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15
Indefinite Trust Region Subproblems And Nonsymmetric Eigenvalue Perturbations
, 1995
"... This paper extends the theory of trust region subproblems in two ways: (i) it allows indefinite inner products in the quadratic constraint and (ii) it uses a two sided (upper and lower bound) quadratic constraint. Characterizations of optimality are presented, which have no gap between necessity and ..."
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Cited by 57 (17 self)
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This paper extends the theory of trust region subproblems in two ways: (i) it allows indefinite inner products in the quadratic constraint and (ii) it uses a two sided (upper and lower bound) quadratic constraint. Characterizations of optimality are presented, which have no gap between necessity and sufficiency. Conditions for the existence of solutions are given in terms of the definiteness of a matrix pencil. A simple dual program is intro...
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 39 (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.
Strong Duality in Nonconvex Quadratic Optimization with Two Quadratic Constraints
 SIAM Journal on Optimization
"... Abstract. We consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints. When the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions. We then develop a connecti ..."
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Cited by 18 (10 self)
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Abstract. We consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints. When the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions. We then develop a connection between the image of the real and complex spaces under a quadratic mapping, which together with the results in the complex case lead to a condition that ensures strong duality in the real setting. Preliminary numerical simulations suggest that for random instances of the extended trust region subproblem, the sufficient condition is satisfied with a high probability. Furthermore, we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems. Finally, we discuss an application of our results to robust least squares problems.
Generalizations Of The Trust Region Problem
 OPTIMIZATION METHODS AND SOFTWARE
, 1993
"... The trust region problem requires the global minimum of a general quadratic function subject to an ellipsoidal constraint. The development of algorithms for the solution of this problem has found applications in nonlinear and combinatorial optimization. In this paper we generalize the trust region p ..."
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Cited by 16 (0 self)
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The trust region problem requires the global minimum of a general quadratic function subject to an ellipsoidal constraint. The development of algorithms for the solution of this problem has found applications in nonlinear and combinatorial optimization. In this paper we generalize the trust region problem by allowing a general quadratic constraint. The main results are a characterization of the global minimizer of the generalized trust region problem, and the development of an algorithm that finds an approximate global minimizer in a finite number of iterations.
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...
Strong duality for the CDT subproblem: A necessary and sufficient condition
 Mathematics of Operations Research, 33(4): 965  975
"... Abstract. In this paper we consider the problem of minimizing a nonconvex quadratic function, subject to two quadratic inequality constraints. As an application, such quadratic program plays an important role in the trust region method for nonlinear optimization; such problem is known as the CDT sub ..."
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Cited by 3 (1 self)
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Abstract. In this paper we consider the problem of minimizing a nonconvex quadratic function, subject to two quadratic inequality constraints. As an application, such quadratic program plays an important role in the trust region method for nonlinear optimization; such problem is known as the CDT subproblem in the literature. The Lagrangian dual of the CDT subproblem is a Semidefinite Program (SDP), hence convex and solvable. However, a positive duality gap may exist between the CDT subproblem and its Lagrangian dual because the CDT subproblem itself is nonconvex. In this paper, we present a necessary and sufficient condition to characterize when the CDT subproblem and its Lagrangian dual admits no duality gap (i.e., the strong duality holds). This necessary and sufficient condition is easy verifiable and involves only one (any) optimal solution of the SDP relaxation for the CDT subproblem. Moreover, the condition reveals that it is actually rare to render a positive duality gap for the CDT subproblems in general. Moreover, if the strong duality holds then an optimal solution for the CDT problem can be retrieved from an optimal solution of the SDP relaxation, by means of a matrix rankone decomposition procedure. The same analysis is extended to the framework where the necessary and sufficient condition is presented in terms of the Lagrangian multipliers at a KKT point. Furthermore, we show that the condition is numerically easy to work with approximatively. Key words. Quadratically constrained quadratic programming, strong Lagrangian duality, CDT subproblem, SDP relaxation. AMS subject classifications. 90C33, 90C51, 90C05 1. Introduction. In
A Survey of the Trust Region Subproblem within a Semidefinite Framework
, 2000
"... Trust region subproblems arise within a class of unconstrained methods called trust region methods. The subproblems consist of minimizing a quadratic function subject to a norm constraint. This thesis is a survey of dierent methods developed to nd an approximate solution to the subproblem. We study ..."
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Cited by 3 (0 self)
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Trust region subproblems arise within a class of unconstrained methods called trust region methods. The subproblems consist of minimizing a quadratic function subject to a norm constraint. This thesis is a survey of dierent methods developed to nd an approximate solution to the subproblem. We study the wellknown method of More and Sorensen [18] and two recent methods for large sparse subproblems: the socalled Lanczos method of Gould et al. [7] and the Rendl and Wolkowicz algorithm [31]. The common ground to explore these methods will be semidenite programming. This approach has been used by Rendl and Wolkowicz [31] to explain their method and the More and Sorensen algorithm; we extend this work to the Lanczos method. The last chapter of this thesis is dedicated to some improvements done to the Rendl and Wolkowicz algorithm and to comparisons between the Lanczos method and the Rendl and Wolkowicz algorithm. In particular, we show some weakness of the Lanczos method and show that ...
On Maxima Of Dual Function Of The Cdt Subproblem
 Institute of Computational Mathematics and Scientific/ Engineering Computing, Chinese Academy of Sciences
, 2001
"... In this paper, we show the geometry meaning of the maxima of the CDT subproblem's dual function. We also studied the continuity of the global solution of the trust region subproblem. Based on an approximation model, we prove that the global solution of the CDT subproblem is given with the Hessian of ..."
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Cited by 2 (1 self)
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In this paper, we show the geometry meaning of the maxima of the CDT subproblem's dual function. We also studied the continuity of the global solution of the trust region subproblem. Based on an approximation model, we prove that the global solution of the CDT subproblem is given with the Hessian of Lagrangian positive semide nite by some speciallylocated dual maxima and by restricting the location region of the multipliers which corresponding a global solution in other cases.
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...