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Sequential Quadratic Programming
, 1995
"... this paper we examine the underlying ideas of the SQP method and the theory that establishes it as a framework from which effective algorithms can ..."
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Cited by 121 (3 self)
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this paper we examine the underlying ideas of the SQP method and the theory that establishes it as a framework from which effective algorithms can
A New Trust Region Algorithm For Equality Constrained Optimization
, 1995
"... . We present a new trust region algorithm for solving nonlinear equality constrained optimization problems. At each iterate a change of variables is performed to improve the ability of the algorithm to follow the constraint level sets. The algorithm employs L 2 penalty functions for obtaining global ..."
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Cited by 53 (7 self)
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. We present a new trust region algorithm for solving nonlinear equality constrained optimization problems. At each iterate a change of variables is performed to improve the ability of the algorithm to follow the constraint level sets. The algorithm employs L 2 penalty functions for obtaining global convergence. Under certain assumptions we prove that this algorithm globally converges to a point satisfying the second order necessary optimality conditions; the local convergence rate is quadratic. Results of preliminary numerical experiments are presented. 1. Introduction. We consider the equality constrained optimization problem minimize f(x) subject to c(x) = 0 (1:1) where x 2 ! n and f : ! n ! !, and c : ! n ! ! m are smooth nonlinear functions. Problem (1.1) is often solved by successive quadratic programming (SQP) methods. At a current point x k 2 ! n , SQP methods determine a search direction d k by solving a quadratic programming problem minimize rf(x k ) T d + 1 2 ...
Arclength method for frictional contact problems using mathematical programming with complementarity constraints
 Journal of Optimization Theory and Applications
"... Abstract. A new formulation as well as a new solution technique is proposed for an equilibrium pathfollowing method in twodimensional quasistatic frictional contact problems. We consider the Coulomb friction law as well as a geometrical nonlinearity explicitly. Based on a criterion of maximum dis ..."
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Abstract. A new formulation as well as a new solution technique is proposed for an equilibrium pathfollowing method in twodimensional quasistatic frictional contact problems. We consider the Coulomb friction law as well as a geometrical nonlinearity explicitly. Based on a criterion of maximum dissipation of energy, we propose a formulation as a mathematical program with complementarity constraints (MPEC) in order to avoid unloading solutions in which most contact candidate nodes become stuck. A regularization scheme for the MPEC is proposed, which can be solved by using a conventional nonlinear programming approach. The equilibrium paths of various structures are computed in cases such that there exist some limit points and/or infinite number of successive bifurcation points. Key Words. Contact problems, Coulombâ€™s friction, arclength method, mathematical program with complementarity constraints, maximum dissipation. 1.
A QuasiNewton Quadratic Penalty Method For Minimization Subject To Nonlinear Equality Constraints
"... . We present a modified quadratic penalty function method for equality constrained optimization problems. The pivotal feature of our algorithm is that at every iterate we invoke a special change of variables to improve the ability of the algorithm to follow the constraint level sets. This change of ..."
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Cited by 1 (0 self)
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. We present a modified quadratic penalty function method for equality constrained optimization problems. The pivotal feature of our algorithm is that at every iterate we invoke a special change of variables to improve the ability of the algorithm to follow the constraint level sets. This change of variables gives rise to a suitable block diagonal approximation to the Hessian which is then used to construct a quasiNewton method. We show that the complete algorithm is globally convergent. Preliminary computational results are reported. Key words. nonlinearly constrained optimization, equality constraints, quasiNewton methods, BFGS, quadratic penalty function, reduced Hessian approximation AMS(MOS) subject classifications. 65K05, 65K10, 65H10, 90C30, 90C05, 68L10 1. Introduction. One of the great success stories in continuous optimization is the development of effective quasiNewton methods for unconstrained minimization (at least for problems of moderate size). Three important reas...
A Stablized SQP Method via Linear Equations
, 2000
"... . In this paper, we propose a stablized sequential quadratic programming (SSQP) method for solving general nonlinear constrained optimization problems (NLP). Most present locally quadratically convergent algorithms for solving NLP need to assume at least the linear independence constraint qualicatio ..."
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. In this paper, we propose a stablized sequential quadratic programming (SSQP) method for solving general nonlinear constrained optimization problems (NLP). Most present locally quadratically convergent algorithms for solving NLP need to assume at least the linear independence constraint qualication and sometimes also the strict complementarity condition. Some recently developed SSQP methods can achieve local quadratic convergence without assuming these two conditions. But at each step of these SSQP methods, one or several quadratic programming problems need to be solved. For the NewtonSSQP method proposed in this paper, not only it achieves local quadratic convergence without assuming the abovementioned two conditions, but also only one or several systems of linear equations need to be solved at each step. A characterization theorem for superlinear convergence of quasiNewtonSSQP methods is also given. Key Words. Constrained optimization problem, SSQP method, superlinear/quadrat...
A QuasiNewton L2Penalty Method for Minimization Subject to Nonlinear Equality Constraints
"... . We present a modified L 2 penalty function method for equality constrained optimization problems. The pivotal feature of our algorithm is that at every iterate we invoke a special change of variables to improve the ability of the algorithm to follow the constraint level sets. This change of variab ..."
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. We present a modified L 2 penalty function method for equality constrained optimization problems. The pivotal feature of our algorithm is that at every iterate we invoke a special change of variables to improve the ability of the algorithm to follow the constraint level sets. This change of variables gives rise to a suitable block diagonal approximation to the Hessian which is then used to construct a quasiNewton method. We show that the complete algorithm is globally convergent with a local Qsuperlinearly convergence rate. Preliminary computational results are given for a few problems. 1. Introduction. We construct a quasiNewton L 2 penalty method for solving the equality constrained optimization problem minimize f(x) subject to c(x) = 0 (1:1) where x 2 ! n , and f : ! n ! ! and c : ! n ! ! m are smooth nonlinear functions. This method possesses both strong global convergence properties and a local superlinear convergence rate by combining an L 2 penalty function method ...
POINT QUASINEWTON METHODS F O R CONSTRAINED OPTIMIZATION
, 1996
"... Abstract This paper analyzes local convergence rates of primaldual interior point methods for general nonlinearly constrained optimization problems. For this purpose, we first discuss modified Newton methods and modified quasiNewton methods for solving a nonlinear system of equations, and show loc ..."
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Abstract This paper analyzes local convergence rates of primaldual interior point methods for general nonlinearly constrained optimization problems. For this purpose, we first discuss modified Newton methods and modified quasiNewton methods for solving a nonlinear system of equations, and show local and Qquadratic/Qsuperlinear convergence of these methods. These methods are characterized by a perturbation of the righthand side of the Newton equation applied to the system, an approximation of the Jacobian matrix by some matrix, and componentwise dampings of the step. By applying these convergence results for the nonlinear system of equations to the primaldual interior point methods for nonlinear optimization, we obtain convergence results of the primaldual interior point Newton and quasiNewton methods. A necessary and sufficient condition for Qsuperlinear convergence of the latter methods corresponds to the DennisMore condition. Furthermore, we present some quasiNewton updating formulae. Finally, we give an analysis of the Qrate in a part of variables for the primaldual interior point quasiNewton methods, and obtain a necessary and sufficient condition for the Qrate. This condition is a generalization of the result given by Martinez, Parada and Tapia (1995), which was done independently. 1.