King Saud Univ Coll Sci Dept Math Riyadh 11451 Saudi Arabia Chinese Univ Hong Kong Dept Math Hong Kong Hong Kong Peoples R China Chinese Univ Hong Kong IMS Hong Kong Hong Kong Peoples R China

For an inequality system defined by an infinite family of proper convex functions (not necessarily lower semicontinuous), we introduce some new notions of constraint qualifications. Under the new constraint qualifications, we provide necessary and/or sufficient conditions for the KKT rules to hold. Similarly, we provide characterizations for constrained minimization problems to have total Lagrangian dualities. Several known results in the conic programming problem are extended and improved. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： Convex inequality system Optimality condition Total Lagrangian duality conic programming

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conic mixed-integer rounding cuts

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MATHEMATICAL programming 2010年第1期122卷 1-20页

作者： Atamtuerk, Alper Narayanan, Vishnu Univ Calif Berkeley Dept Ind Engn & Operat Res Berkeley CA 94720 USA

A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined by second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral conic substructures of second-order conic sets. These cuts can be readily incorporated in branch-and-bound algorithms that solve either second-order conic programming or linear programming relaxations of conic integer programs at the nodes of the branch-and-bound tree. Central to our approach is a reformulation of the second-order conic constraints with polyhedral second-order conic constraints in a higher dimensional space. In this representation the cuts we develop are linear, even though they are nonlinear in the original space of variables. This feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed-integer programming. The reformulation also allows the use of polyhedral methods for conic integer programming. We report computational results on solving unstructured second-order conic mixed-integer problems as well as mean-variance capital budgeting problems and least-squares estimation problems with binary inputs. Our computational experiments show that conic mixed-integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed-integer programs and, hence, improving their solvability.

关键词： conic programming Integer programming Mixed-integer rounding Branch-and-cut algorithms

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A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2010年第3期207卷 1210-1220页

作者： Sun, Jie Zhang, Su Natl Univ Singapore Dept Decis Sci Singapore 119245 Singapore Natl Univ Singapore Risk Management Inst Singapore 119245 Singapore Nankai Univ Sch Business Inst Modern Management Tianjin Peoples R China

We propose a modified alternating direction method for solving convex quadratically constrained quadratic semidefinite optimization problems. The method is a first-order method, therefore requires much less computational effort per iteration than the second-order approaches such as the interior point methods or the smoothing Newton methods. In fact, only a single inexact metric projection onto the positive semidefinite cone is required at each iteration. We prove global convergence and provide numerical evidence to show the effectiveness of this method. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Alternating direction method conic programming Quadratic semidefinite optimization

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