Maintaining the positive definiteness of the matrices in reduced secant methods for equality constrained optimization

Gilbert, Jean Charles

doi:10.1007/bf01594922

Cited by 10 publications

(4 citation statements)

References 28 publications

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“…For large problems, however, computing this QR factorization is often too expensive. Therefore many researchers, including Gabay (1982), Gilbert (1991), Fletcher (1987), Murray and Prieto (1992), and Xie When A(x) is large and sparse, a sparse LU decomposition of C(x) can often be computed e ciently, and this approach will be considerably less expensive than the QR factorization of A. Note that from the assumed nonsingularity o f C(x) both Y (x) and Z(x) v ary smoothly with x, provided the same partition of the variables is maintained.…”

Section: The Basis Matricesmentioning

confidence: 99%

A Reduced Hessian Method for Large-Scale Constrained Optimization

Biegler¹,

Nocedal²,

Schmid³

1995

SIAM J. Optim.

148

114

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We propose a quasi-Newton algorithm for solving large optimization problems with nonlinear equality constraints. It is designed for problems with few degrees of freedom and is motivated by the need to use sparse matrix factorizations. The algorithm incorporates a correction vector that approximates the cross term Z T W Y p Y in order to estimate the curvature in both the range and null spaces of the constraints. The algorithm can be considered to be, in some sense, a practical implementation of an algorithm of Coleman and Conn. We give conditions under which local and superlinear convergence is obtained.

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Section: The Basis Matricesmentioning

confidence: 99%

A Reduced Hessian Method for Large-Scale Constrained Optimization

Biegler¹,

Nocedal²,

Schmid³

1995

SIAM J. Optim.

148

114

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“…Full-space methods and simplex space methods are often used to deal with problems with equation constraints. For large-scale optimization problems with a large number of equation constraints and limited degrees of freedom, the simplex space method can be used to reduce the dimensionality of the SQP subproblem through a spatial decomposition strategy, thereby reducing the computational effort of the Hessian matrix and improving the solution efficiency of the algorithm [7][8][9]. However, the spatial decomposition process adds additional computational cost and its convergence is not as good as the full-space method based on exact second-order derivative information.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Optimization of Variable Load Process for Combined Heat and Power Unit Based on Sequential Quadratic Programming and Interior Point Method Alternating Solution Method

et al. 2023

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Aiming at the problem that the modeling and solving method of combined heat and power (CHP) unit variable load control process is challenging to meet the demand for efficient analysis of complex systems, this paper proposes a method based on sequential quadratic programming and interior point method (SQP-IPM) alternating solution for dynamic optimization of the CHP unit variable load process. Firstly, by constructing the CHP unit mechanism model, multi-variable coordination control constraints, and output variable process constraints, the dynamic optimization proposition of the CHP unit variable load control process is formed. Then, the large-scale nonlinear programming (NLP) problem formed by using the orthogonal configuration method to discrete the state and control variables is optimally solved using the IPM-SQP alternating solution method. Further, from the perspective of balancing the accuracy of the solution and computational efficiency, the flexible convergence depth control (CDC) strategy is introduced into the alternative solution method to improve the real-time performance of the algorithm. Finally, the variable load control process of 300MW extraction CHP unit is simulated to verify that the proposed method reduces the calculation time for 12 consecutive variable load scenarios by about 70%, effectively improving the real-time performance of scenario applications.

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“…This is the motivation for reduced Hessian methods. Reduced Hessian methods (for equality constrained problems) have been proposed by Murray and Wright [15], Gabay [10], Gilbert [11], Coleman and Corm [5], Nocedal and Overton [16], and Byrd and Nocedal [31. Although reduced Hessian methods are well developed for equality constrained problems, there is little work on generalizing this method for *Research supported in part by NSF, AFORS and ONR through NSF grant DMS-8920550.…”

Section: Introductionmentioning

confidence: 99%