Quasi-Newtonian methods of constrained optimization

Lagrangian functions are the basis of many of the more successful methods for nonlinear constraints in optimization calculations. Sometimes they are used in conjunction with linear approximations to the constraints and sometimes penalty terms are included to allow the use of algorithms for unconstrained optimization. Much has been discovered about these techniques during the last eight years and this paper gives a view of the progress and understanding that has been achieved and its relevance to practical algorithms. A particular method is recommended that seems to be more powerful than the author believed to be possible at the beginning of 1976.

Algorithms for nonlinear constraints that use lagrangian functions

This paper presents an algorithm for solving nonlinearly constrained nonlinear programming problems. The algorithm reduces the original problem to a sequence of linearly-constrained minimization problems, for which efficient algorithms are available. A convergence theorem is given which states that if the process is started sufficiently close to a strict second-order Kuhn—Tucker point, then the sequence produced by the algorithm exists and convergesR-quadratically to that point.

A quadratically-convergent algorithm for general nonlinear programming problems

Methods of searching saddle points

QUASI-NEWTON algorithms for minimizing a smooth convex function f(x), x ϵ En, are investigated. The feature of the algorithms is that the matrix Ak−1→[f″(xk)]−1 is constructed with the aid of a system of conjugate vectors, obtained from a linearly independent system.

Quasi-Newtonian methods of constrained optimization

References

Algorithms for nonlinear constraints that use lagrangian functions

A quadratically-convergent algorithm for general nonlinear programming problems

Methods of searching saddle points

Quasi-Newton minimization algorithms, based on the construction of systems of conjugate vectors

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