Active Set Strategy Research Articles

응답량 재사용을 통한 순차 근사최적설계

A general approximate optimization technique by sequential design domain(SDD) did not save response values for getting an approximate functions in each step. It has a disadvantage at aspect of an expense. In this paper, previous response values are recycled for constructing an approximate functions. For this reason, approximation functions is more accurate. Accordingly, even if we did not determine move limit, a system is converged to the optimal design. Size an shape optimization using approximate optimization technique is carried out with SDD. Algorithm executing Pro/Engineer and ANSYS are automatically adopted in the approximate optimization program by SDD. Convergence criterion is defined such that optimal point must be located within SDD during the three steps. The PLBA(Pshenichny-Lim-Belegundu-Arora) algorithm is used to solve approximate optimization problems. This algorithm uses the second-order information in the directions finding problem and used the active set strategy.

A Primal-Dual Active Set Strategy for Optimal Boundary Control of a Nonlinear Reaction-Diffusion System

This paper is concerned with optimal boundary control of an instationary reaction-diffusion system in three spatial dimensions. This problem involves a coupled nonlinear system of parabolic differential equations with bilateral as well as integral control constraints. We include the integral constraint in the cost by a penalty term whereas the bilateral control constraints are handled explicitly. First- and second-order conditions for the optimization problem are analyzed. A primal-dual active set strategy is utilized to compute optimal solutions numerically. The algorithm is compared to a semismooth Newton method.

A priori error estimates and an inexact primal-dual active set strategy for linear and quadratic finite elements applied to multibody contact problems

Nonconforming domain decomposition methods and their application to the numerical simulation of non-linear multibody contact problems play an important role in many applications in mechanics. To handle the non-linearity of the contact conditions, we apply a primal-dual active set strategy based on dual Lagrange multipliers. Combining this method with an optimal multigrid for the resulting linear algebraic problems and using inexact strategies, our algorithm yields an efficient iterative solver. Furthermore, we establish, under some regularity assumptions on the solution, optimal convergence orders for the discretization errors for the displacement and the Lagrange multiplier for linear and quadratic finite element spaces; we combine quadratic finite elements with linear and quadratic dual Lagrange multipliers. Several numerical examples confirm our theoretical results. In the last section, we extend our algorithm to a dynamic non-linear multibody contact problem.

A primal–dual active set strategy for non-linear multibody contact problems

Non-conforming domain decomposition methods provide a powerful tool for the numerical approximation of partial differential equations. For the discretization of a non-linear multibody contact problem, we use the mortar approach with a dual Lagrange multiplier space. To handle the non-linearity of the contact conditions, we apply a primal–dual active set strategy to find the actual contact zone. The algorithm can be easily generalized to multibody contact problems. A suitable basis transformation guarantees the same algebraic structure in the multibody situation as in the one body case. Using an inexact primal–dual active set strategy in combination with a multigrid method yields an efficient iterative solver. Different numerical examples for one and multibody contact problems illustrate the performance of the method.

Solving rank one revised linear systems by the scaled ABS method

AbstractIn mathematical programming, an important tool is the use of active set strategies to update the current solution of a linear system after a rank one change in the constraint matrix. We show how to update the general solution of a linear system obtained by use of the scaled ABS method when the matrix coefficient is subjected to a rank one change.

Active Set Support Vector Regression

This paper presents active set support vector regression (ASVR), a new active set strategy to solve a straightforward reformulation of the standard support vector regression problem. This new algorithm is based on the successful ASVM algorithm for classification problems, and consists of solving a finite number of linear equations with a typically large dimensionality equal to the number of points to be approximated. However, by making use of the Sherman-Morrison-Woodbury formula, a much smaller matrix of the order of the original input space is inverted at each step. The algorithm requires no specialized quadratic or linear programming code, but merely a linear equation solver which is publicly available. ASVR is extremely fast, produces comparable generalization error to other popular algorithms, and is available on the web for download.

Active-Set Projected Trust-Region Algorithm for Box-Constrained Nonsmooth Equations

In this paper, by means of an active-set strategy, we present a trust-region method for solving box-constrained nonsmooth equations. Nice properties of the proposed method include: (a) all iterates remain feasible; (b) the search direction, as adequate combination of the projected gradient direction and the trust-region direction, is an asymptotic Newton direction under mild conditions; (c) the subproblem of the proposed method, possessing the form of an unconstrained trust-region subproblem, can be solved by existing methods; (d) the subproblem of the proposed method is of reduced dimension, which is potentially cheaper when applied to solve large-scale problems. Under appropriate conditions, we establish global and local superlinear/quadratic convergence of the method. Preliminary numerical results are given.

Approximate Optimization Using RBF Network and Genetic Range Genetic Algorithms (Proposal of Base Function and Basic Consideration)

In this paper, we will propose integrated optimization system that is composed by RBF network as approximation tool and Genetic Range Genetic Algorithms as optimizer. This system has function of optimizing radius of basis function in optimization, indication of next searching points by using basis function and optimization by using Genetic Range Genetic Algorithms that is a step further of Adaptive Range Genetic Algorithms. Key success in approximation by using RBF network lies in giving good data sets. Here, “good data sets” means that they can update accuracy around local optimum points, and also they can give global information of original function's response surface. In that sense, indication function is quite important in this system. For that purpose, we introduce base function which is composed by approximation RBF network. We demonstrated the proposed system to a bench mark problem to show efficiency of the proposed method. In constraint optimization case, it works something like active set strategy, and we can reduce the number of function calls.

Adaptive Approximation of Young Measure Solutions in Scalar Nonconvex Variational Problems

This paper addresses the numerical approximation of Young measures appearing as generalized solutions to scalar nonconvex variational problems. We prove a priori and a posteriori error estimates for a macroscopic quantity, the stress. For a scalar three-well problem we show convergence of other quantities such as Young measure support and microstructure region. Numerical experiments indicate that the computational effort in the solution of the large optimization problem is significantly reduced by using an adaptive mesh refinement strategy based on a posteriori error estimates in combination with an active set strategy due to Carstensen and Roubicek [Numer. Math., 84 (2000), pp. 395--414].

Semismooth Newton and Augmented Lagrangian Methods for a Simplified Friction Problem

In this paper a simplified friction problem and iterative second-order algorithms for its solution are analyzed in infinite dimensional function spaces. Motivated from the dual formulation, a primal-dual active set strategy and a semismooth Newton method for a regularized problem as well as an augmented Lagrangian method for the original problem are presented and their close relation is analyzed. Local as well as global convergence results are given. By means of numerical tests, we discuss among others convergence properties, the dependence on the mesh, and the role of the regularization and illustrate the efficiency of the proposed methodologies.

Augmented Lagrangian Active Set Methods for Obstacle Problems

Active set strategies for two-dimensional and three-dimensional, unilateral and bilateral obstacle problems are described. Emphasis is given to algorithms resulting from the augmented Lagrangian (i.e., primal-dual formulation of the discretized obstacle problems), for which convergence and rate of convergence are considered. For the bilateral case, modifications of the basic primal-dual algorithm are also introduced and analyzed. Finally, efficient computer realizations that are based on multigrid and multilevel methods are suggested and different aspects of the proposed techniques are investigated through numerical experiments.

Active set strategies in an ellipsoid algorithm for nonlinear programming

The classical ellipsoid algorithm solves convex nonlinear programming problems having feasible sets of full dimension. Convergence is certain only for the convex case (Math. Oper. Res. 10 (1985) 515), but the algorithm often works in practice for nonconvex problems as well (SIAM J. Control Optim. 23 (1985) 657). Shah's algorithm (Comput. Oper. Res. 20 (2001) 85) modifies the classical method to permit the solution of nonlinear programs including equality constraints. This paper describes a robust restarting procedure for Shah's algorithm and investigates two active set strategies to improve computational efficiency. Experimental results are presented to show the new algorithm is effective, and usually faster than Shah's algorithm, for a wide variety of convex and nonconvex nonlinear programs with inequality and equality constraints. We also demonstrate that the algorithm can be used to solve systems of nonlinear equations and inequalities, including Karush–Kuhn–Tucker conditions. Scope and purpose The purpose of this paper is to describe an ellipsoid algorithm using two active set strategies, and to provide a careful experimental characterization of its performance on a significant set of general nonlinear programming test problems.

A global convergence theory for an active-trust-region algorithm for solving the general nonlinear programing problem

A new trust-region active-set algorithm for solving minimizing a nonlinear function subject to nonlinear equality and inequality constraints is described. In this algorithm, an active set strategy is used together with a projected Hessian technique to compute the trial step. A convergence theory for this algorithm is presented. Under important assumptions, it is shown that the algorithm is globally convergent. In particular, it is shown that a subsequence of the iteration sequence is not bounded away from either Fritz–John points or KKT points.

Semi–Smooth Newton Methods for Variational Inequalities of the First Kind

Semi–smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions. It is shown that they are equivalent to certain active set strategies. Global and local super-linear convergence are proved. To overcome the phenomenon of finite speed of propagation of discretized problems a penalty version is used as the basis for a continuation procedure to speed up convergence. The choice of the penalty parameter can be made on the basis of an L ∞ estimate for the penalized solutions. Unilateral as well as bilateral problems are considered.

Tube Methods for BV Regularization

In this paper tube methods for reconstructing discontinuous data from noisy and blurred observation data are considered. It is shown that discrete bounded variation (BV)-regularization (commonly used in inverse problems and image processing) and the taut-string algorithm (commonly used in statistics) select reconstructions in a tube. A version of the taut-string algorithm applicable for higher dimensional data is proposed. This formulation results in a bilateral contact problem which can be solved very efficiently using an active set strategy. As a by-product it is shown that the Lagrange multiplier of the active set strategy is an efficient parameter for edge detection.

Lagrange-SQP Techniques for the Control Constrained Optimal Boundary Control for the Burgers Equation

A Lagrange-Newton-SQP method is analyzed for the optimal control of the Burgers equation. Boundary controls are given, which are restricted by pointwise lower and upper bounds. The convergence of the method is proved in appropriate Banach spaces. This proof is based on a second-order sufficient optimality condition and the theory of Newton methods for generalized equations in Banach spaces. For the numerical realization a primal-dual active set strategy is applied. To illustrate the theoretical investigations, numerical examples are included. Moreover, a globalization technique for the SQP method is tested numerically.

New projection-type methods for monotone LCP with finite termination

In this paper we establish two new projection-type methods for the solution of monotone linear complementarity problem (LCP). The methods are a combination of the extragradient method and the Newton method, in which the active set strategy is used and only one linear system of equations with lower dimension is solved at each iteration. It is shown that under the assumption of monotonicity, these two methods are globally and linearly convergent. Furthermore, under a nondegeneracy condition they have a finite termination property. At last, the methods are extended to solving monotone affine variational inequality problem.

A Dual‐Type Method Based Algorithm for Nonlinear Large Network Optimization Problems

ABSTRACTIn previous research, we have proposed a Dual Projected Pseudo Quasi Newton (DPPQN) method which differs from the conventional Lagrange relaxation method by treating the inequality constraints as the domain of the primal variables in the dual function and using Projection Theory to handle the inequality constraints. We have combined this dual‐type method with a Projected Jacobi (PJ) method to solve nonlinear large network optimization problems with decomposable inequality constraints, and have achieved several attractive features.To retain the attractive features and to remedy the flaw of the previous method, in the current paper, we propose an active set strategy based DPPQN method to solve the projection problem formed by coupling functional inequality constraints. This method associated with the DPPQN method and the PJ method can be used to solve general nonlinear large network optimization problems. We present this algorithm, demonstrate its computational efficiency through numerical simulations and compare it with the previous method.

Globally and Superlinearly Convergent QP-Free Algorithm for Nonlinear Constrained Optimization

A new, infeasible QP-free algorithm for nonlinear constrained optimization problems is proposed. The algorithm is based on a continuously differentiable exact penalty function and on active-set strategy. After a finite number of iterations, the algorithm requires only the solution of two linear systems at each iteration. We prove that the algorithm is globally convergent toward the KKT points and that, if the second-order sufficiency condition and the strict complementarity condition hold, then the rate of convergence is superlinear or even quadratic. Moreover, we incorporate two automatic adjustment rules for the choice of the penalty parameter and make use of an approximated direction as derivative of the merit function so that only first-order derivatives of the objective and constraint functions are used.

Improvements on the duality based method used in solving optimal power flow problems

To improve the previously developed dual-type (DT) method used in solving optimal power flow (OPF) problems with large number of thermal-limit constraints, the authors propose two new techniques in this paper. The first one is a graph-method based decomposition technique which can decompose the large-dimension projection problem, caused by the large number of thermal-limit constraints, into several independent medium-dimension projection subproblems at the expense of slight increment of the dual problem's dimension. The second technique is an active-set strategy based DT method which can solve the medium-dimension projection subproblems efficiently. They have used the DT method embedded with these two new techniques in solving numerous OPFs with large number of thermal-limit constraints. The test results show that the proposed techniques are very efficient and effectively improve the DT method for handling large number of thermal-limit constraints.