Active Set Technique Research Articles

An infeasible QP-free method without a penalty function for nonlinear inequality constrained optimization

In this paper, we present a new QP-free method without using a penalty function for inequality constrained optimization. It is an infeasible method. At each iteration, three linear equations with the same coefficient matrix are solved. The nearly active set technique is used in the algorithm, which eliminates some inactive constraints and reduces the dimension of coefficient matrix, and thereby reduces the amount of computational work. Moreover, the algorithm reduces the value of objective function or the measure of constraints violation according to the relationship between optimality and feasibility. Under common conditions, we prove that the proposed method has global and superlinear local convergence. Lastly, preliminary numerical results are reported.

Numeric treatment of nonlinear second order multi-point boundary value problems using ANN, GAs and sequential quadratic programming technique

In this paper, computational intelligence technique are presented for solving multi - point nonlinear boundary value problems based on artificial neural networks, evolutionary computing approach, and active-set technique. The neural network is to provide convenient methods for obtaining useful model based on unsupervised error for the differential equations. The motivation for presenting this work comes actually from the aim of introducing a reliable framework that combines the powerful features of ANN optimized with soft computing frameworks to cope with such challenging system. The applicability and reliability of such methods have been monitored thoroughly for various boundary value problems arises in science, engineering and biotechnology as well. Comprehensive numerical experimentations have been performed to validate the accuracy, convergence, and robustness of the designed scheme. Comparative studies have also been made with available standard solution to analyze the correctness of the proposed scheme.

A Spline Smoothing Newton Method for Semi-Infinite Minimax Problems

Based on discretization methods for solving semi-infinite programming problems, this paper presents a spline smoothing Newton method for semi-infinite minimax problems. The spline smoothing technique uses a smooth cubic spline instead of max function and only few components in the max function are computed; that is, it introduces an active set technique, so it is more efficient for solving large-scale minimax problems arising from the discretization of semi-infinite minimax problems. Numerical tests show that the new method is very efficient.

A Spline Smoothing Newton Method for L∞ Distance Regression with Bound Constraints

Orthogonal distance regression is arguably the most common criterion for fitting a model to data with errors in the observations. It is not appropriate to force the distances to be orthogonal, when angular information is available about the measured data points. We consider here a natural generalization of a particular formulation of that problem which involves the replacement of l2 norm by l∞ norm. This criterion may be a more appropriate one in the context of accept/reject decisions for manufacture parts. For l∞ distance regression with bound constraints, we give a smoothing Newton method which uses cubic spline and aggregate function, to smooth max function. The main spline smoothing technique uses a smooth cubic spline instead of max function and only few components in the max function are computed; hence it acts also as an active set technique, so it is more efficient for the problem with large amounts of measured data. Numerical tests in comparison to some other methods show that the new method is very efficient.

A Semismooth Active-set Algorithm for Degenerate Nonlinear Complementarity Problems

We propose a semismooth active-set Newton algorithm for solving the nonlinear complementarity problems with degenerate solutions. This method introduces the active-set technique to identify the degenerate set. At each iteration, the search direction is obtained by two reduced linear systems. Instead of employing gradient steps as adjustments to guarantee the sufficient reduction of the merit function, the algorithm employs a Newton-type direction, which is more efficient than gradient direction, in the adjustment step. This method has globally convergence. When near the solution, the degenerate set will be identified correctly, and only one reduced linear system is solved at each iteration. Under some mild assumptions, locally superlinear convergence is obtained as well. Numerical experiments on MATLAB shows the efficiency of the method.

A new convergence theory for trust-region algorithm for solving constrained optimization problems

In this paper, we propose a new trust-region algorithm for solving a constrained optimization problem with equality and inequality constraints. In this algorithm, an active-set technique is used to convert the constrained optimization problem with equality and inequality constraints to equality constrained optimization problem. A projected Hessian technique is used together with a conjugate gradient method to compute the trial step. Global convergence results are established under important assumptions and it is shown that a subsequence of the iteration sequence is not bounded away from KKT points. Preliminary numerical experiment on the algorithm is presented. The performance of the algorithm is reported. The numerical results show that our approach is of value and merit further investigation.

Fault-Tolerant Control of a Nonlinear Process with Input Constraints

A Fault-Tolerant Control (FTC) methodology has been presented for nonlinear processes being imposed by control input constraints. The proposed methodology uses a combination of Feedback Linearization and Model Predictive Control (FLMPC) schemes. The resulting constraints in the transformed process will be dependent on the actual evolving states, making their incorporation in the design context a non-trivial task. A feasible direction method has been integrated in the design procedure based on active set technique to resolve the challenging constraint–based FLMPC problem. The formulated FLMPC design method is utilized to develop a FTC scheme by providing a set of backup control configurations for a CSTR benchmark process. The successful performance of the proposed FTC methodology has been demonstrated via a category of common fault scenarios by exercising an arbitrary replacement of control configurations through a supervisor to maintain the CSTR operation at an unstable desired steady-state point.

AN ACTIVE SET SQP-FILTER METHOD FOR SOLVING NONLINEAR PROGRAMMING

Sequential quadratic programming (SQP) has been one of the most important methods for solving nonlinear constrained optimization problems. Recently, filter method, proposed by Fletcher and Leyffer, has been extensively applied for its promising numerical results. In this paper, we present and study an active set SQP-filter algorithm for inequality constrained optimization. The active set technique reduces the size of quadratic programming (QP) subproblem. While by the filter method, there is no penalty parameter estimate. Moreover, Maratos effect can be overcome by filter technique. Global convergence property of the proposed algorithm are established under suitable conditions. Some numerical results are reported in this paper.

AN INFEASIBLE SSLE FILTER ALGORITHM FOR GENERAL CONSTRAINED OPTIMIZATION WITHOUT STRICT COMPLEMENTARITY

In this paper, we propose a new sequential systems of linear equations (SSLE) filter algorithm, which is an infeasible QP-free method. The new algorithm needs to solve a few reduced systems of linear equations with the same nonsingular coefficient matrix, and after finitely many iterations, only two linear systems need to be solved. Furthermore, the nearly active set technique is used to improve the computational effect. Under the linear independence condition, the global convergence is proved. In particular, the rate of convergence is proved to be one-step superlinear without assuming the strict complementarity condition. Numerical results and comparison with other algorithms indicate that the new algorithm is promising.

A New Trust-Region Algorithm for Nonlinear Constrained Optimization

We propose a new trust region algorithm for nonlinear constrained optimization problems. In each iteration of our algorithm, the trial step is computed by minimizing a quadratic approximation to the augmented Lagrange function in the trust region. The augmented Lagrange function is also used as a merit function to decide whether the trial step should be accepted. Our method extends the traditional trust region approach by combining a fllter technique into the rules for accepting trial steps so that a trial step could still be accepted even when it is rejected by the traditional rule based on merit function reduction. An estimate of the Lagrange multiplier is updated at each iteration, and the penalty parameter is updated to force su‐cient reduction in the norm of the constraint violations. Active set technique is used to handle the inequality constraints. Numerical results for a set of constrained problems from the CUTEr collection are also reported.

A Moving Balls Approximation Method for a Class of Smooth Constrained Minimization Problems

We introduce a new algorithm for a class of smooth constrained minimization problems which is an iterative scheme that generates a sequence of feasible points that approximates the constraints set by a sequence of balls and is accordingly called the Moving Balls Approximation algorithm (MBA). The computational simplicity of MBA, which uses first order data information, makes it suitable for large scale problems. Theoretical and computational properties of MBA in its primal and dual forms are studied, and convergence and global rate of convergence results are established for nonconvex and convex problems. We then introduce a variant of MBA that includes an active set technique and is particularly suitable for problems with a large number of constraints. This variant is as simple as MBA and is proven to preserve the convergence properties of MBA. Extension of MBA is also developed for a class of variational inequalities. Initial numerical experiments on quadratically constrained problems demonstrate the viability and performance of our methods when compared to some existing state-of-the-art optimization methods/software such as a sequential quadratic programming solver from the IMSL Library and the CVXOPT software package for convex optimization.

Global convergence on an active set SQP for inequality constrained optimization

Sequential quadratic programming (SQP) has been one of the most important methods for solving nonlinearly constrained optimization problems. In this paper, we present and study an active set SQP algorithm for inequality constrained optimization. The active set technique is introduced which results in the size reduction of quadratic programming (QP) subproblems. The algorithm is proved to be globally convergent. Thus, the results show that the global convergence of SQP is still guaranteed by deleting some “redundant” constraints.

Error Estimates for the Adaptive Computation of a Scalar Three Well Problem

We investigate the numerical approximation of Young measure solutions appearing as generalised solutions in scalar non-convex variational problems. A priori and a posteriori error estimates for a macroscopic quantity, i.e., the stress, are given. Numerical experiments for a scalar three well problem, occurring as a subproblem in the theory of phase transitions in crystalline solids, show that the computational effort can be significantly reduced using an adaptive mesh-refinement strategy combined with an active set technique by Carstensen and Roubíček.