Constraint Matrices Research Articles

On Conditioning of Constraints Arising from Variationally Consistent Discretization of Contact Problems and Duality Based Solvers

Abstract Variationally consistent approximation of the non-penetration conditions and friction laws was introduced by Wohlmuth [Acta Numer. 20 (2011) 569–734] as a powerful tool for the effective discretization of contact problems. This approach is especially useful when the potential contact interface is large and curved or when non-matching grids are applied. The point of this note is to give bounds on the singular values of the constraint matrices arising from the variationally consistent approximation and to indicate their application in the analysis of the scalable duality based domain decomposition algorithms for the solution of multi-body contact problems [Comput. Methods Appl. Mech. Eng. 247–248 (2012), 37–50.]. The theoretical results are illustrated by numerical experiments.

An Active-Set Method for Quadratic Programming Based On Sequential Hot-Starts

A new method for solving sequences of quadratic programs (QPs) is presented. For each new QP in the sequence, the method utilizes hot-starts that employ information computed by an active-set QP solver during the solution of the first QP. This avoids the computation and factorization of the full constraint and Hessian matrices for all but the first problem in the sequence. The proposed algorithm can be seen as an extension of the iterative refinement procedure for linear systems to QP problems, coupled with the application of an accelerated linear solver method that employs hot-started QP solves as a preconditioner. Local convergence results are presented. The practical performance of the proposed method is demonstrated on a sequence of QPs arising in nonlinear model predictive control and during the solution of a set of randomly generated nonlinear optimization problems using sequential quadratic programming. In these experiments, the method proves to be fairly reliable, despite the lack of global convergence guarantees. The results also show a significant reduction in the computation time for large problems with dense constraint matrices, as well as in the number of matrix-vector products.

Totally unimodular multistage stochastic programs

We consider totally unimodular multistage stochastic programs, that is, multistage stochastic programs whose extensive-form constraint matrices are totally unimodular. We establish several sufficient conditions and identify examples that have arisen in the literature.

Integrality in Stochastic Inventory Models

We study several finite‐horizon, discrete‐time, dynamic, stochastic inventory control models with integer demands: the newsvendor model, its multi‐period extension, and a single‐product, multi‐echelon assembly model. Equivalent linear programs are formulated for the corresponding stochastic dynamic programs, and integrality results are derived based on the total unimodularity of the constraint matrices. Specifically, for all these models, starting with integer inventory levels, we show that there exist optimal policies that are integral. For the most general single‐product, multi‐echelon assembly system model, integrality results are also derived for a practical alternative to stochastic dynamic programming, namely, rolling‐horizon optimization by a similar argument. We also present a different approach to prove integrality results for stochastic inventory models. This new approach is based on a generalization we propose for the one‐dimensional notion of piecewise linearity with integer breakpoints to higher dimensions. The usefulness of this new approach is illustrated by establishing the integrality of both the dynamic programming and rolling‐horizon optimization models of a two‐product capacitated stochastic inventory control system.

Origami-Inspired Integrated Planar-Spherical Overconstrained Mechanisms

This paper presents two integrated planar-spherical overconstrained mechanisms that are inspired and evolved from origami cartons with a crash-lock base. Investigating the crash-lock base of the origami cartons, the first overconstrained mechanism is evolved by integrating a planar four-bar linkage with two spherical linkages in the diagonal corners. The mechanism has mobility one and the overconstraint was exerted by the two spherical linkages. This mechanism is then evolved into another integrated planar-spherical overconstrained mechanism with two double-spherical linkages at the diagonal corners. The evolved mechanism has mobility one. It is interesting to find that the double-spherical linkage at the corner of this new mechanism is an overconstrained 6R linkage. The geometry evolution is presented and the constraint matrices of the mechanisms are formulated using screw-loop equations verifying mobility of the mechanisms. The paper further reveals the assembly conditions and geometric constraint of the two overconstrained mechanisms. Further, with mechanism decomposition, geometry and kinematics of the mechanisms are investigated with closed-form equations, leading to comparison of these two mechanisms with numerical simulation. The paper further proposes that the evolved overconstrained mechanism can in reverse lead to new origami folds and crease patterns. The paper hence not only lays the groundwork for kinematic investigation of origami-inspired mechanisms but also sheds light on the investigation of integrated overconstrained mechanisms.

Construction Method of Constraint Matrices Corresponded by an Optimal Solution

Through deep analysis of the solvability, which is based on interval linear equations and inequalities systems, for a given optimal solution to interval linear programming problem, we propose the construction method of constraint matrices corresponded by the optimal solution in this paper.

On matrix algebras associated to sum-of-squares semidefinite programs

To each semidefinite program (SDP) in primal form, we associate the matrix algebra generated by its constraint matrices. In this note, we show that this algebra is always a full matrix algebra for SDPs arising from (commutative or non-commutative) sum of squares (SOS) problems. For SDPs arising from non-commutative SOS and commutators problems, the situation is less clear. We identify an exceptional case, where the corresponding matrix algebra is not the full matrix algebra, and use it to reprove the Burgdorf–Klep non-commutative variant of Hilbert’s ternary quartics theorem: a bivariate non-commutative polynomial of degree at most is trace positive if it is a sum of four squares and commutators.

EM-type method for measuring graph dissimilarity

Graph dissimilarity (or similarity) computation is one of the key intermediate operations for intelligent information processing. In this paper, a new dissimilarity measure is proposed to gauge the pairwise structural difference between graphs through constrained graph matching. The measure is obtained as a solution to an optimization problem that we formalize by a new objective function in matrix representation, based on the adjacency matrices of graphs and binary constraints. An EM-type algorithm based on the minimum-weight bipartite graph method is proposed to optimize the matrix in polynomial time. The suitability of the proposed measure is evaluated in nearest-neighbor classification on four real-world graph datasets and on synthetic datasets.

Space-Time-Frequency (STF) MIMO Communication Systems With Blind Receiver Based on a Generalized PARATUCK2 Model

In this paper, we first propose a generalized fourth-order PARATUCK2 tensor model for multiple-input multiple-output (MIMO) communication systems with space-time-frequency (STF) spreading-multiplexing. The core of the proposed PARATUCK2 model is composed of two third-order interaction tensors that define a joint time and frequency allocation of the data streams to the transmit antennas, thus allowing to adjust the multiplexing degree and spreading redundancy in three domains: space (transmit antennas), time (blocks) and frequency (subcarriers). Then, we investigate the identifiability of the PARATUCK2-STF MIMO system by deriving sufficient conditions which are translated into design recommendations for the STF allocation structure. In particular, essential uniqueness is discussed by interpreting the generalized fourth-order PARATUCK2 model as an equivalent third-order constrained factor (CONFAC) model with two fixed constraint matrices and one variable constraint matrix that depends on the stream-to-antenna allocation structure. We also present a blind receiver using the Levenberg-Marquardt (LM) algorithm based on the generalized fourth-order PARATUCK2 model. Numerical results are provided for a bit-error-rate performance evaluation and a comparison with some competing algorithms.

Multi-objective optimization integration of query interfaces for the Deep Web based on attribute constraints

In order to query and retrieve the rich and useful information hidden in the Deep Web efficiently, extensive research on domain-specific Deep Web Data Integration Systems (DWDIS) has been carried out in recent years. In DWDIS, large-scale automatic integration of query interfaces of domain-specific Web Databases (WDBs) remains a serious challenge due to the scale of the problem and the great diversity of the WDBs' query interfaces. To address this challenge, in this paper, we first give a definition of the constraint matrix which can accurately describe three types of constraints (hierarchical constraints, group constraints and precedence constraints) and the strengths of attributes of a query interface, and then prove that the schema tree of the query interface corresponds to only one constraint matrix, and vice versa. Furthermore, we transform the problem of integrating domain-specific query interfaces into a problem of integrating the constraint matrices and set up a multi-objective optimization problem model. To effectively solve the optimization model, some strategies to extend and merge the constraint matrices are designed. A method for automatically detecting and filtering abnormal data (noises) in the query interfaces is also proposed. More importantly, a novel and efficient algorithm applicable to large-scale automatic integration of domain-specific query interfaces is developed. Finally, the proposed algorithm is evaluated by experiments on the real query interface data set. Our theoretical analysis and experimental results show that the proposed algorithm outperforms existing state-of-the-art integration algorithms of domain-specific query interfaces.

Integrality in Stochastic Inventory Models

We study several dynamic, stochastic inventory control models with integer demands: the newsvendor model, its multi-period extension and a single-product, multi-echelon assembly model. Equivalent linear programs are formulated for the corresponding stochastic dynamic programs, and integrality results are derived based on the total unimodularity of the constraint matrices. Specifically, for all these models, starting with integer inventory levels, we show that there exist optimal policies that are integral. For the most general singleproduct, multi-echelon assembly system model, integrality results are also derived for a practical alternative to stochastic dynamic programming, namely rolling-horizon optimization by a similar argument. We also present a different approach to prove integrality results for stochastic inventory models. This new approach is based on a generalization we propose for the one dimensional notion of piecewise linearity with integer breakpoints to higher dimensions. The usefulness of this new approach is illustrated by showing an integrality result for the rolling-horizon optimization model of a two-product capacitated stochastic inventory control system.

A new switching strategy for addressing Euler parameters in dynamic modeling and simulation of rigid multibody systems

This paper presents a new numerical strategy to address the extraneous coordinates in the Euler parameters. These parameters are unit quaternions with four variables, which have been used to describe the three rotational degrees-of-freedom (DOF) of a rigid body. This redundancy avoids the singularity issue that appears in the other rotational description methods including, Euler angles (3 parameters), direction cosines (9 parameters), and the Rodrigues parameters (3 parameters). However, these parameters must satisfy a normality constraint. A numerical and online constraint embedding method is invoked to address this holonomic constraint implicitly. It leads to a reduction of the equations of motion to a minimal form. However, the proposed method requires a procedure for the selection of the dependent Euler parameters. Several alternatives were examined and the best one is proposed as the method to use in this situation. They check specific conditions including, condition number of the mass and constraint matrices, to make a decision about retaining or switching the current dependent parameter. The effectiveness of the three different selection algorithms are examined using a 3D double pendulum with ball-and-socket joints. The results show the ability of the proposed strategy to handle drifting and singularity issues.

Proving Termination by Policy Iteration

In this paper, we explore the adaptation of policy iteration techniques to compute greatest fixpoint approximations, in order to find sufficient conditions for program termination. Restricting ourselves to affine programs and the abstract domain of template constraint matrices, we show that the abstract greatest fixpoint can be computed exactly using linear programming, and that strategies are related to the template constraint matrix used. We also present a first result on the relationships between this approach and methods which use ranking functions.

Numerical aspects in the dynamic simulation of geometrically exact rods

Classical geometrically exact Kirchhoff and Cosserat models are used to study the nonlinear deformation of rods. Extension, bending and torsion of the rod may be represented by the Kirchhoff model. The Cosserat model additionally takes into account shearing effects. Second order finite differences on a staggered grid define discrete viscoelastic versions of these classical models. Since the rotations are parametrised by unit quaternions, the space discretisation results in differential-algebraic equations that are solved numerically by standard techniques like index reduction and projection methods. Using absolute coordinates, the mass and constraint matrices are sparse and this sparsity may be exploited to speed-up time integration. Further improvements are possible in the Cosserat model, because the constraints are just the normalisation conditions for unit quaternions such that the null space of the constraint matrix can be given analytically. The results of the theoretical investigations are illustrated by numerical tests.

A Two-Stage Method for the Approximate Solution of General Multiparametric Mixed-Integer Linear Programming Problems

In this work, we focus on the approximate solution of multiparametric mixed-integer linear programming (mp-MILP) problems involving uncertainty in the objective function coefficients and in the entries of the constraint matrices and vectors. A two-stage algorithmic procedure is proposed. In the first stage, the model is partially immunized against uncertainty using the worst-case oriented approach which leads to a partially robust mp-MILP model, whereas in the second stage explicit solutions of the robust model are derived by applying a suitable multiparametric programming algorithm for mp-MILP problems. Computational studies are presented, demonstrating that the proposed two-stage robust optimization/multiparametric programming procedure is computationally efficient and that it provides an upper bound on the overall solution of the general mp-MILP problem.

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.

Solving L 1-CTA in 3D tables by an interior-point method for primal block-angular problems

The purpose of the field of statistical disclosure control is to avoid that confidential information could be derived from statistical data released by, mainly, national statistical agencies. Controlled tabular adjustment (CTA) is an emerging technique for the protection of statistical tabular data. Given a table with sensitive information, CTA looks for the closest safe table. In this work we focus on CTA for three-dimensional tables using the L1 norm for the distance between the original and protected tables. Three L1-CTA models are presented, giving rise to six different primal block-angular structures of the constraint matrices. The resulting linear programming problems are solved by a specialized interior-point algorithm for this constraints structure which solves the normal equations by a combination of Cholesky factorization and preconditioned conjugate gradients (PCG). In the past this algorithm was shown to be one of the most efficient approaches for some classes of block-angular problems. The effect of quadratic regularizations is also analyzed, showing that for three of the six primal block-angular structures the performance of PCG is guaranteed to improve. Computational results are reported for a set of large instances, which provide linear optimization problems of up to 50 million variables and 25 million constraints. The specialized interior-point algorithm is compared with the state-of-the-art barrier solver of the CPLEX 12.1 package, showing to be a more efficient choice for very large L1-CTA instances.

Some remarks upon the characteristics of the explicit representation of the MPC problem

In this paper we revisit the explicit MPC representation and related notions. We point to the special structure of the constraint matrices and exploit it in order to provide novel results. We give an upper bound for the collection of admissible active sets with use in the mixed integer representation of the KKT problem and a partial recursive description of the explicit partitioning of the MPC problem. The results are tested over illustrative examples.

A factorization with update procedures for a KKT matrix arising in direct optimal control

Quadratic programs obtained for optimal control problems of dynamic or discrete-time processes usually involve highly block structured Hessian and constraints matrices, to be exploited by efficient numerical methods. In interior point methods, this is elegantly achieved by the widespread availability of advanced sparse symmetric indefinite factorization codes. For active set methods, however, conventional dense matrix techniques suffer from the need to update base matrices in every active set iteration, thereby loosing the sparsity structure after a few updates. This contribution presents a new factorization of a KKT matrix arising in active set methods for optimal control. It fully respects the block structure without any fill-in. For this factorization, matrix updates are derived for all cases of active set changes. This allows for the design of a highly efficient block structured active set method for optimal control and model predictive control problems with long horizons or many control parameters.

A new method for automatic synthesis of tolerances for complex assemblies based on polychromatic sets

In computer-aided tolerance design (CAT), integrated design of dimensional and geometric tolerances is still one of the research hotspots. Polychromatic sets theory (PST) is a new mathematic tool, which is especially suitable for formal hierarchical structure models. Based on PST, in this article, a new hierarchical representation model for tolerance synthesis is presented to realise integrated design of dimensional and geometric tolerances. According to the inference relations between unified and individual colours of PST, the synthesis matrices of variational geometric constraints (VGC) are established in the VGC tier of the model, and the synthesis matrices of tolerance types are established in the tolerance type tier of the model. On this basis, the synthesis processes from the feature tier to the VGC tier and from the VGC tier to the tolerance type tier can be realised. VGCs, which are achieved by the synthesis matrices of VGCs, can be combined together to establish a well-constrained VGC network (VGCN). Tolerance types, which are achieved by the synthesis matrices of tolerance types, can be added to the well-constrained VGCN to construct a well-constrained tolerance network. An application example is given in the article to illustrate the synthesis steps.