Second-order Cone Programming Problem Research Articles

Peak reduction in OFDM using second-order cone programming relaxation

In this paper, we address the problem of peak-to-average power ratio (PAPR) reduction in orthogonal frequency-division multiplexing (OFDM) systems. We formulate the problem as an error vector magnitude (EVM) optimization task with constraints on PAPR and free carrier power overhead (FCPO). This problem, which is known to be NP hard, is shown to be approximated by a second-order cone programming (SOCP) problem using a sequential convex programming approach, making it much easier to handle. This approach can be extended to the more general problem when PAPR, EVM, and FCPO are constrained simultaneously. Our performance results show the effectiveness of the proposed approach, which allows good performance with lower computational complexity and infeasibility rate than state-of-the-art PAPR-reduction convex approaches. Moreover, in the case when all the three system parameters are constrained simultaneously, the proposed approach outperforms the convex approaches in terms of infeasibility rate, PAPR, and bit error rate (BER) performances.

Comments on: Support Vector Machines Maximizing Geometric Margins for Multi-class Classification

The article deals with multi-class discrimination with support vector machines (SVMs). The authors present multi-class SVMs (MSVMs) which they have introduced in recent years: multiobjective MSVMs (MMSVMs). Those machines are based on the same functional class as that of the standard MSVMs (Guermeur 2012). They differ in the nature of the learning problem, which is no longer a standard optimization problem (convex quadratic programming problem), but a multiobjective optimization problem (taking the form of a second-order cone programming problem). The aim is to maximize exactly all geometric margins, so as to improve generalization performance. This performance is assessed empirically, through experiments performed on data sets from the UCI benchmark repository. In our comments, we make use of the latest results of the statistical theory of large margin multi-category classifiers to study the connection between the (width of the) geometric margins and the generalization performance.

Comments on: Support vector machines maximizing geometric margins for multi-class classification

The paper provides an overview of the multi-objective multi-class support vector machine (MMSVM) based on a series of research articles written by the authors and their collaborators. As an extension of the binary SVM, the MMSVM takes an alltogether approach to classification with multiple categories. Along with the description of the original multi-objective optimization problem for the method, the paper lays out computational strategies for its implementation and further approximation of the solutions to the initial problem via alternative single-objective second-order cone programming problems. The progression from the initial problem to various alternative formulations is methodically presented with summary of the relation between the two sets of corresponding solutions.

Support vector machines maximizing geometric margins for multi-class classification

Machine learning is a very interesting and important branch of artificial intelligence. Among many learning models, the support vector machine is a popular model with high classification ability which can be trained by mathematical programming methods. Since the model was originally formulated for binary classification, various kinds of extensions have been investigated for multi-class classification. In this paper, we review some existing models, and introduce new models which we recently proposed. The models are derived from the viewpoint of multi-objective maximization of geometric margins for a discriminant function, and each model can be trained by solving a second-order cone programming problem. We show that discriminant functions with high generalization ability can be obtained by these models through some numerical experiments.

Seismic stability of earth-rock dams using finite element limit analysis

In this study, a finite element limit analysis method is developed to assess the seismic stability of earth-rock dams. A pseudo-static approach is employed within the limit analysis framework to determine the lower and upper bounds on the critical seismic coefficients of dams. The interlocking force in the soil is considered, and the rockfill material is assumed to follow the Mohr–Coulomb failure criterion and an associated flow rule. Based on the native form of the failure criterion, the lower and upper bound theorems are formulated as second-order cone programming problems. The nonlinear shear strength properties of rockfill materials are also considered. The developed finite element limit analysis is applied to two different types of earth-rock dams. The results indicate that the rigorous lower and upper bounds are very close even for rockfill materials with large internal friction angles. The failure surfaces are easily predicted using the contour of the yield function and the displacement field obtained by the limit analysis method. In addition, the pore water pressures are modelled as external forces in the limit analysis to assess the seismic stability of earth-rock dams in the reservoir filling stage.

Systematic Method for Designing Constellations for Intensity-Modulated Optical Systems

We consider the design of compact multisubcarrier constellations for intensity-modulated direct-detected optical systems. The constellations are designed to minimize the average electrical, average optical, and peak optical power for a given minimum distance between constellation points. We formulate the constellation design as a nonconvex optimization problem with second-order cone constraints, (nonconvex) quadratic constraints, and a convex objective function. We show that this problem can be relaxed to a (convex) second-order cone programming (SOCP) problem. We introduce a simple iterative method in which the SOCP relaxation is improved in each iteration. Several numerical simulation examples are provided to illustrate the effectiveness of our method. For the single-subcarrier case, the new constellations are compared with the best known formats in terms of power and spectral efficiency. Our new constellations outperform the corresponding face-centered cubic lattice and quadrature-amplitude-modulation-based constellations, with average electrical and optical power gains in the vicinity of 0.5 dB, for low symbol error rates. The corresponding peak optical power gains are also in the vicinity of 0.5 dB. By studying the mutual information inherent to the new constellations, we show that the potentials are still valid for coded systems. For the two-subcarrier case, we still outperform two-subcarrier schemes based on conventional constellations and optimized single-subcarrier constellations with the same dimensions.

A homotopy method for nonlinear second-order cone programming

This paper is concerned with a homotopy method for solving nonlinear second-order cone programming problems. The method extends to this setting a combined homotopy interior point method, recently introduced for solving nonlinear programming problems. Global convergence of a smooth curve determined by constructed homotopy is proven under mild conditions. Some numerical results are reported and show that the considered algorithm is applicable and efficient.

SPACE-BORNE HEXAGONAL ARRAY ELEMENT FAILURE CORRECTION USING ITERATIVE CONVEX OPTIMIZTION

Element failure distorts the main-lobe pattern and increases side-lobe power level, which is almost impossible to be corrected artiflcially for space-borne array. It might be solved by redistributing the excitations of the left functional elements; however, this is a nonlinear, non-convex, and NP-hard problem. In this paper, two efiective approaches are proposed for failure correction, which is performed for space-borne hexagonal array using digital beamforming (DBF). One method, a modifled real-code genetic algorithm (RCGA), is employed that uses reinsertion and worst-elimination schemes, but it pays the high computation complexity. The other approach based on convex optimization chooses the excitations synthesized by RCGA as the initial points, and transforms the non-convex problem into a sequence of second-order cone programming (SOCP) problem, which can be solved iteratively by e-cient optimization tool. Numerical results conflrm that after the correction based on iterative convex optimization, the average root-mean-square error (RMSE) is reduced by 36%, and the relative side-lobe level (RSLL) is decreased by 6.7dB, with respect to the RCGA-based correction pattern.

Transceiver Designs for Multipoint-to-Multipoint MIMO Amplify-and-Forward Relaying Systems

In this paper, we consider multipoint-to-multipoint multi-input multi-output relay systems where multiple source-destination pairs simultaneously communicate through a single multiple antenna relay. In this system configuration, we optimize two metrics when designing the source, relay and destination filters. The first one is sum mean-squared-error (MSE) and the second one is maximum pairwise MSE. These two problems are shown to be non-convex, and thus it is quite complex to find an analytical solution. To address this issue, we first introduce the sum MSE minimization scheme based on global channel state information (CSI) at all nodes. Then, a method which reduces complexity and CSI requirements is proposed by utilizing error covariance decomposition and MSE duality between broadcast channel and multiple access channel. Second, the pairwise MSE balancing algorithms are developed to guarantee fairness for all pairs. Similar to the case of the sum MSE minimization problem, we simplify the original pairwise MSE balancing problem to a second-order cone programming problem which can be solved using standard optimization tools. Through numerical simulations, we confirm the effectiveness of our proposed schemes.

1211 混合整数2次錐計画を用いたトラスの離散最適化法(最適化手法・理論)

This paper presents a truss topology optimization method for finding the minimum compliance design in which only a limited number of different cross-section sizes are used. The member cross-sectional areas are considered either discrete design variables or continuous design variables. In both cases the optimization problem of interest can be reduced to a mixed-integer second-order cone programming problem. The global optimal solution is then computed by using an existing algorithm based on a branch-and-cut method.

Homogeneous Self-dual Algorithms for Stochastic Second-Order Cone Programming

Jin et al. (in J. Optim. Theory Appl. 155:1073---1083, 2012) proposed homogeneous self-dual algorithms for stochastic semidefinite programs with finite event space. In this paper, we utilize their work to derive homogeneous self-dual algorithms for stochastic second-order cone programs with finite event space. We also show how the structure in the stochastic second-order cone programming problems may be exploited so that the algorithms developed for these problems have less complexity than the algorithms developed for stochastic semidefinite programs mentioned above.

An Enhanced DFT-Based Channel Estimator for LTE-A Uplink

In this paper, we investigate the popular discrete Fourier transform (DFT)-based channel estimator (D-bCE) for Long-Term Evolution (LTE)-Advanced uplink. Due to limitations of the current pilot pattern, orthogonality among channels of multiple antennas at the time domain does not exactly hold. This degrades the performance of a conventional D-bCE severely. Therefore, we propose a novel enhancement of the D-bCE by using the frequency-domain weighting scheme. By analyzing the mean square error, we present a heuristic way to design weights and reformulate the resulting optimization into a second-order cone programming problem, which can be easily solved. The proposed scheme does not rely on the channel covariance matrix, and the weights can be generated offline. Thus, it is easy to use this scheme in practice. Simulation results validate the effectiveness of our proposed method.

A majorized penalty approach to inverse linear second order cone programming problems

This paper focuses on a type of inverse linear second order cone programming (LSOCP) problems which require us to adjust the parameters in both the objective function and the constraint set of a given LSOCP problem as little as possible so that a known feasible solution becomes optimal one. This inverse problem can be formulated as a linear second order cone complementarity constrained optimization problem and is difficult to solve due to the presence of second order cone complementarity constraint. To solve this difficult problem, we first partially penalize the inverse problem and then propose the majorization approach to the penalized problem by solving a sequence of convex optimization problems with quadratic objective function and simple second order cone constraints. Numerical results demonstrate the efficiency of our approach to inverse LSOCP problems.

Upper bound finite element analysis of slope stability using a nonlinear failure criterion

In this study, upper bound finite element (FE) limit analysis is applied to stability problems of slopes using a nonlinear criterion. After formulating the upper bound analysis as the dual form of a second-order cone programming (SOCP) problem, the stress field and corresponding shear strength parameters can be determined iteratively. Thus, the nonlinear failure criterion is represented by the shear strength parameters associated with stress so that the analysis of slope stability using a nonlinear failure criterion can be transformed into the traditional upper bound method with a linear Mohr–Coulomb failure criterion. Comparison with published solutions illustrates the accuracy and feasibility of the proposed method for a simple homogeneous slope stability problem. The proposed approach is also applied to a seismic stability problem for a rockfill dam to study the influence of different failure criterions on the upper bound solutions. The results show that the seismic stability coefficients obtained using two different nonlinear failure criteria are similar but that the convergence differs significantly.

On p-norm linear discrimination

We consider a p-norm linear discrimination model that generalizes the model of Bennett and Mangasarian (1992) and reduces to a linear programming problem with p-order cone constraints. The proposed approach for handling linear programming problems with p-order cone constraints is based on reformulation of p-order cone optimization problems as second order cone programming (SOCP) problems when p is rational. Since such reformulations typically lead to SOCP problems with large numbers of second order cones, an “economical” representation that minimizes the number of second order cones is proposed. A case study illustrating the developed model on several popular data sets is conducted.

Axisymmetric Lower-Bound Limit Analysis Using Finite Elements and Second-Order Cone Programming

AbstractIn this paper, the formulation of a lower-bound limit analysis for axisymmetric problems by means of finite elements leads to an optimization problem with a large number of variables and constraints. For the Mohr-Coulomb criterion, it is shown that these axisymmetric problems can be solved by second-order cone programming (SOCP). First, a brief introduction to SOCP is given and how axisymmetric lower-bound limit analysis can be formulated in this way is described. Through the use of an efficient toolbox (MOSEK or SDPT3), large-scale SOCP problems can be solved in minutes on a desktop computer. The method is then applied to estimate the collapse load of circular footings and uplift capacity of single or multiplate circular anchors. By comparing the present analysis with the results reported in the literature, it is shown that the results obtained from the proposed method are accurate and computationally more efficient than the numerical lower-bound limit analysis incorporated with linear programming.

Hybrid relaying and jamming for wireless physical layer security based on cooperative beamforming

To improve the information-theoretic security of two-way wireless cooperative communication systems, we propose in this study a hybrid relaying and jamming mechanism based on multi-node cooperative beamforming to enhance the security of the physical layer transmission of information. The basic concept is that in the cooperative transmission phase, some intermediate nodes adopt distributed beamforming to help the transmitter send messages to legitimate users and simultaneously others jam the eavesdropper, which can significantly improve the security performance of the system in the broadcast phase. Two different cooperative beamforming schemes are proposed with and without the instantaneous channel state information (CSI) of the eavesdropper. We show that optimizing both the schemes requires the mathematical solving of one or a series of second-order convex cone programming (SOCP) problems, which can be executed by efficiently using the interior point method. Simulation and comparisons show that our proposed hybrid relaying, jamming mechanism and beamforming approaches significantly increase the security sum rate of the two source terminals and outperform both the relays-only and jammers-only mechanisms when the number of intermediate nodes is relatively large.

Strong duality for robust minimax fractional programming problems

We develop a duality theory for minimax fractional programming problems in the face of data uncertainty both in the objective and constraints. Following the framework of robust optimization, we establish strong duality between the robust counterpart of an uncertain minimax convex–concave fractional program, termed as robust minimax fractional program, and the optimistic counterpart of its uncertain conventional dual program, called optimistic dual. In the case of a robust minimax linear fractional program with scenario uncertainty in the numerator of the objective function, we show that the optimistic dual is a simple linear program when the constraint uncertainty is expressed as bounded intervals. We also show that the dual can be reformulated as a second-order cone programming problem when the constraint uncertainty is given by ellipsoids. In these cases, the optimistic dual problems are computationally tractable and their solutions can be validated in polynomial time. We further show that, for robust minimax linear fractional programs with interval uncertainty, the conventional dual of its robust counterpart and the optimistic dual are equivalent.

Autonomous Trajectory Planning for Rendezvous and Proximity Operations by Conic Optimization

Autonomous rendezvous and proximity operations of spacecraft require the capability of onboard planning and executing highly constrained trajectories without ground support. This paper presents a general and rigorous methodology and algorithmic procedure toward this goal with a target vehicle that can be in an arbitrary orbit. The rendezvous and proximity operations problem is formulated as a nonlinear optimal control problem, subject to various state and control inequality constraints and equality constraints on interior points and terminal conditions. By a lossless relaxation technique, a relaxed problem is formed, the solution of which is proven to be equivalent to that of the original rendezvous and proximity operations problem. The relaxed problem is then solved by a novel successive solution process, in which the solutions of a sequence of constrained subproblems with linear, time-varying dynamics are sought. After discretization, each of these problems becomes a second-order cone programming problem. Their solutions, if they exist, are guaranteed to be found by a primal-dual interior-point algorithm. The efficacy of the proposed methodology is strongly supported by numerical experiments.

COMPUTATION OF LIMIT LOAD USING EDGE-BASED SMOOTHED FINITE ELEMENT METHOD AND SECOND-ORDER CONE PROGRAMMING

This paper presents a novel numerical procedure for limit analysis of plane problems using edge-based smoothed finite element method (ES-FEM) in combination with second-order cone programming. In the ES-FEM, the discrete weak form is obtained based on the strain smoothing technique over smoothing domains associated with the edges of the elements. Using constant smoothing functions, the incompressibility condition only needs to be enforced at one point in each smoothing domain, and only one Gaussian point is required, ensuring that the size of the resulting optimization problem is kept to a minimum. The discretization problem is transformed into the form of a second-order cone programming problem which can be solved using highly efficient interior-point solvers. Finally, the efficacy of the procedure is demonstrated by applying it to various benchmark plane stress and strain problems.