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第40届中国控制会议

作者： Gang Wu Shuangshan Lv Wanqiang Wu School of Information & Engineering Shenyang University

In wireless sensor networks, the problem of sensor node location is a very important indicator. The accuracy of localization depends on the elimination of errors. Based on Euclidean distance, an optimization algorithm for semidefinite programming(SDP) is proposed in this paper. Combining the effective anchor node position selection and ratio range setting, the SDP algorithm was used to relax the non-convex constraints, and effectively reduced the impact of errors and get the actual position of measurement nodes. This algorithm effectively solved the problem of inaccurate estimation of nodes outside the convex hull of the anchor node. The numerical results show that the proposed SDP algorithm achieves high accuracy in the location estimation of unknown nodes.

关键词： semi-definite programming localization sensor node

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SDP-based Benders decomposition for solving p-median quadratic facility location problems

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COMPUTERS & OPERATIONS RESEARCH 2025年 182卷

作者： Yang, Yingying Bui, Hoa T. Loxton, Ryan Curtin Univ ARC Ctr Transforming Maintenance Data Sci Perth Australia Curtin Univ Curtin Ctr Optimisat & Decis Sci Perth Australia

The p-median facility location problem involves selecting the locations for p facilities from a set of potential locations, to balance the trade-off between facility establishment costs and distance to end users. In this paper, we introduce an extension, named the p-median quadratic facility location problem, that also considers inter-facility distance-important in applications where the facilities serve as intermediate hubs linking sites in different regions. This creates a quadratic term in the objective that makes the problem nonlinear. We develop a Benders decomposition method that uses a tight semi-definite programming relaxation of the Benders master problem, instead of solving the nonlinear master problem directly. We incorporate this decomposition strategy into a branch and bound framework to ensure convergence. Our numerical results show that this method outperforms the linear reformulation and classical Benders decomposition techniques, working independently or in tandem.

关键词： Facility location problem Binary quadratic programming Benders decomposition semi-definite programming

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semidefinite APPROXIMATIONS OF REACHABLE SETS FOR DISCRETE-TIME POLYNOMIAL SYSTEMS

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2019年第4期57卷 2799-2820页

作者： Magron, Victor Garoche, Pierre-Loic Henrion, Didier Thirioux, Xavier LAAS CNRS F-31400 Toulouse France Off Natl Etud & Rech Aerosp F-31000 Toulouse France Univ Toulouse LAAS F-31400 Toulouse France IRIT F-31071 Toulouse France

We consider the problem of approximating the reachable set of a discrete-time polynomial system from a semialgebraic set of initial conditions under general semialgebraic set constraints. Assuming inclusion in a given simple set like a box or an ellipsoid, we provide a method to compute certified outer approximations of the reachable set. The proposed method consists of building a hierarchy of relaxations for an infinite-dimensional moment problem. Under certain assumptions, the optimal value of this problem is the volume of the reachable set and the optimum solution is the restriction of the Lebesgue measure on this set. Then, one can outer approximate the reachable set as closely as desired with a hierarchy of super level sets of increasing degree polynomials. For each fixed degree, finding the coefficients of the polynomial boils down to computing the optimal solution of a convex semidefinite program. When the degree of the polynomial approximation tends to infinity, we provide strong convergence guarantees of the super level sets to the reachable set. We also present some application examples together with numerical results.

关键词： reachable set discrete-time polynomial systems polynomial optimization semi-definite programming moment relaxations sums of squares

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State Feedback Controller Design for General Polynomial-Fuzzy-Model-Based Systems with Time-Varying Delay

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INTERNATIONAL JOURNAL OF FUZZY SYSTEMS 2019年第2期21卷 583-591页

作者： Ye, Dan Li, Xiehuan Northeastern Univ Coll Informat Sci & Engn State Key Lab Synthet Automat Proc Ind Shenyang 110189 Liaoning Peoples R China Northeastern Univ Coll Informat Sci & Engn Shenyang 110189 Liaoning Peoples R China

This paper mainly investigates the state feedback controller design and the stability analysis for the general polynomial-fuzzy-model-based systems with time-varying delay. A Lyapunov-Krasovskii (L-K) function with the delay information in integral parts is used in the stability analysis process, and the time-delay terms are solved by an introduced lemma. Here, the assumption that the input matrices and the delay matrices have zero rows at the same time is not needed. A novel square of sum (SOS) optimization method successfully solves the non-convex problem by transforming the nonlinear terms and the time-varying delay matrix items into indices, which are optimized to zero by a semi-definite programming. The resultant SOS conditions are delay dependent, and more flexible stability conditions are obtained by introducing the slack matrices. Numerical simulation proves the effectiveness of the method.

关键词： semi-definite programming Polynomial-fuzzy-model-based systems Slack matrix Time-varying delay Lyapunov-Krasovskii function

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A new approximation hierarchy for polynomial conic optimization

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2019年第1期73卷 37-67页

作者： Dickinson, Peter J. C. Povh, Janez Univ Groningen Groningen Netherlands Univ Vienna Vienna Austria Univ Twente Dept Appl Math POB 217 NL-7500 AE Enschede Netherlands Univ Ljubljana Fac Mech Engn Askerceva Ulica 6 Ljubljana 1000 Slovenia

In this paper we consider polynomial conic optimization problems, where the feasible set is defined by constraints in the form of given polynomial vectors belonging to given nonempty closed convex cones, and we assume that all the feasible solutions are non-negative. This family of problems captures in particular polynomial optimization problems (POPs), polynomial semi-definite polynomial optimization problems (PSDPs) and polynomial second-order cone-optimization problems (PSOCPs). We propose a new general hierarchy of linear conic optimization relaxations inspired by an extension of Polya's Positivstellensatz for homogeneous polynomials being positive over a basic semi-algebraic cone contained in the non-negative orthant, introduced in Dickinson and Povh (J Glob Optim 61(4):615-625, 2015). We prove that based on some classic assumptions, these relaxations converge monotonically to the optimal value of the original problem. Adding a redundant polynomial positive semi-definite constraint to the original problem drastically improves the bounds produced by our method. We provide an extensive list of numerical examples that clearly indicate the advantages and disadvantages of our hierarchy. In particular, in comparison to the classic approach of sum-of-squares, our new method provides reasonable bounds on the optimal value for POPs, and strong bounds for PSDPs and PSOCPs, even outperforming the sum-of-squares approach in these latter two cases.

关键词： Polynomial conic optimization Polynomial semi-definite programming Polynomial second-order cone programming Approximation hierarchy Linear programming semi-definite programming

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Efficient two-dimensional line spectrum estimation based on decoupled atomic norm minimization

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SIGNAL PROCESSING 2019年 163卷 95-106页

作者： Zhang, Zhe Wang, Yue Tian, Zhi George Mason Univ Elect & Comp Engn Dept 4400 Univ Dr Fairfax VA 22030 USA

This paper presents an efficient optimization technique for gridless 2-D line spectrum estimation, named decoupled atomic norm minimization (D-ANM). The framework of atomic norm minimization (ANM) is considered, which has been successfully applied in 1-D problems to allow super-resolution frequency estimation for correlated sources even when the number of snapshots is highly limited. The state-of-the-art 2-D ANM approach vectorizes the 2-D measurements to their 1-D equivalence, which incurs huge computational cost and may become too costly for practical applications. We develop a novel decoupled approach of 2-D ANM via semi-definite programming (SDP), which introduces a new matrix-form atom set to naturally decouple the joint observations in both dimensions without loss of optimality. Accordingly, the original large-scale 2-D problem is equivalently reformulated via two decoupled one-level Toeplitz matrices, which can be solved by simple 1-D frequency estimation with pairing. Compared with the conventional vectorized approach, the proposed D-ANM technique reduces the computational complexity by several orders of magnitude with respect to the problem size, at no loss of optimality. It also retains the benefits of ANM in terms of precise signal recovery, small number of required measurements, and robustness to source correlation. The complexity benefits are particularly attractive for large-scale antenna systems such as massive MIMO, radar signal processing and radio astronomy. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Two-dimensional Line spectrum estimation Atomic norm minimization semi-definite programming Decoupled ANM

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NEW UPPER BOUNDS FOR EQUIANGULAR LINES BY PILLAR DECOMPOSITION

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SIAM JOURNAL ON DISCRETE MATHEMATICS 2019年第4期33卷 2479-2508页

作者： King, Emily J. Tang, Xiaoxian Colorado State Univ Dept Math Ft Collins CO 80523 USA Univ Bremen Fac Math Comp Sci 3 D-28359 Bremen Germany Beihang Univ Sch Math Sci Beijing 100083 Peoples R China Texas A&M Univ Dept Math College Stn TX 77843 USA

We derive a procedure for computing an upper bound on the number of equiangular lines in various Euclidean vector spaces by generalizing the classical pillar decomposition developed by Lemmens and Seidel;namely, we use linear algebra and combinatorial arguments to bound the number of vectors within an equiangular set which have inner products of certain signs with a negative clique. After projection and rescaling, such sets are also certain spherical two-distance sets, and semidefinite programming techniques may be used to bound the size of the set. Applying our method, we prove new relative bounds for the angle arccos(1/5). Experiments show that our relative bounds for all possible angles are considerably smaller than the known semidefinite programming bounds for a range of larger dimensions. Our computational results also show an explicit bound on the size of a set of equiangular lines in R-r regardless of angle, which is strictly less than the well-known Gerzon's bound if r + 2 is not a square of an odd number.

关键词： equiangular lines pillar decomposition Lemmens and Seidel's conjecture semi-definite programming spherical two-distance sets

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CVX-based algorithms for constructing various optimal regression designs

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CANADIAN JOURNAL OF STATISTICS-REVUE CANADIENNE DE STATISTIQUE 2019年第3期47卷 374-391页

作者： Wong, Weng Kee Zhou, Julie Univ Calif Los Angeles Dept Biostat Los Angeles CA 90095 USA Univ Victoria Dept Math & Stat Victoria BC V8W 2Y2 Canada

CVX-based numerical algorithms are widely and freely available for solving convex optimization problems but their applications to solve optimal design problems are limited. Using the CVX programs in MATLAB, we demonstrate their utility and flexibility over traditional algorithms in statistics for finding different types of optimal approximate designs under a convex criterion for nonlinear models. They are generally fast and easy to implement for any model and any convex optimality criterion. We derive theoretical properties of the algorithms and use them to generate new A-, c-, D- and E-optimal designs for various nonlinear models, including multi-stage and multi-objective optimal designs. We report properties of the optimal designs and provide sample CVX program codes for some of our examples that users can amend to find tailored optimal designs for their problems. The Canadian Journal of Statistics 47: 374-391;2019 (c) 2019 Statistical Society of Canada

关键词： Convex optimization dose-response model equivalence theorem multi-objective design multi-stage design semi-definite programming

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Rational Optimization for Nonlinear Reconstruction With Approximate l0 Penalization

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2019年第6期67卷 1407-1417页

作者： Castella, Marc Pesquet, Jean-Christophe Marmin, Arthur Univ Paris Saclay CNRS Telecom SudParis SAMOVAR F-91011 Evry France INRIA Ctr Visual Comp Cent Supelec F-91192 Gif Sur Yvette France Univ Paris Saclay F-91192 Gif Sur Yvette France

Recovering nonlinearly degraded signal in the presence of noise is a challenging problem. In this paper, this problem is tackled by minimizing the sum of a nonconvex least-squares fit criterion and a penalty term. We assume that the nonlinearity of the model can be accounted for by a rational function. In addition, we suppose that the signal to be sought is sparse and a rational approximation of the l(0) pseudonorm thus constitutes a suitable penalization. The resulting composite cost function belongs to the broad class of semialgebraic functions. To find a globally optimal solution to such an optimization problem, it can be transformed into a generalized moment problem, for which a hierarchy of semidefinite programming relaxations can be built. Global optimality comes at the expense of an increased dimension and to overcome computational limitations concerning the number of involved variables, the structure of the problem has to be carefully addressed. A situation of practical interest is when the nonlinear model consists of a convolutive transform followed by a componentwise nonlinear rational saturation. We then propose to use a sparse relaxation able to deal with up to several hundreds of optimized variables. In contrast with the naive approach consisting of linearizing the model, our experiments show that the proposed approach offers good performance.

关键词： Signal reconstruction sparse signal nonlinear model polynomial optimization semi-definite programming

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Dynamic network data envelopment analysis based upon technology changes

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INFOR 2019年第2期57卷 242-259页

作者： Zhang, Linyan Nanjing Audit Univ Coll Auditing & Evaluat Nanjing 211815 Jiangsu Peoples R China

The existing dynamic models assume the technology is unchanged in which the same factor should have the same multiplier, no matter which process it is associated with. The internal network structures embedded in a multi-period system are ignored in the literature. The current paper considers that the technology is changed in the dynamic system The same factor may have different multipliers in different periods, except for the variables of intermediate measures connecting two stages in one period and flows connecting two consecutive periods. An additive aggregation dynamic network data envelopment analysis is developed to measure the multi-period systems with a two-stage process embedded in each period. The system efficiency, overall efficiency and stage efficiencies of each period can be derived, and the relationship between the system efficiency and period efficiencies can be identified. The newly developed dynamic network model is nonlinear, and can be transformed to a semi-definite programming problem. A case of high-tech industry in China is illustrated to the approach.

关键词： Data envelopment analysis network dynamics two-stage semi-definite programming

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