Quadratic Convergence Property Research Articles

A New Searching Direction for Linear Complementarity Problems

In this paper, we investigate the properties of a simple function. As an application, we present a full-step interior-point algorithm for linear complementarity problem. The algorithm uses the simple function to determine the searching direction and define the neighborhood of central path. The full-step used in the algorithm has local quadratic convergence property according to the proximity function which is also constructed by this simple function. We derive the iteration complexity for the algorithm and obtain the best-known iteration bounds for linear complementarity problem.

A full-Newton step interior-point algorithm based on modified-Newton direction

By a modification of the classic-Newton direction in scaled version for linear optimization, we give a new interior-point algorithm based on a very simple function. The algorithm uses full modified-Newton step, thus no need to perform line search. In the processing of the algorithm, the simple function is used to control the searching direction and measure the proximity of iterates to the central path. Moreover, the modified-Newton step used in the algorithm has local quadratic convergence property according to the proximity function. The iteration complexity is derived, and which is the best-known.

Local convergence of filter methods for equality constrained non-linear programming

In Gonzaga et al. [A globally convergent filter method for nonlinear programming, SIAM J. Optimiz. 14 (2003), pp. 646–669] we discuss general conditions to ensure global convergence of inexact restoration filter algorithms for non-linear programming. In this article we show how to avoid the Maratos effect by means of a second-order correction. The algorithms are based on feasibility and optimality phases, which can be either independent or not. The optimality phase differs from the original one only when a full Newton step for the tangential minimization of the Lagrangian is efficient but not acceptable by the filter method. In this case a second-order corrector step tries to produce an acceptable point keeping the efficiency of the rejected step. The resulting point is tested by trust region criteria. Under the usual hypotheses, the algorithm inherits the quadratic convergence properties of the feasibility and optimality phases. This article includes a comparison between classical Sequential Quadratic Programming (SQP) and Inexact Restoration (IR) iterations, showing that both methods share the same asymptotic convergence properties.

On Smooth LU Decompositions with Applications to Solutions of Nonlinear Eigenvalue Problems

We study the smooth LU decomposition of a given analytic functional �-matrix A(�) and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided. By using these smooth LU decompositions, we propose two numerical methods for computing multiple nonlinear eigenvalues of A(�), and establish their locally quadratic convergence properties. Several numerical examples are provided to show the feasibility and effectiveness of these new methods.

Sequential quadratically constrained quadratic programming norm-relaxed algorithm of strongly sub-feasible directions

Abstract In this paper, we present a sequential quadratically constrained quadratic programming (SQCQP) norm-relaxed algorithm of strongly sub-feasible directions for the solution of inequality constrained optimization problems. By introducing a new unified line search and making use of the idea of strongly sub-feasible direction method, the proposed algorithm can well combine the phase of finding a feasible point (by finite iterations) and the phase of a feasible descent norm-relaxed SQCQP algorithm. Moreover, the former phase can preserve the “sub-feasibility” of the current iteration, and control the increase of the objective function. At each iteration, only a consistent convex quadratically constrained quadratic programming problem needs to be solved to obtain a search direction. Without any other correctional directions, the global, superlinear and a certain quadratic convergence (which is between 1-step and 2-step quadratic convergence) properties are proved under reasonable assumptions. Finally, some preliminary numerical results show that the proposed algorithm is also encouraging.

Truncated regularized Newton method for convex minimizations

Recently, Li et al. (Comput. Optim. Appl. 26:131---147, 2004) proposed a regularized Newton method for convex minimization problems. The method retains local quadratic convergence property without requirement of the singularity of the Hessian. In this paper, we develop a truncated regularized Newton method and show its global convergence. We also establish a local quadratic convergence theorem for the truncated method under the same conditions as those in Li et al. (Comput. Optim. Appl. 26:131---147, 2004). At last, we test the proposed method through numerical experiments and compare its performance with the regularized Newton method. The results show that the truncated method outperforms the regularized Newton method.

A numerical algorithm for finding solution of sign-indefinite algebraic Riccati equations for general multiparameter singularly perturbed systems

In this paper, a computational algorithm for solving sign-indefinite general multiparameter algebraic Riccati equation (SIGMARE) that arises in the H ∞ filtering problem is investigated. After establishing the asymptotic structure of the solution of the SIGMARE, in order to solve the SIGMARE, Newton’s method and two fixed point algorithms are combined. As a result, the new iterative algorithm achieves the quadratic convergence property and succeeds in reducing the computing workspace dramatically. As another important feature, the convergence criteria for small parameters ε i is derived for the first time. Moreover, it is shown that the uniqueness and positive semidefiniteness of the convergence solutions are guaranteed in the neighborhood of the initial conditions.

Quadratic Convergence of a Primal-Dual Interior Point Method for Degenerate Nonlinear Optimization Problems

Recently studies of numerical methods for degenerate nonlinear optimization problems have been attracted much attention. Several authors have discussed convergence properties without the linear independence constraint qualification and/or the strict complementarity condition. In this paper, we are concerned with quadratic convergence property of a primal-dual interior point method, in which Newton's method is applied to the barrier KKT conditions. We assume that the second order sufficient condition and the linear independence of gradients of equality constraints hold at the solution, and that there exists a solution that satisfies the strict complementarity condition, and that multiplier iterates generated by our method for inequality constraints are uniformly bounded, which relaxes the linear independence constraint qualification. Uniform boundedness of multiplier iterates is satisfied if the Mangasarian-Fromovitz constraint qualification is assumed, for example. By using the stability theorem by Hager and Gowda (1999), and Wright (2001), the distance from the current point to the solution set is related to the residual of the KKT conditions. By controlling a barrier parameter and adopting a suitable line search procedure, we prove the quadratic convergence of the proposed algorithm.

Simulation of light‐weight membrane structures by wrinkling model

AbstractThe computational challenge in dealing with membrane systems is closely connected to the lack of bending stiffness that constitutes the main feature of this category of structures. This manifests numerically in badly conditioned or singular systems requiring the use of stabilized solution procedures, in our case of a ‘pseudo‐dynamic’ approach. The absence of the flexural stiffness makes the membrane very prone to local instabilities which manifest physically in the formation of little ‘waves’ in ‘compressed’ areas. Current work presents an efficient, sub‐iteration free ‘explicit’, penalty material based, wrinkling simulation procedure suitable for the solution of ‘static’ problems. The procedure is stabilized by taking full advantage of the pseudo‐dynamic solution strategy, which allows to retain the elemental quadratic convergence properties inside the single solution step. Results are validated by comparison with published results and by setting up ‘numerical experiments’ based on the solution of test cases using dense meshes. Copyright © 2005 John Wiley & Sons, Ltd.

NUMERICAL COMPUTATION OF PARETO OPTIMAL STRATEGY FOR GENERAL MULTIPARAMETER SINGULARLY PERTURBED SYSTEMS

In this paper, Pareto optimal strategy for general multiparameter singularly perturbed systems is investigated. The main contribution is to propose a new computational method for obtaining the high-order Pareto near-optimal strategy. Newton's method and two fixed point algorithms are combined. As a result, the new iterative algorithm achieves the quadratic convergence property and succeeds in reducing the computing workspace dramatically. It is newly shown that the resulting optimal strategy achieves the cost functional Jj* + O(||μ||2i).

On Geometric Algorithms for Real-Time Grasping Force Optimization

Grasping force optimization with nonlinear friction constraints is a fundamental problem in dextrous manipulation with multifingered robotic hands. Over the last few years, by transforming the problem into convex optimization problems on Riemannian manifolds of symmetric and positive definite matrices, significant advances have been achieved in this area. Five promising algorithms: two gradient algorithms, two Newton algorithms, and one interior point algorithm have been proposed for real-time solutions of the problem. In this paper, we present in a unified geometric framework, the derivation of these five algorithms and the selection of step sizes for each algorithm. Using the geometric structure of the affine-scaling vector fields associated with the optimization problem, we prove that some of these algorithms have quadratic convergence properties, and their continuous versions are exponentially convergent. We evaluate the performance of these algorithms through simulation and experimental studies with the Hong Kong University of Science and Technology (HKUST) three-fingered hand. This study will facilitate selection and implementation of grasping force optimization algorithms for similar applications.

Global optimization of mixed-integer nonlinear programs: A theoretical and computational study

This work addresses the development of an efficient solution strategy for obtaining global optima of continuous, integer, and mixed-integer nonlinear programs. Towards this end, we develop novel relaxation schemes, range reduction tests, and branching strategies which we incorporate into the prototypical branch-and-bound algorithm. In the theoretical/algorithmic part of the paper, we begin by developing novel strategies for constructing linear relaxations of mixed-integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties. We then use Lagrangian/linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming. This theory leads to new range reduction schemes, including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch-and-bound tree. Finally, we incorporate these relaxation and reduction strategies in a branch-and-bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems. In the computational part of the paper, we describe our implementation discussing, wherever appropriate, the use of suitable data structures and associated algorithms. We present computational experience with benchmark separable concave quadratic programs, fractional 0–1 programs, and mixed-integer nonlinear programs from applications in synthesis of chemical processes, engineering design, just-in-time manufacturing, and molecular design.

A revised Kleinman algorithm to solve algebraic Riccati equation of singularly perturbed systems

In this paper, we show that the Kleinman algorithm can be used well to solve the algebraic Riccati equation (ARE) of singularly perturbed systems, where the quadratic term of the ARE may be indefinite. The quadratic convergence property of the Kleinman algorithm is proved by using the Newton–Kantorovich theorem when the initial condition is chosen appropriately. In addition, the numerical method to solve the generalized algebraic Lyapunov equation (GALE) appearing in the Kleinman algorithm is given.

Smoothing Newton and quasi-Newton methods for mixed complementarity problems

The mixed complementarity problem can be reformulated as a nonsmooth equation by using the median operator. In this paper, we first study some useful properties of this reformulation and then derive the Chen-Harker-Kanzow-Smale smoothing function for the mixed complementarity problem. On the basis of this smoothing function, we present a smoothing Newton method for solving the mixed complementarity problem. Under suitable conditions, the method exhibits global and quadratic convergence properties. We also present a smoothing Broyden-like method based on the same smoothing function. Under appropriate conditions, the method converges globally and superlinearly.

Approximation methods for non-convex curves

Based on the triangle sandwich and block sandwich methods, two approximation methods for non-convex curves are considered in this paper. Numerical algorithms corresponding to these methods are developed and quadratic convergence properties of two methods in the original space are established. Application is given to the approximation of an efficient frontier of a bi-criteria optimization problem.

Quadratic and Superlinear Convergence of the Huschens Method for NonlinearLeast Squares Problems

This paper is concerned with quadratic and superlinear convergence of structured quasi-Newton methods for solving nonlinear least squares problems. These methods make use of a special structure of the Hessian matrix of the objective function. Recently, Huschens proposed a new kind of structured quasi-Newton methods and dealt with the convex class of the structured Broyden family, and showed its quadratic and superlinear convergence properties for zero and nonzero residual problems, respectively. In this paper, we extend the results by Huschens to a wider class of the structured Broyden family. We prove local convergence properties of the method in a way different from the proof by Huschens.

New Imperfect Conjugate Gradient Algorithm

We present a method which generates conjugate search directions and maintains the quadratic convergence property, when applied to a quadratic function, even when the line searches are not exact. The method is similar to that given by Dixon, but needs one less vector store. When the method is applied to a number of general functions of different dimensionality, results show the efficiency of the method.

Numerical solutions of the algebraic matrix Riccati equation

Abstract The linear-quadratic control model is one of the most widely used control models in both empirical and theoretical economic modeling. In order to obtain the equilibrium solution of this control model, the so-called algebraic matrix Riccati equation has to be solved. In this note we present a numerical solution method for solving this equation. Our method solves the Riccati equation as a multidimensional fixed-point problem. By establishing the analytical derivative of the Riccati equation we have been able to construct a very efficient Newton-type solution method with quadratic convergence properties. Our method is an extension for the Newton-Raphson method described in Kwakernaak and Sivan (1972) and does not require any special conditions on the transition matrix as in the nonrecursive method of Vaughan (1970).

A method for convex curve approximation

In this paper, a new sandwich method is introduced to approximate a convex curve in R2. This method requires only function evaluation and the solution of a number of scalar optimization problems. A quadratic convergence property of the method is established, that is, the total number of optimization problems solved is bounded by a constant multiple of the square root of the inverse of the given error. An application to approximation of the efficient frontier of a bi-criteria convex quadratic network program is given.

A semismooth equation approach to the solution of nonlinear complementarity problems

In this paper we present a new algorithm for the solution of nonlinear complementarity problems. The algorithm is based on a semismooth equation reformulation of the complementarity problem. We exploit the recent extension of Newton's method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions. Other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration. We include numerical tests which show the viability of the approach.