Modified Barrier Method Research Articles

Hierarchical coordinated control distribution and experimental verification for six-wheeled unmanned ground vehicles

In recent years, the all-wheel independent drive has been the most promising form of drive configuration in unmanned ground vehicles. Considering the difficulties in the control allocation for this kind of vehicle, this paper presents a hierarchical control coordination strategy with three layers to distribute control in real time effectively and accurately. In the upper layer, a hybrid instruction parsing method is proposed, which converts commands of the control panel into driving force requirement and target steering yaw rate, respectively, to prioritize steering command to maintain the trajectory based on the motor properties. Subsequently, a sliding mode controller is employed to convert the target yaw rate into the required yaw moment. The state estimation layer receives data from the sensors and estimates different properties/parameters required in other layers. The lower-level control layer receives commands from the upper layer and allocates respective control to wheels. The control allocation problem has been formulated as an optimization problem and later has been converted into a quadratic programming problem, in which a novel modified barrier method with the combination of reduced equation dimension has been adopted to minimize the computational effort and complexity for implementation on the embedded platform. Computer simulation and field experiment have been conducted, which verify the performance of the proposed strategy.

Exploiting sensor mobility and covariance sparsity for distributed tracking of multiple sparse targets

The problem of distributed tracking of multiple targets is tackled by exploiting sensor mobility and the presence of sparsity in the sensor data covariance matrix. Sparse matrix decomposition relying on norm-one/two regularization is integrated with a kinematic framework to identify informative sensors, associate them with the targets, and enable them to follow closely the moving targets. Coordinate descent techniques are employed to determine in a distributed way the target-informative sensors, while the modified barrier method is employed to minimize proper error covariance matrices acquired by extended Kalman filtering. Different from existing approaches which force all sensors to move, here, local updating recursive rules are obtained only for the target-informative sensors that can update their location and follow closely the corresponding targets while staying connected. Simulations advocate that the proposed scheme outperforms alternative tracking schemes while accurately tracks multiple targets by imposing mobility only on the target-informative sensors.

Loss minimization by the predictor–corrector modified barrier approach

This paper presents a new approach, predictor–corrector modified barrier approach (PCMBA), to minimize the active losses in power system planning studies. In the PCMBA, the inequality constraints are transformed into equalities by introducing positive auxiliary variables, which are perturbed by the barrier parameter, and treated by the modified barrier method. The first-order necessary conditions of the Lagrangian function are solved by predictor–corrector Newton's method. The perturbation of the auxiliary variables results in an expansion of the feasible set of the original problem, reaching the limits of the inequality constraints. The feasibility of the proposed approach is demonstrated using various IEEE test systems and a realistic power system of 2256-bus corresponding to the Brazilian South-Southeastern interconnected system. The results show that the utilization of the predictor–corrector method with the pure modified barrier approach accelerates the convergence of the problem in terms of the number of iterations and computational time.

On the solution of large-scale SDP problems by the modified barrier method using iterative solvers

The limiting factors of second-order methods for large-scale semidefinite optimization are the storage and factorization of the Newton matrix. For a particular algorithm based on the modified barri...

On the solution of large-scale SDP problems by the modified barrier method using iterative solvers

The limiting factors of second-order methods for large-scale semidefinite optimization are the storage and factorization of the Newton matrix. For a particular algorithm based on the modified barrier method, we propose to use iterative solvers instead of the routinely used direct factorization techniques. The preconditioned conjugate gradient method proves to be a viable alternative for problems with a large number of variables and modest size of the constrained matrix. We further propose to avoid explicit calculation of the Newton matrix either by an implicit scheme in the matrix---vector product or using a finite-difference formula. This leads to huge savings in memory requirements and, for certain problems, to further speed-up of the algorithm.

Application of the modified barrier method in large-scale quadratic programming problems

The application of the penalty/modified barrier function method (PE/MBF) is presented for the solution of large-scale positive-semidefinite quadratic programming problems (QP). A review of the recent literature on QP methods is presented and the choice of the PE/MBF method for QP problems is justified by previous experience in very large-scale bound-constrained problems. The proposed algorithm performs two types of iterations: an outer iteration in which the Lagrange multipliers of the bounds are adjusted, and an inner iteration for the solution of an equality constrained subproblem. The inner iteration solves a modified problem, containing penalty/modified barrier terms for the bounds in the objective, and is subject to equality constraints only. The equality constraints are handled directly via the use of additional Lagrange multipliers during the inner iteration and thus, instead of an unconstrained problem, the inner iteration solves a modified equality constrained problem. Any inequality constraints, other than bounds, are formulated as equalities via the use of slack variables. Computational results show this method to be promising, and motivate further investigation for the general case of nonlinear programming problems.

The Newton modified barrier method for QP problems

The Modified Barrier Functions (MBF) have elements of both Classical Lagrangians (CL) and Classical Barrier Functions (CBF). The MBF methods find an unconstrained minimizer of some smooth barrier function in primal space and then update the Lagrange multipliers, while the barrier parameter either remains fixed or can be updated at each step. The numerical realization of the MBF method leads to the Newton MBF method, where the primal minimizer is found by using Newton's method. This minimizer is then used to update the Lagrange multipliers. In this paper, we examine the Newton MBF method for the Quadratic Programming (QP) problem. It will be shown that under standard second-order optimality conditions, there is a ball around the primal solution and a cut cone in the dual space such that for a set of Lagrange multipliers in this cut cone, the method converges quadratically to the primal minimizer from any point in the aforementioned ball, and continues, to do so after each Lagrange multiplier update. The Lagrange multipliers remain within the cut cone and converge linearly to their optimal values. Any point in this ball will be called a “hot start”. Starting at such a “hot start”, at mostO(In Inɛ-1) Newton steps are sufficient to perform the primal minimization which is necessary for the Lagrange multiplier update. Here, ɛ>0 is the desired accuracy. Because of the linear convergence of the Lagrange multipliers, this means that onlyO(Inɛ-1)O(In Inɛ-1) Newton steps are required to reach an ɛ-approximation to the solution from any “hot start”. In order to reach the “hot start”, one has to perform\(\mathcal{O}(\sqrt m {\text{ }}In{\text{ }}C)\) Newton steps, wherem characterizes the size of the problem andC>0 is the condition number of the QP problem. This condition number will be characterized explicitly in terms of key parameters of the QP problem, which in turn depend on the input data and the size of the problem.

Application of the modified barrier method in large-scale Quadratic Programming problems

In this paper we present the application of the Penalty/Modified Barrier Function method (PE/MBF) in the solution of large-scale Quadratic Programming problems (QP). A review of the recent literature on PE/MBF methods is presented and the choice of this method for QP problems is justified by previous experience in very large-scale bound-constrained problems. The proposed algorithm performs two types of iterations: an outer iteration in which the Lagrange multipliers of the bounds are adjusted, and an inner iteration for the solution of an equality constrained subproblem. The inner iteration solves a modified problem, containing penalty/modified barrier terms for the bounds in the objective and is subject to equality constraints only. The equality constraints are handled directly via the use of additional Lagrange multipliers during the inner iteration and thus, instead of an unconstrained problem, the inner iteration solves a modified equality constrained problem. Any inequality constraints, other than bounds, are formulated as equalities via the use of slack variables. Computational results show this method to be promising, and motivate further investigation for the general case of nonlinear programming problems.

Structural optimization using the Newton modified barrier method

The Newton Modified Barrier Method (NMBM) is applied to structural optimization problems with large a number of design variables and constraints. This nonlinear mathematical programming algorithm was based on the Modified Barrier Function (MBF) theory and the Newton method for unconstrained optimization. The distinctive feature of the NMBM method is the rate of convergence that is due to the fact that the design remains in the Newton area after each Lagrange multiplier update. This convergence characteristic is illustrated by application to structural problems with a varying number of design variables and constraints. The results are compared with those obtained by optimality criteria (OC) methods and by the ASTROS program.

The convergence of a modified barrier method for convex programming

We show, using elementary considerations, that a modified barrier function method for the solution of convex programming problems converges for any fixed positive setting of the barrier parameter. With mild conditions on the primal and dual feasible regions, we show how to use the modified barrier function method to obtain primal and dual optimal solutions, even in the presence of degeneracy. We illustrate the argument for convergence in the case of linear programming, and then generalize it to the convex programming case.

Application of newton modified barrier method to structural optimization

This paper presents the application of the Newton modified barrier method to obtain a minimum weight structure with constraints on displacements and minimum sizes. The solution to the problem is obtained via minimizing the modified barrier function (MBF) at each step by using the Newton method and updating Lagrange multipliers. The Lagrange multipliers are updated by using the value of the constraints at the minimum of the MBF. Three truss problems with a different number of design variables are solved. The convergence to the minimum weight design was found to be monotonic and the algorithm is potentially robust for solving problems with a large number of design variables.

Modified barrier functions (theory and methods)

The nonlinear rescaling principle employs monotone and sufficiently smooth functions to transform the constraints and/or the objective function into an equivalent problem, the classical Lagrangian which has important properties on the primal and the dual spaces. The application of the nonlinear rescaling principle to constrained optimization problems leads to a class of modified barrier functions (MBF's) and MBF Methods (MBFM's). Being classical Lagrangians (CL's) for an equivalent problem, the MBF's combine the best properties of the CL's and classical barrier functions (CBF's) but at the same time are free of their most essential deficiencies. Due to the excellent MBF properties, new characteristics of the dual pair convex programming problems have been found and the duality theory for nonconvex constrained optimization has been developed. The MBFM have up to a superlinear rate of convergence and are to the classical barrier functions (CBF's) method as the Multipliers Method for Augmented Lagrangians is to the Classical Penalty Function Method. Based on the dual theory associated with MBF, the method for the simultaneous solution of the dual pair convex programming problems with up to quadratic rates of convergence have been developed. The application of the MBF to linear (LP) and quadratic (QP) programming leads to a new type of multipliers methods which have a much better rate of convergence under lower computational complexity at each step as compared to the CBF methods. The numerical realization of the MBFM leads to the Newton Modified Barrier Method (NMBM). The excellent MBF properties allow us to discover that for any nondegenerate constrained optimization problem, there exists a "hot" start, from which the NMBM has a better rate of convergence, a better complexity bound, and is more stable than the interior point methods, which are based on the classical barrier functions.