Asymptotics for Constrained Dirichlet Distributions

Currently used methods for the analysis of contingency tables with log-linear models and their extensions are essentially a way to subject the parameters of the saturated model to a set of linear constraints. However, in many contexts, expecially when a subset of the variables are ordinal, it would be very useful to be able to consider also linear inequality constraints. In this paper we show how to extend previous models for contingency tables by considering simultaneously both equality constraints. We also give a self contained discussion of the problem of testing equality and inequality constraints in the context of contingency tables analysis.

Fully constrained linear spectral mixture analysis (FCLSMA) has been used for material quantification in remotely sensed imagery. In order to implement FCLSMA, two constraints are imposed on abundance fractions, referred to as Abundance Sum-to-one Constraint (ASC) and Abundance Nonnegativity Constraint (ANC). While the ASC is linear equality constraint, the ANC is a linear inequality constraint. A direct approach to imposing the ASC and ANC has been recently investigated and is called fully constrained least-squares (FCLS) method. Since there is no analytical solution resulting from the ANC, a modified fully constrained least-squares method (MFCLS) which replaces the ANC with an Absolute Abundance Sum-to-one Constraint (AASC) was proposed to convert a set of inequality constraints to a quality constraint. The results produced by these two approaches have been shown to be very close. In this paper, we take an oopposite approach to the MFCLS method, called least-squares with linear inequality constraints (LSLIC) method which also solves FCLSMA, but replaces the ASC with two linear inequalities. The proposed LSLIC transforms the FCLSMA to a linear distance programming problem which can be solved easily by a numerical algorithm. In order to demonstrate its utility in solving FCLSMA, the LSLIC method is compared to the FCLS and MFCLS methods. The experimental results show that these three methods perform very similarly with only subtle differences resulting from their problem formations.

This paper presents an efficient algorithm which solves the mathematical programming linear problem of finding the maximum of a real variable subject to a set of linear and quadratic inequality constraints. The proposed algorithm is based on an analytical procedure which exploits the linear and quadratic form of the inequality constraints and gives the solution, if it exists, in a finite number of steps with a light computational burden. The application of the proposed algorithm is shown in a numerical example.

The present study pertains to the finding of the lower bound solution, formulating it as a non‐linear programming problem using the generalized method developed by Lysmer with certain variations to incorporate the non‐linear no‐yield condition constraints directly in the analysis. The method considers the family of plane stress fields having the property that all stresses vary linearly within each triangular element of some mesh which covers the soil mass under study. For this type of stress field it is possible to express all equilibrium conditions as a set of linear constraints and the no‐yield as a set of non‐linear constraints. The boundary condition constraints may be of linear equality or inequality type. By expressing some of the design variables in terms of the remaining variables the linear equality constraints are implicity satisfied. Such a technique minimizes the complexity of the problem by eliminating the equality constraints and reduces the dimensionality of the problem, saving much, computational effort. The optimal lower bound is isolated by formulating it as a non‐linear programming (NLP) problem subjected to both linear and non‐linear inequality constraints. The sequential unconstrained minimization technique using the extended penalty function method as suggested by Kavlie has been used to isolate the optimal lower bound. The method has successfully been applied to the passive earth pressure and bearing capacity problem.Numerical results are obtained and compared with Lysmer's solution to show the effectiveness of the present approach.

This paper considers a general nonlinear econometric model framework that contains a large class of estimators defined as solutions to optimization problems. For this framework we derive several asymptotically equivalent forms of a test statistic for the local (in a way made precise in the paper) multivariate nonlinear inequality constraints testH:h(β) ≥ 0 versusK: β ∈RK. We extend these results to consider local hypotheses tests of the formH:h1(β) ≥ 0 andh2(β) = 0 versusK: β ∈RK. For each test we derive the asymptotic distribution for any size test as a weighted sum of χ2-distributions. We contrast local as opposed to global inequality constraints testing and give conditions on the model and constraints when each is possible. This paper also extends the well-known duality results in testing multivariate equality constraints to the case of nonlinear multivariate inequality constraints and combinations of nonlinear inequality and equality constraints.

In this paper, the development and application of a new upper bound limit method for two- and three-dimensional (2D and 3D) slope stability problems is presented. Rigid finite elements are used to construct a kinematically admissible velocity field. Kinematically admissible velocity discontinuities are permitted to occur at all inter-element boundaries. The proposed method formulates the slope stability problem as an optimization problem based on the upper bound theorem. The objective function for determination of the minimum value of the factor of safety has a number of unknowns that are subject to a set of linear and nonlinear equality constraints as well as linear inequality constraints. The objective function and constrain equations are derived from an energy–work balance equation, the Mohr–Coulomb failure (yield) criterion, an associated flow rule, and a number of boundary conditions. The objective function with constraints leads to a standard nonlinear programming problem, which can be solved by a sequential quadratic algorithm. A computer program has been developed for finding the factor of safety of a slope, which makes the present method simple to implement. Four typical 2D and 3D slope stability problems are selected from the literature and are analysed using the present method. The results of the present limit analysis are compared with those produced by other approaches reported in the literature.Key words: limit analysis, upper bound, rigid finite element, nonlinear programming, sequential quadratic algorithm, slope stability.

An optimal force distribution scheme of multiple cooperating robots is proposed. General formulations of the optimal force distribution problem are built into quadratic programming with linear equality and quadratic inequality constraints. The quadratic constraints are due to the norm constraints of grasping forces and the approximation of maximum joint torque constraints which is used for simplification of the problem. This paper presents a technique for solving these general optimal force distribution problems. The original problem is transformed into a compact problem by eliminating the linear equality constraints. Then, the compact problem is transformed into a dual problem to be solved by using Lagrange multipliers and nonlinear programming dual method. This technique reduces the problem size considerably and makes the problem almost unconstrained so that no initial feasible points are needed. These features exhibit the efficiency of this technique. Simulation results for two cooperating PUMA robots indicating the ability of real time applications of the proposed technique are presented.

This paper explores several approaches for articulated-pose estimation, assuming that video-rate depth information is available, from either stereo cameras or other sensors. We use these depth measurements in the traditional linear brightness constraint equation, as well as in a depth constraint equation. To capture the joint constraints, we combine the brightness and depth constraints with twist mathematics. We address several important issues in the formation of the constraint equations, including updating the body rotation matrix without using a first-order matrix approximation and removing the coupling between the rotation and translation updates. The resulting constraint equations are linear on a modified parameter set. After solving these linear constraints, there is a single closed-form non-linear transformation to return the updates to the original pose parameters. We show results for tracking body pose in oblique views of synthetic walking sequences and in moving-camera views of synthetic jumping-jack sequences. We also show results for tracking body pose in side views of a real walking sequence.

Linear constraint equations for continuous support conditions in finite element analysis

We consider the problem of minimizing a convex quadratic function over a region defined by a linear equality or inequality constraint and two-sided bounds on the variables (box constraints) in R n . Such problems are interesting from both theoretical and practical points of view because they arise as subproblems of some mathematical programming problems as well as in various practical problems. Polynomial algorithms are proposed for solving problems of this form and their convergence is proved. Some examples and results of numerical experiments are also presented.

This paper presents a new method for generating motion profiles of rotating bodies, for example, the joint angle motion of mechanical systems such as space manipulators and antennas. In the planning of a motion from the given initial conditions to the final conditions, there are many constraints that should be considered, for example, actuator torque limits, rate limits, jerk limits, etc. Furthermore, flexibility of a system cannot be ignored in general. Vibration suppression is of great importance in some applications. For stationary boundary conditions (rest-to-rest cases), if the only constraint is the torque limit, a simple bang-bang type profile is the optimal one. However, if we consider more general boundary conditions, complex constraints, and vibration suppression conditions, the problem becomes considerably complicated. In this study, motion planning for a fixed motion time is considered. The control input or the acceleration of the motion is expressed as a linear superposition of triangle waves whose weight coefficients are determined in order to optimize the motion profile. In this formulation, all the boundary conditions and several constraints, including vibration suppression conditions, are expressed as linear equality and inequality constraints. Therefore, by appropriately setting the performance index, the problem becomes a linear programming problem and can be solved efficiently.

We consider the problem of optimal allocation of measurement resources, when: (1) the total measurement budget and time duration of measurements are fixed, and (2) the cost of an individual measurement varies inversely with the (controllable) measurement accuracy. The objective is to determine the time-distribution of measurement variances that minimizes a measure of error in estimating a discrete-time, vector stochastic process with known auto-correlation matrix using a linear estimator. The metric of estimation error is the trace of weighted sum of estimation error covariance matrices at various time indices. We show that this problem reduces to a nonlinear optimization problem with linear equality and inequality constraints. The solution to this problem is obtained via a variation of the projected Newton method. For the special case when the vector stochastic process is the state of a linear, finite-dimensional stochastic system, the problem reduces to the solution of a nonlinear optimal control problem. In this case, the gradient and Hessian with respect to the measurement costs are obtained via the solution of a two-point boundary value problem and the resulting optimization problem is solved via a variation of the projected Newton method. The proposed method is illustrated using four examples. >

We consider least squares semidefinite programming (LSSDP) where the primal matrix variable must satisfy given linear equality and inequality constraints, and must also lie in the intersection of the cone of symmetric positive semidefinite matrices and a simple polyhedral set. We propose an inexact accelerated block coordinate descent (ABCD) method for solving LSSDP via its dual, which can be reformulated as a convex composite minimization problem whose objective is the sum of a coupled quadratic function involving four blocks of variables and two separable nonsmooth functions involving only the first and second block, respectively. Our inexact ABCD method has the attractive $O(1/k^2)$ iteration complexity if the subproblems are solved progressively more accurately. The design of our ABCD method relies on recent advances in the symmetric Gauss--Seidel technique for solving a convex minimization problem whose objective is the sum of a multiblock quadratic function and a nonsmooth function involving only th...

Theorems of the alternative for linear algebraic equations and inequalities are considered in this paper. Classical theorems of the alternative, such as Gordan’s theorem, Farkas’ theorem, Stiemke’s theorem, Motzkin’s theorem, Slater’s theorem, Tucker’s theorem, Duffin’s theorem, Gale’s theorems, etc. are recalled. Theorems of this form are important for both linear algebra and mathematical programming, especially for mathematical programming problems with linear equality and/or inequality constraints. Some new theorems of the alternative are formulated and proved.

1-bit compressing sensing (CS) is an important class of sparse optimization problems. This paper focuses on the stability theory for 1-bit CS with quadratic constraint. The model is rebuilt by reformulating sign measurements by linear equality and inequality constraints, and the quadratic constraint with noise is approximated by polytopes to any level of accuracy. A new concept called restricted weak RSP of a transposed sensing matrix with respect to the measurement vector is introduced. Our results show that this concept is a sufficient and necessary condition for the stability of 1-bit CS without noise and is a sufficient condition if the noise is available.