Linear constraint equations for continuous support conditions in finite element analysis

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.

For a class of nonsmooth composite optimization problems with linear equality constraints, we utilize a Lyapunov-based approach to establish the global exponential stability of the primal-dual gradient flow dynamics based on the proximal augmented Lagrangian. The result holds when the differentiable part of the objective function is strongly convex with a Lipschitz continuous gradient; the non-differentiable part is proper, lower semi-continuous, and convex; and the matrix in the linear constraint is full row rank. Our quadratic Lyapunov function generalizes recent result from strongly convex problems with either affine equality or inequality constraints to a broader class of composite optimization problems with nonsmooth regularizers and it provides a worst-case lower bound of the exponential decay rate. Finally, we use computational experiments to demonstrate that our convergence rate estimate is less conservative than the existing alternatives.

Influence of welding sequence on welding deformation and residual stress of a stiffened plate structure

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.

We derive the asymptotic approximation for the posterior distribution when the data are multinomial and the prior is Dirichlet conditioned on satisfying a finite set of linear equality and inequality constraints so the posterior is also Dirichlet conditioned on satisfying these same constraints. When only equality constraints are imposed, the asymptotic approximation is normal. Otherwise it is normal conditioned on satisfying the inequality constraints. In both cases the posterior is a root-n-consistent estimator of the parameter vector of the multinomial distribution. As an application we consider the constrained Polya posterior which is a non-informative stepwise Bayes posterior for finite population sampling which incorporates prior information involving auxiliary variables. The constrained Polya posterior is a root-n-consistent estimator of the population distribution, hence yields a root-n-consistent estimator of the population mean or any other differentiable function of the vector of population probabilities.

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.

This paper develops a novel algorithm that tackles a general distributed convex optimization problem with a nonsmooth objective function and a series of coupling nonlinear inequality and linear equality constraints. To address such a problem, we first design a distributed proximal algorithm for problems with coupling equality constraints only, and then extend this algorithm to problems with both inequality and equality constraints by incorporating a virtual-queue-based method. The resulting algorithm, referred to as Proximal Primal-Dual Algorithm (PPDA), is shown to achieve O(1/k) rates of convergence with respect to both optimality and feasibility. Compared to the alternative methods in the literature, PPDA eliminates their assumptions on the smoothness of the objective function and achieves a stronger convergence rate result. In addition, PPDA exhibits faster convergence in a couple of numerical examples.

On a basic multivariate EIV model with linear equality constraints

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.

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.

We propose two limited-memory BFGS (L-BFGS) trust-region methods for large-scale optimization with linear equality constraints. The methods are intended for problems where the number of equality constraints is small. By exploiting the structure of the quasi-Newton compact representation, both proposed methods solve the trust-region subproblems nearly exactly, even for large problems. We derive theoretical global convergence results of the proposed algorithms, and compare their numerical effectiveness and performance on a variety of large-scale problems.

Systems of linear fuzzy constraints

The thermo-mechanical stresses analysis in an internally cooled first stage bucket, applying the finite element analysis (FEA) and computational fluid dynamics (CFD) is presented. In this analysis two cases are considered; first one where the model is solved using constraint equations in two zones, blade-platform and platform-root, obtaining the distribution of stresses which is in agreement with the physical problem, nevertheless due to the application of the constraint equations it was observed that the model has numerical limitations to obtain results. For the second case, without constraint equations, a geometry that include all the zones of the blade is achieved, generating a new mesh on which the same analysis was realized (thermo-mechanic) obtaining similar magnitudes of stresses like the first one model, but without numerical limitations. In both cases, the analysis was done using a complete model in 3D including the internal cooling channels. To carry out this analysis it was necessary to apply an interface FLUENT-ANSYS (FAI) to transfer the temperature data from an analysis obtained with a CFD model in steady state to the finite element model. A methodology was developed to eliminate to constraint equations in the finite element model across skills of union elements to achieve a uniform mesh in the whole model. The results show that an analysis without constraint equations, to avoid geometric simplifications and numerical limitations, permitting to obtain better results in the analysis of thermo-mechanics stress in blades of a gas turbine.

This paper studies a class of network optimization problems where the objective function is the summation of individual agents' convex functions and their decision variables are coupled by linear equality constraints. These constraints are not sparse, meaning that they do not match the pattern of the network adjacency matrix. We propose two approaches to design efficient distributed algorithms to solve the network optimization problem. Our first approach consists of transforming the non-sparse equality constraints into sparse ones by increasing the number of the agents' decision variables, yielding an exact reformulation of the original optimization problem. We discuss two reformulations, based on the addition of consensus variables and of constraint-mismatch variables, and discuss the scalability of the strategies resulting from them. Our second approach consists instead of sparsifying the non-sparse constraints by zeroing some coefficients, yielding an approximate reformulation of the original problem. We formally characterize the gap on the distance between the optimizers of the original and approximated problems as a function of the number of entries made zero in the constraints. Various simulations illustrate our results.

Static optimization with linear equality constraints and separable structures is studied by using the mixed coordination method. The idea is to relax equality constraints via Lagrange multipliers, and create a hierarchy where the Lagrange multipliers and part of the decision variables are selected as high-level variables. The method was proposed about ten years ago with a simple high-level updating scheme. We show that the solution of the high-level problem is a saddle point, and the simple updating scheme has a linear convergence rate under appropriate conditions. To obtain faster convergence, the modified Newton method is adopted at the high level. There are two difficulties associated with this approach. One is how to obtain the hessian matrix in determining the Newton direction, since second-order derivatives of the objective function with respect to all high-level variables are needed. The second is when to stop in performing a line search along the Newton direction, as the high-level problem is a maxmini problem looking for a saddle point. In this paper, the hessian matrix is obtained by using a kind of sensitivity analysis. The line search stopping criterion, on the other hand, is based on the norm of the gradient vector. Extensive numerical testings show that our approach performs much better than the simple high-level updating scheme. Since the low level consists of a set of independent subproblems, this method is well suited for parallel implementation in solving large-scale problems. Simulated parallel-processing results show that our method outperforms the one-level Lagrange relaxation method for all the test problems. Furthermore, since convexification terms can be added while maintaining the separability of low-level subproblems, the method is very promising for non-convex problems.