Constraint Matrices Research Articles

Nonconvex quadratically constrained quadratic programming: best D.C. decompositions and their SDP representations

We propose in this paper a general D.C. decomposition scheme for constructing SDP relaxation formulations for a class of nonconvex quadratic programs with a nonconvex quadratic objective function and convex quadratic constraints. More specifically, we use rank-one matrices and constraint matrices to decompose the indefinite quadratic objective into a D.C. form and underestimate the concave terms in the D.C. decomposition formulation in order to get a convex relaxation of the original problem. We show that the best D.C. decomposition can be identified by solving an SDP problem. By suitably choosing the rank-one matrices and the linear underestimation, we are able to construct convex relaxations that dominate Shor's SDP relaxation and the strengthened SDP relaxation. We then propose an extension of the D.C. decomposition to generate an SDP bound that is tighter than the SDP+RLT bound when additional box constraints are present. We demonstrate via computational results that the optimal D.C. decomposition schemes can generate both tight SDP bounds and feasible solutions with good approximation ratio for nonconvex quadratically constrained quadratic problems.

Mobility and Geometric Analysis of the Hoberman Switch-Pitch Ball and Its Variant

Abstract This paper investigates the mobility and kinematics of the Hoberman switch-pitch ball, and particularly, its variant that does not resort to bevel gears. The ball variant is a general case of the Hoberman switch-pitch ball and constitutes the ball. This paper starts from examining the geometry of the ball variant and its composition, and decomposes it into loops containing eight-bar radially foldable linkages. To investigate the eight-bar radially foldable linkage, constraint matrices are developed using the screw-loop equation. This paper extends the study to the ball variant and investigates the singularity and various configurations based on the geometry and kinematics of the ball variant. This leads to the investigation of the Hoberman switch-pitch ball as a special case of the ball variant with bevel gears to simultaneously drive three joints in every vertex of the ball mechanism. The analysis is then followed by a numerical demonstration of the kinematic characteristics of the Hoberman switch-pitch ball.

New matrix iterative methods for constraint solutions of the matrix equation [formula omitted

In this paper, two new matrix iterative methods are presented to solve the matrix equation A X B = C , the minimum residual problem min X ∈ S ‖ A X B − C ‖ and the matrix nearness problem min X ∈ S E ‖ X − X ∗ ‖ , where S is the set of constraint matrices, such as symmetric, symmetric R -symmetric and ( R , S ) -symmetric, and S E is the solution set of above matrix equation or minimum residual problem. These matrix iterative methods have faster convergence rate and higher accuracy than the matrix iterative methods proposed in Deng et al. (2006) [13], Huang et al. (2008) [15], Peng (2005) [16] and Lei and Liao (2007) [17]. Paige’s algorithms are used as the frame method for deriving these matrix iterative methods. Numerical examples are used to illustrate the efficiency of these new methods.

State-space modeling of the radio frequency inductively-coupled plasma generator

Computational fluid dynamics models of RF-ICP are useful in understanding the basic transport phenomenon in an ICP torch under a wide variety of operating conditions. However, these models lack the ability to evaluate the effects of the plasma condition on the RF generator. In this paper, simulation of an induction plasma generator has been done using state space modelling by considering inductively coupled plasma as a part of RF network .The time dependent response of the RF-ICP generator circuit to given input excitation has been computed by extracting the circuit's state-space variables and their constraint matrices. MATLAB 7.1 software has been used to solve the state equations. The values of RF coil current, frequency and plasma power has been measured experimentally also at different plate bias voltage. The simulated model is able to predict RF coil current, frequency, plasma power, overall efficiency of the generator. The simulated and measured values are in agreement with each other. This model can prove useful as a design tool for the Induction plasma generator.

A new library of structured semidefinite programming instances

Solvers for semidefinite programming (SDP) have evolved a great deal in the last decade, and their development continues. In order to further support and encourage this development, we present a new test set of SDP instances. These instances arise from recent applications of SDP in the coding theory, computational geometry, graph theory and structural design. Most of these instances have a special structure that may be exploited during a pre-processing phase, e.g. algebraic symmetry, or low rank in the constraint matrices.

A Hierarchy of Bounds for Stochastic Mixed-Integer Programs

Strong relaxations are critical for solving deterministic mixed-integer programs. As solving stochastic mixed-integer programs (SMIPs) is even harder, it is likely that strong relaxations will also prove essential for SMIPs. We consider general two-stage SMIPs with recourse, where integer variables are allowed in both stages of the problem and randomness is allowed in the objective function, the constraint matrices (i.e., the technology matrix and the recourse matrix), and the right-hand side. We develop a hierarchy of lower and upper bounds for the optimal objective value of an SMIP by generalizing the wait-and-see (WS) solution and the expected result of using the expected value (EEV) solution. These bounds become progressively stronger but, generally, more difficult to compute. Our numerical study indicates that the bounds developed in this paper can be very strong relative to those provided by stochastic linear programming relaxations.

Weakly-relational shapes for numeric abstractions: improved algorithms and proofs of correctness

Weakly-relational numeric constraints provide a compromise between complexity and expressivity that is adequate for several applications in the field of formal analysis and verification of software and hardware systems. We address the problems to be solved for the construction of full-fledged, efficient and provably correct abstract domains based on such constraints. We first propose to work with semantic abstract domains, whose elements are geometric shapes, instead of the (more concrete) syntactic abstract domains of constraint networks and matrices on which the previous proposals are based. This allows to solve, once and for all, the problem whereby closure by entailment, a crucial operation for the realization of such domains, seemed to impede the realization of proper widening operators. In our approach, the implementation of widenings relies on the availability of an effective reduction procedure for the considered constraint description: one for the domain of bounded difference shapes already exists in the literature; we provide algorithms for the significantly more complex cases of rational and integer octagonal shapes. We also improve upon the state-of-the-art by presenting, along with their proof of correctness, closure by entailment algorithms of reduced complexity for domains based on rational and integer octagonal constraints. The consequences of implementing weakly-relational numerical domains with floating point numbers are also discussed.

A new model of phase shifter for its efficient integration in interior point optimal power flow

AbstractWhen phase shifters are used as control devices in optimal power flow (OPF), the symmetry of the admittance matrix is destroyed; this introduces some difficulties in the implementation of interior point OPF. This paper presents an efficient approach to handle phase shifters in an interior point OPF algorithm. First, a new power/current injection model of phase shifters is proposed. Then, a detailed explanation of how to integrate this model into the existing interior point OPF code is presented. This integration of phase shifters only requires minor modifications to the existing interior point OPF code by updating Jacobian/Hessian matrices of objective and constraints in each iterate. The distinguished features of the proposed approach are: (i) box constraints are used instead of implicit constraints; (ii) no additional buses are introduced, so the dimensions of Jacobian and Hessian matrices remain the same; and (iii) the admittance matrix is constant, and remains symmetric. Numerical studies on several power systems up to 2383 buses indicate that the proposed approach is simple and efficient. Copyright © 2009 John Wiley & Sons, Ltd.

Constrained Tucker-3 model for blind beamforming

This paper presents a constrained Tucker-3 model for blind beamforming. The constrained structure of this model is parameterized by two constraint matrices that capture the spatial and temporal structure of the channel. We first present a generalized Tucker-3 model that considers a multipath propagation scenario with large delay spread and different number of paths per source. This tensor model generalizes the models of [N.D. Sidiropoulos, G.Z. Dimic, Blind multiuser detection in WCDMA systems with large delay spread, IEEE Signal Process. Lett. 8 (3) (2001) 87–89] and [A. de Baynast, L. De Lathauwer, Détection autodidacte pour des systèmes à accès multiple basée sur l’analyse PARAFAC, in: Proceedings of XIX GRETSI Symposium on Signal and Image Processing, Paris, France, 2003] by considering a more general multipath propagation scenario. From this general model, two particular cases can be derived by adjusting the constrained structure of the Tucker-3 core tensor. Identifiability of the proposed tensor model is studied and some simulation results are presented to illustrate the application of this model to blind beamforming.

A Constrained Factor Decomposition With Application to MIMO Antenna Systems

In this paper, we formulate a new tensor decomposition herein called constrained factor (CONFAC) decomposition. It consists in decomposing a third-order tensor into a triple sum of rank-one tensor factors, where interactions involving the components of different tensor factors are allowed. The interaction pattern is controlled by three constraint matrices the columns of which are canonical vectors. Each constraint matrix is associated with a given dimension, or mode, of the tensor. The explicit use of these constraint matrices provides degrees of freedom to the CONFAC decomposition for modeling tensor signals with constrained structures which cannot be handled with the standard parallel factor (PARAFAC) decomposition. The uniqueness of this decomposition is discussed and an application to multiple-input multiple-output (MIMO) antenna systems is presented. A new transmission structure is proposed, the core of which consists of a precoder tensor decomposed as a function of the CONFAC constraint matrices. By adjusting the precoder constraint matrices, we can control the allocation of data streams and spreading codes to transmit antennas. Based on a CONFAC model of the received signal, blind symbol/code/channel recovery is possible using the alternating least squares algorithm. For illustrating this application, we evaluate the bit-error-rate (BER) performance for some configurations of the precoder constraint matrices.

Constrained Tensor Modeling Approach to Blind Multiple-Antenna CDMA Schemes

In this paper, we consider an uplink multiple-antenna code-division multiple-access (CDMA) system linking several multiple-antenna mobile users to one multiple-antenna base station. For this system, a constrained third-order tensor decomposition is introduced for modeling the multiple-antenna transmitter as well as the received signal. The constrained structure of the proposed tensor decomposition is characterized by two constraint matrices that have a meaningful physical interpretation in our context. They can be viewed as canonical allocation matrices that define the allocation of users' data streams and spreading codes to the transmit antennas. The distinguishing features of the proposed tensor modeling with respect to the already existing tensor-based CDMA models are: i) it copes with multiple transmit antennas and spreading codes per user and ii) it models several spatial spreading/multiplexing schemes with multiple spreading codes by controlling the constrained structure of the tensor signal model. A systematic design procedure for the canonical allocation matrices leading to a unique blind symbol (or joint blind symbol-code) recovery is proposed which allows us to derive a finite set of multiple-antenna schemes for a fixed number of transmit antennas. Identifiability of the proposed tensor model is discussed, and a blind multiuser detection receiver based on the alternating least squares algorithm is considered for performance evaluation of several multiple-antenna CDMA schemes derived from the constrained modeling approach.

Multiuser MIMO system using block space–time spreading and tensor modeling

In this paper, we consider a point-to-multipoint downlink multiuser wireless communication system, where a multiple-antenna base station simultaneously transmits data to several users equipped with multiple receive antennas. The transmit antenna array is partitioned into transmission blocks, each one being associated with a given user. Space–time spreading is performed within each block using a transmit antenna subset. We formulate block space–time spreading using a tensor modeling. We show that the tensor-based block space–time spreading model has the distinguishing feature of modeling a multiuser space–time transmission with different spatial spreading factors (diversity gains) as well as different multiplexing factors (code rates) for the users. The space–time spreading structure is chosen to allow a deterministic multiuser interference (MUI) elimination by each user. A block-constrained tensor model is then presented for the received signal, which is characterized by fixed constraint matrices that reveal the overall space–time spreading pattern. At each receiver, blind joint channel and symbol recovery is performed using an alternating least squares algorithm. Simulation results illustrate the performance of the proposed transceiver model in terms of bit-error-rate, channel/symbol estimation accuracy and link-level throughput.

On a decision procedure for quantified linear programs

Quantified linear programming is the problem of checking whether a polyhedron specified by a linear system of inequalities is non-empty, with respect to a specified quantifier string. Quantified linear programming subsumes traditional linear programming, since in traditional linear programming, all the program variables are existentially quantified (implicitly), whereas, in quantified linear programming, a program variable may be existentially quantified or universally quantified over a continuous range. In this paper, the term linear programming is used to describe the problem of checking whether a system of linear inequalities has a feasible solution. On account of the alternation of quantifiers in the specification of a quantified linear program (QLP), this problem is non-trivial. QLPs represent a class of declarative constraint logic programs (CLPs) that are extremely rich in their expressive power. The complexity of quantified linear programming for arbitrary constraint matrices is unknown. In this paper, we show that polynomial time decision procedures exist for the case in which the constraint matrix satisfies certain structural properties. We also provide a taxonomy of quantified linear programs, based on the structure of the quantifier string and discuss the computational complexities of the constituent classes.

A full modal decomposition of thin-walled, single-branched open cross-section members via the constrained finite strip method

This paper derives a new method for fully decomposing the elastic stability solution, of a thin-walled single-branched open cross-section member, into mechanically consistent buckling classes associated with global, local, distortional, and shear and transverse extension buckling modes. The method requires a set of formal mechanical definitions for each of the buckling classes. For global and distortional buckling the definitions employed successfully by generalized beam theory are utilized herein, while for local and other (shear and transverse extension) buckling, new definitions are provided. The mechanical definitions for a given buckling class represent a series of constraint conditions on the general deformations that the thin-walled cross-section may undergo. These constraint conditions are derived as explicit constraint matrices within the context of the finite strip method, and provide the desired decomposition of the buckling deformations of the member. The decomposition is full in the sense that the union of the deformation spaces of the decomposed buckling classes is the same as the general deformation space in the original finite strip method. The resulting method is termed the constrained finite strip method (cFSM). The two primary applications for cFSM are modal decomposition and modal identification. Modal decomposition reduces the general finite strip solution to a desired set of buckling classes and performs a useful model reduction that allows the results to focus on a particular buckling class, e.g., distortional buckling. Modal identification provides a means to quantify the extent to which a given buckling class is contributing to a general buckling deformation. Application of cFSM, including graphical representation of the buckling classes, and the advantages of modal decomposition and modal identification, are provided in a series of numerical examples.

Parallel implementation of a semidefinite programming solver based on CSDP on a distributed memory cluster

In this paper, we present the algorithmic framework and practical aspects of implementing a parallel version of a primal–dual semidefinite programming solver on a distributed memory computer cluster. Our implementation is based on the CSDP solver and uses a message passing interface and the ScaLAPACK library. A new feature is implemented to deal with problems that have rank-one constraint matrices. We show that significant improvement is obtained for a test set of problems with rank-one constraint matrices. Moreover, we show that very good parallel efficiency is obtained for large-scale problems where the number of linear equality constraints is very large compared to the block sizes of the positive semidefinite matrix variables.

저자유도 병렬형 로봇의 강성 모델링

This paper presents a stiffness modeling of a low-DOF parallel robot, which takes into account of elastic deformations of joints and links, A low-DOF parallel robot is defined as a spatial parallel robot which has less than six degrees of freedom. Differently from serial chains in a full 6-DOF parallel robot, some of those in a low-DOF parallel robot may be subject to constraint forces as well as actuation forces. The reaction forces due to actuations and constraints in each serial chain can be determined by making use of the theory of reciprocal screws. It is shown that the stiffness of an F-DOF parallel robot can be modeled such that the moving platform is supported by 6 springs related to the reciprocal screws of actuations (F) and constraints (6-F). A general <TEX>$6{\times}6$</TEX> stiffness matrix is derived, which is the sum of the stiffness matrices of actuations and constraints, The compliance of each spring can be precisely determined by modeling the compliance of joints and links in a serial chain as follows; a link is modeled as an Euler beam and the compliance matrix of rotational or prismatic joint is modeled as a <TEX>$6{\times}6$</TEX> diagonal matrix, where one diagonal element about the rotation axis or along the sliding direction is infinite. By summing joint and link compliance matrices with respect to a reference frame and applying unit reciprocal screw to the resulting compliance matrix of a serial chain, the compliance of a spring is determined by the resulting infinitesimal displacement. In order to illustrate this methodology, the stiffness of a Tricept parallel robot has been analyzed. Finally, a numerical example of the optimal design to maximize stiffness in a specified box-shape workspace is presented.

Quadratic approximation of penalty functions for solving large-scale linear programs

Using the least squares, modified Lagrangian function, and some other methods as examples, the capabilities of the new optimization technique based on the quadratic approximation of penalty functions that has been recently proposed by O. Mangasarian for a special class of linear programming problems are demonstrated. The application of this technique makes it possible to use unified matrix operations and standard linear algebra packages (including parallel ones) for solving large-scale problems with sparse strongly structured constraint matrices. With this technique, the computational schemes of some well-known algorithms can take an unexpected form.

Parallel Implementation of a Semidefinite Programming Solver Based on CSDP on a Distributed Memory Cluster

In this paper we present the algorithmic framework and practical aspects of implementing a parallel version of a primal-dual semidefinite programming solver on a distributed memory computer cluster. Our implementation is based on the CSDP solver and uses a message passing interface (MPI), and the ScaLAPACK library. A new feature is implemented to deal with problems that have rank-one constraint matrices. We show that significant improvement is obtained for a test set of problems with rank one constraint matrices. Moreover, we show that very good parallel efficiency is obtained for large-scale problems where the number of linear equality constraints is very large compared to the block sizes of the positive semidefinite matrix variables.

Upper bound on the number of vertices of polyhedra with [formula omitted]-constraint matrices

In this note we give upper bounds for the number of vertices of the polyhedron P ( A , b ) = { x ∈ R d : A x ⩽ b } when the m × d constraint matrix A is subjected to certain restriction. For instance, if A is a 0/1-matrix, then there can be at most d! vertices and this bound is tight, or if the entries of A are non-negative integers so that each row sums to at most C, then there can be at most C d vertices. These bounds are consequences of a more general theorem that the number of vertices of P ( A , b ) is at most d ! ⋅ W / D , where W is the volume of the convex hull of the zero vector and the row vectors of A, and D is the smallest absolute value of any non-zero d × d subdeterminant of A.

Universal duality in conic convex optimization

Given a primal-dual pair of linear programs, it is well known that if their optimal values are viewed as lying on the extended real line, then the duality gap is zero, unless both problems are infeasible, in which case the optimal values are +∞ and −∞. In contrast, for optimization problems over nonpolyhedral convex cones, a nonzero duality gap can exist when either the primal or the dual is feasible.For a pair of dual conic convex programs, we provide simple conditions on the ``constraint matrices'' and cone under which the duality gap is zero for every choice of linear objective function and constraint right-hand side. We refer to this property as ``universal duality''. Our conditions possess the following properties: (i) they are necessary and sufficient, in the sense that if (and only if) they do not hold, the duality gap is nonzero for some linear objective function and constraint right-hand side; (ii) they are metrically and topologically generic; and (iii) they can be verified by solving a single conic convex program. We relate to universal duality the fact that the feasible sets of a primal convex program and its dual cannot both be bounded, unless they are both empty. Finally we illustrate our theory on a class of semidefinite programs that appear in control theory applications.