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Algorithms for Constrained Minimization of Smooth Nonlinear by A.G. Buckley, J-.L. Goffin

By A.G. Buckley, J-.L. Goffin

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Mukai and E. Polak, "On the use of approximations in algorithms for optimization problems with equality and inequality constraints", SIAM Journal on Numerical Analysis 15 (1978) 674-693. T. Rockafellar, "'Augmented Lagrange multiplier functions and duality in nor~cor~vex programming", SIAM Journal on Control 12 (1973) 555-562. P. Bertsekas, "Multiplier methods: A survey", Automatica 12 (1976) 133-145. M. L. Mangasarian, "Superlinearly convergent quasi-Newton algorithms for nonlinearly constrained optimization problems", Mathematical Programming 11 (1976) 1-13.

Polak/ Superlinearly convergent algorithm Proposition 1. Let {p, A,/z} be a K u h n - T u c k e r triple I for QP(x, H). Then p is a descent direction for ~,(x, c) if: (i) H is a positive definite, tel j=l $=1 This result is easily established from the fact that if {p, ;t, #} is a Kuhn-Tucker triple for QP(x, H), then: (11) i=t j=l The algorithm presented in this paper has the following components: a procedure for choosing the penalty parameter; a procedure for determining a search direction, a procedure for determining step length and a mechanism for avoiding truncation of the step length near a solution.

1. 3b) c k = c(xE). 3c) Assume that the sequence {x k} converges to a local minimum x* of f on C and let A* be the corresponding Lagrange multipliers vector associated with the constraint equations. Assume moreover that (x*, I*) satisfies the second-order sufficient optimality condition. 4a) and that the matrices HE remain positive definite and such that m2llpll z <- (p, n k p ) <--m3llpll 2 for all p E R "-m and all k. 4b) D. e. 3a) is also the image of x k by a map H satisfying IIH(x) - n ( y ) l [ - m~llx - yll for all x, y ~ R".

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