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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.
Trust Regions and Relaxations for the Quadratic Assignment Problem
 In Quadratic assignment and related problems (New
, 1993
"... . General quadratic matrix minimization problems, with orthogonal constraints, arise in continuous relaxations for the (discrete) quadratic assignment problem (QAP). Currently, bounds for QAP are obtained by treating the quadratic and linear parts of the objective function, of the relaxations, separ ..."
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Cited by 6 (5 self)
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. General quadratic matrix minimization problems, with orthogonal constraints, arise in continuous relaxations for the (discrete) quadratic assignment problem (QAP). Currently, bounds for QAP are obtained by treating the quadratic and linear parts of the objective function, of the relaxations, separately. This paper handles general objectives as one function. The objectives can be both nonhomogeneous and nonconvex. The constraints are orthogonal or Loewner partial order (positive semidefinite) constraints. Comparisons are made to standard trust region subproblems. Numerical results are obtained using a parametric eigenvalue technique. Contents 1. Introduction 1 2. Preliminary Notations and Motivation 2 2.1. Notations 2.2. A Survey on Eigenvalue Bounds for the QAP 2.3. Loewner Partial Order 3. Optimality Conditions 6 3.1. First Order Conditions 3.2. Second Order Conditions 1991 Mathematics Subject Classification. Primary 90B80, 90C20, 90C35, 90C27; Secondary 65H20, 65K05. Key words...
A Global Convergence Theory for a General Class of TrustRegionBased Algorithms for Constrained Optimization Without Assuming Regularity
 SIAM Journal on Optimization
, 1997
"... This work presents a convergence theory for a general class of trustregionbased algorithms for solving the smooth nonlinear programming problem with equality constraints. The results are proved under very mild conditions on the quasinormal and tangential components of the trial steps. The Lagrang ..."
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Cited by 3 (0 self)
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This work presents a convergence theory for a general class of trustregionbased algorithms for solving the smooth nonlinear programming problem with equality constraints. The results are proved under very mild conditions on the quasinormal and tangential components of the trial steps. The Lagrange multiplier estimates and the Hessian estimates are assumed to be bounded. In addition, the regularity assumption is not made. In particular, the linear independence of the gradients of the constraints is not assumed. The theory proves global convergence for the class. In particular, it shows that a subsequence of the iteration sequence satisfies one of four types of MayerBliss stationary conditions in the limit. This theory holds for Dennis, ElAlem, and Maciel's class of trustregionbased algorithms. Key Words: Nonlinear programming, equality constrained problems, constrained optimization, global convergence, regularity assumption, augmented Lagrangian, MayerBliss points, stationary p...
Trust Regions and the Quadratic Assignment Problem
 In Proceedings of the DIMACS Workshop on Quadratic Assignment Problems
, 1992
"... The quadratic assignment problem (QAP) belongs to the class of NP hard combinatorial optimization problems. Although it has been studied extensively over the past 35 years, problems of size n 15 still prove to be intractable. Since good lower bounds are necessary to solve larger problem instances, ..."
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Cited by 2 (2 self)
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The quadratic assignment problem (QAP) belongs to the class of NP hard combinatorial optimization problems. Although it has been studied extensively over the past 35 years, problems of size n 15 still prove to be intractable. Since good lower bounds are necessary to solve larger problem instances, we discuss the class of eigenvalue bounds for QAP and extend previous results of these continuous optimization approaches for QAP. Since these approaches include trust region techniques and optimization over partial orders, new results for these areas are obtained, as well. i Acknowledgements My deepest appreciation goes to my supervisor, Henry Wolkowicz, for his guidance, patience and encouragement during my studies. No question or idea did seem to be too trivial, to not to be answered and considered by him. I was happy to be his student. I would like to acknowledge the Austrian Bundesministerium fur Wissenschaft und Forschung, the Exportakademie der Osterreichischen Bundeswirtschaftsk...
Convergence to a SecondOrder Point of a TrustRegion Algorithm with a Nonmonotonic Penalty Parameter for Constrained Optimization
 Rice University
, 1996
"... In a recent paper, the author (Ref. 1) proposed a trustregion algorithm for solving the problem of minimizing a nonlinear function subject to a set of equality constraints. The main feature of the algorithm is that the penalty parameter in the merit function can be decreased whenever it is warrant ..."
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Cited by 2 (0 self)
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In a recent paper, the author (Ref. 1) proposed a trustregion algorithm for solving the problem of minimizing a nonlinear function subject to a set of equality constraints. The main feature of the algorithm is that the penalty parameter in the merit function can be decreased whenever it is warranted. He studied the behavior of the penalty parameter and proved several global and local convergence results. One of these results is that there exists a subsequence of the iterates generated by the algorithm, that converges to a point that satisfies the firstorder necessary conditions. In the current paper, we show that, for this algorithm, there exists a subsequence of iterates that converges to a point that satisfies both the firstorder and the secondorder necessary conditions. Key Words : Constrained optimization, equality constrained, penalty parameter, nonmonotonic penalty parameter, convergence, trustregion methods, firstorder point, secondorder point, necessary conditions. B 1...
SQ^P, Sequential Quadratic Constrained Quadratic Programming
, 1998
"... We follow the popular approach for unconstrained minimization, i.e. we develop a local quadratic model at a current approximate minimizer in conjunction with a trust region. We then minimize this local model in order to find the next approximate minimizer. Asymptotically, finding the local minimizer ..."
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We follow the popular approach for unconstrained minimization, i.e. we develop a local quadratic model at a current approximate minimizer in conjunction with a trust region. We then minimize this local model in order to find the next approximate minimizer. Asymptotically, finding the local minimizer of the quadratic model is equivalent to applying Newton's method to the stationarity condition. For constrained problems, the local quadratic model corresponds to minimizinga quadratic approximation of the objective subject to quadratic approximations of the constraints (Q 2 P), with an additional trust region. This quadratic model is intractable in general and is usually handled by using linear approximations of the constraints and modifying the Hessian of the objective using the Hessian of the Lagrangean, i.e. a SQP approach. Instead, we solve the Lagrangean relaxation of Q 2 P using semidefinite programming. We develop this framework and present an example which illustrates the adva...