Constrained Optimization and Lagrange Multiplier Methods by Dimitri P. Bertsekas

By Dimitri P. Bertsekas

This extensively referenced textbook, first released in 1982 by means of educational Press, is the authoritative and accomplished remedy of a few of the main well-known limited optimization tools, together with the augmented Lagrangian/multiplier and sequential quadratic programming tools. between its exact good points, the booklet: 1) treats broadly augmented Lagrangian equipment, together with an exhaustive research of the linked convergence and cost of convergence homes 2) develops comprehensively sequential quadratic programming and different Lagrangian tools three) presents an in depth research of differentiable and nondifferentiable specific penalty equipment four) offers nondifferentiable and minimax optimization equipment according to smoothing five) comprises a lot extensive study now not present in the other textbook

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Direct calculation using (8) yields that if A,-, have A, = L, L;, where = Li- ,Lf- INTRODUCTION ,,then we also and a, is given by (9). Thus, to show that the factorization given above is valid, it will be sufficient to show that a,, - lf 1, > 0, and thus A,, is well defined as a real number from (13). Then because A, is positive definite, we have = Thus, A,, as defined by (13) is well defined as a positive real number. In order to show uniqueness of the factorization, a similar induction argument may be used.

On the other hand, Newton's method in its pure form has several serious drawbacks. First, the inverse [V2f (xk)]-' may fail to exist, in which case the method breaks down. This may happen, for example, iff is linear within some region in which case V2f = 0. Second, iteration (40) is not a descent method in the sense that it may easily happen that f (xk+,) > f (x,). Third, the method tends to be attracted by local maxima just as much 1. 44 INTRODUCTION as it is attracted by local minima. 17 where it is assumed that Vg(x*) is invertible but not necessarily positive definite.

Thus, d, will be the Newton direction (49) for all xk sufficiently close to x. 3 45 UNCONSTRAINED MINIMIZATION a, will equal unity. 17. A variation of this modification scheme is given by the iteration x, + 1 = Xk + a,[@, 4' - (1 - ~ k P (f~ k ) l , where D is a positive definite matrix and d r is the Newton direction ' if [V2f(xk)] - exists. Otherwise df = - DVf (x,). The stepsize a, is chosen by an Armijo-type rule with initial stepsize unity whereby a, = Pmkand m, is the first nonnegative integer m for which f (xk) - f r x k + fimdk(Prn>I 2 -opm lVf ( ~ k12, > where o E (0, 31,p E (0, I), and This is a line search along the curve of points of the form z, = x[xdf - (1 - c)DVf (x,)] with a E [0, 11.

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