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39
On the convergence of reflective Newton methods for largescale nonlinear minimization subject to bounds
, 1992
"... . We consider a new algorithm, a reflective Newton method, for the problem of minimizing a smooth nonlinear function of many variables, subject to upper and/or lower bounds on some of the variables. This approach generates strictly feasible iterates by following piecewise linear paths ("reflection" ..."
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Cited by 60 (4 self)
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. We consider a new algorithm, a reflective Newton method, for the problem of minimizing a smooth nonlinear function of many variables, subject to upper and/or lower bounds on some of the variables. This approach generates strictly feasible iterates by following piecewise linear paths ("reflection" paths) to generate improved iterates. The reflective Newton approach does not require identification of an "activity set". In this report we establish that the reflective Newton approach is globally and quadratically convergent. Moreover, we develop a specific example of this general reflective path approach suitable for largescale and sparse problems. 1 Research partially supported by the Applied Mathematical Sciences Research Program (KC04 02) of the Office of Energy Research of the U.S. Department of Energy under grant DEFG0286ER25013. A000, and in part by NSF, AFOSR, and ONR through grant DMS8920550, and by the Cornell Theory Center which receives major funding from the National Sci...
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 51 (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 ...
A reflective Newton method for minimizing a quadratic function subject to bounds on some of the variables
, 1992
"... . We propose a new algorithm, a reflective Newton method, for the minimization of a quadratic function of many variables subject to upper and lower bounds on some of the variables. The method applies to a general (indefinite) quadratic function, for which a local minimizer subject to bounds is requi ..."
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Cited by 49 (1 self)
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. We propose a new algorithm, a reflective Newton method, for the minimization of a quadratic function of many variables subject to upper and lower bounds on some of the variables. The method applies to a general (indefinite) quadratic function, for which a local minimizer subject to bounds is required, and is particularily suitable for the largescale problem. Our new method exhibits strong convergence properties, global and quadratic convergence, and appears to have significant practical potential. Strictly feasible points are generated. Experimental results on moderately large and sparse problems support the claim of practicality for largescale problems. 1 Research partially supported by the Applied Mathematical Sciences Research Program (KC04 02) of the Office of Energy Research of the U.S. Department of Energy under grant DEFG0286ER25013. A000, and by the Computational Mathematics Program of the National Science Foundation under grant DMS8706133, and by the Cornell Theory Cen...
A Global Convergence Theory for General TrustRegionBased Algorithms for Equality Constrained Optimization
 SIAM Journal on Optimization
, 1992
"... This work presents a global convergence theory for a broad class of trustregion algorithms for the smooth nonlinear progro.mmln S problem with equality constraints. The main result generalizes Powell's 1975 result for unconstrained trustregion algorithms. ..."
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Cited by 42 (10 self)
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This work presents a global convergence theory for a broad class of trustregion algorithms for the smooth nonlinear progro.mmln S problem with equality constraints. The main result generalizes Powell's 1975 result for unconstrained trustregion algorithms.
On the implementation of an algorithm for largescale equality constrained optimization
 SIAM Journal on Optimization
, 1998
"... Abstract. This paper describes a software implementation of Byrd and Omojokun’s trust region algorithm for solving nonlinear equality constrained optimization problems. The code is designed for the efficient solution of large problems and provides the user with a variety of linear algebra techniques ..."
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Cited by 38 (11 self)
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Abstract. This paper describes a software implementation of Byrd and Omojokun’s trust region algorithm for solving nonlinear equality constrained optimization problems. The code is designed for the efficient solution of large problems and provides the user with a variety of linear algebra techniques for solving the subproblems occurring in the algorithm. Second derivative information can be used, but when it is not available, limited memory quasiNewton approximations are made. The performance of the code is studied using a set of difficult test problems from the CUTE collection.
A Subspace, Interior, and Conjugate Gradient Method for LargeScale BoundConstrained Minimization Problems
 SIAM Journal on Scientific Computing
, 1999
"... A subspace adaptation of the ColemanLi trust region and interior method is proposed for solving largescale boundconstrained minimization problems. This method can be implemented with either sparse Cholesky factorization or conjugate gradient computation. Under reasonable conditions the convergenc ..."
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Cited by 35 (1 self)
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A subspace adaptation of the ColemanLi trust region and interior method is proposed for solving largescale boundconstrained minimization problems. This method can be implemented with either sparse Cholesky factorization or conjugate gradient computation. Under reasonable conditions the convergence properties of this subspace trust region method are as strong as those of its fullspace version.
Thermodynamic analysis of interacting nucleic acid strands
 SIAM Rev
, 2007
"... Abstract. Motivated by the analysis of natural and engineered DNA and RNA systems, we present the first algorithm for calculating the partition function of an unpseudoknotted complex of multiple interacting nucleic acid strands. This dynamic program is based on a rigorous extension of secondary stru ..."
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Cited by 31 (4 self)
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Abstract. Motivated by the analysis of natural and engineered DNA and RNA systems, we present the first algorithm for calculating the partition function of an unpseudoknotted complex of multiple interacting nucleic acid strands. This dynamic program is based on a rigorous extension of secondary structure models to the multistranded case, addressing representation and distinguishability issues that do not arise for singlestranded structures. We then derive the form of the partition function for a fixed volume containing a dilute solution of nucleic acid complexes. This expression can be evaluated explicitly for small numbers of strands, allowing the calculation of the equilibrium population distribution for each species of complex. Alternatively, for large systems (e.g., a test tube), we show that the unique complex concentrations corresponding to thermodynamic equilibrium can be obtained by solving a convex programming problem. Partition function and concentration information can then be used to calculate equilibrium basepairing observables. The underlying physics and mathematical formulation of these problems lead to an interesting blend of approaches, including ideas from graph theory, group theory, dynamic programming, combinatorics, convex optimization, and Lagrange duality.
Solving the trustregion subproblem using the Lanczos method
, 1997
"... The approximate minimization of a quadratic function within an ellipsoidal trust region is an important subproblem for many nonlinear programming methods. When the number of variables is large, the most widelyused strategy is to trace the path of conjugate gradient iterates either to convergence or ..."
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Cited by 21 (1 self)
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The approximate minimization of a quadratic function within an ellipsoidal trust region is an important subproblem for many nonlinear programming methods. When the number of variables is large, the most widelyused strategy is to trace the path of conjugate gradient iterates either to convergence or until it reaches the trustregion boundary. In this paper, we investigate ways of continuing the process once the boundary has been encountered. The key is to observe that the trustregion problem within the currently generated Krylov subspace has very special structure which enables it to be solved very efficiently. We compare the new strategy with existing methods. The resulting software package is available as HSL VF05 within the Harwell Subroutine Library. 1 Department for Computation and Information, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, England, EU Email : n.gould@rl.ac.uk 2 Current reports available by anonymous ftp from joyousgard.cc.rl.ac.uk (internet ...
Nonmonotone Curvilinear Line Search Methods for Unconstrained Optimization
 Computational Optimization and Applications
, 1995
"... We present a new algorithmic framework for solving unconstrained minimization problems that incorporates a curvilinear linesearch. The search direction used in our framework is a combination of an approximate Newton direction and a direction of negative curvature. Global convergence to a stationary ..."
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Cited by 12 (4 self)
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We present a new algorithmic framework for solving unconstrained minimization problems that incorporates a curvilinear linesearch. The search direction used in our framework is a combination of an approximate Newton direction and a direction of negative curvature. Global convergence to a stationary point where the Hessian matrix is positive semidefinite is exhibited for this class of algorithms by means of a nonmonotone stabilization strategy. An implementation using the BunchParlett decomposition is shown to outperform several other techniques on a large class of test problems. 1 Introduction In this work we consider the unconstrained minimization problem min x2IR n f(x); where f is a real valued function on IR n . We assume throughout that both the gradient g(x) := rf(x) and the Hessian matrix H(x) := r 2 f(x) of f exist and are continuous. Many iterative methods for solving this problem have been proposed; they are usually descent methods that generate a sequence fx k g su...
Multilevel Algorithms for Nonlinear Optimization
 in Optimal Design and Control
, 1994
"... Multidisciplinary design optimization (MDO) gives rise to nonlinear optimization problems characterized by a large number of constraints that naturally occur in blocks. We propose a class of multilevel optimization methods motivated by the structure and number of constraints and by the expense of th ..."
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Cited by 11 (5 self)
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Multidisciplinary design optimization (MDO) gives rise to nonlinear optimization problems characterized by a large number of constraints that naturally occur in blocks. We propose a class of multilevel optimization methods motivated by the structure and number of constraints and by the expense of the derivative computations for MDO. The algorithms are an extension to the nonlinear programming problem of the successful class of local BrownBrent algorithms for nonlinear equations. Our extensions allow the user to partition constraints into arbitrary blocks to fit the application, and they separately process each block and the objective function, restricted to certain subspaces. The methods use trust regions as a globalization startegy, and they have been shown to be globally convergent under reasonable assumptions. The multilevel algorithms can be applied to all classes of MDO formulations. Multilevel algorithms for solving nonlinear systems of equations are a special case of the multil...