A.R. Conn, N.I.M. Gould and Ph.L. Toint. A note on using alternative second-order models for the subproblems arising in barrier function methods for minimization, Numer. Math. 68, pp 17--33, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....convergence of this approach is obtained by decreasing k superlinearly to zero (that is, lim k 1 k 1 = k = 0) while taking no more than a fixed number of Newton steps at each value of k . In the case of LICQ, rapid convergence of this type has been investigated by Conn, Gould, and Toint [7], Benchakroun, Dussault, and Mansouri [4] Wright and Jarre [29] and Wright [26] We anticipate that similar results will continue to hold when LICQ is replaced by MFCQ, because the central path continues to be smooth and the convergence domain (46) for Newton s method is similar in both cases. A ....

A. R. Conn, N. I. M. Gould, and Ph. L. Toint. A note on using alternative secondorder models for the subproblems arising in barrier function methods for minimization. Numerische Mathematik, 68:17--33, 1994.

.... s i 0; i = 1; m; s i = 0; i = 1; m: For discussions on the use of shift variables in interior methods, see, e.g. Gill et al. GMSW88] Gay, Overton and Wright [GOW98, Section 7] Jarre and Saunders [JS95] Todd [Tod94] Polyak [Pol92] Freund [Fre91] and Conn, Gould and Toint [CGT94]. See also Powell [Pow69] for a discussion on the use of shift variables in a penalty method. An acceptable initial iterate can be obtained by choosing x arbitrary and letting s i = maxfc i (x ) 0g, i = 1; m, for some positive . For ease of notation, we introduce one shift ....

A. R. Conn, N. I. M. Gould, and Ph. L. Toint. A note on using alternative secondorder models for the subproblems arising in barrier function methods for minimization. Numer. Math., 68, 17-33, 1994.

....known that the first Newton step for each value of the one taken immediately after is decreased from its previous value usually must be curtailed sharply to avoid leaving the feasible region. That is, a step length ff considerably smaller than 1 is usually needed (see Conn, Gould, and Toint [1], Wright [4] and Wright and Jarre [6] Often, however, steplengths of 1 can be taken safely after a few Newton iterations, yielding quadratic convergence toward x( This phenomenon may not seem surprising, because it is also well known that, under typical nondegeneracy and secondorder ....

A. R. Conn, N. I. M. Gould, and P. L. Toint, A note on using alternative second-order models for the subproblems arising in barrier function methods for minimization, Numerische Mathematik, 68 (1994), pp. 17--33.

....ill conditioning is not the only defect of primal barrier methods. Even if the Newton direction is calculated with perfect accuracy, primal barrier methods suffer from inherently poor scaling of the search direction during the early iterations following a reduction of the barrier parameter; see [35, 5]. Thus, unless special precautions are taken, a full Newton step cannot be taken immediately after the barrier parameter is reduced. This fundamentally undesirable property implies that the classical primal barrier method will be unavoidably inefficient. A fascinating but unresolvable question is ....

A. R. Conn, N. I. M. Gould, and P. L. Toint (1994). A note on using alternative second-order models for the subproblems arising in barrier function methods for minimization, Num. Math. 68, 17--33.

.... as a starting point for Newton s method applied to P ( Delta; it is well known that the first Newton step for each value of usually is a poor search direction, and a step length ff considerably smaller than 1 usually is needed to remain feasible at this iteration (see Conn, Gould, and Toint [5], M. Wright [26] and S. Wright and Jarre [29] Often, however, subsequent iterations of Newton s method converge rapidly to x( Although the Hessian P xx (x; is positive definite near x = x( see the proof of [10, Theorem 12] the observed rate of convergence of Newton s method is better ....

....(2) that take other kinds of steps has been described in a number of papers. In the main, these papers address the poor performance of Newton s method at the step taken immediately after a reduction of the barrier parameter by using some other method to generate this step. Conn, Gould, and Toint [5] advocate taking a primal dual step, obtained by applying Newton s method for nonlinear equations to the system (9) treating x and as independent variables. Gould [15] applies a similar strategy to the quadratic penalty function for equality constrained optimization. The related approach is to ....

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A. R. Conn, N. I. M. Gould, and Ph. L. Toint. A note on using alternative secondorder models for the subproblems arising in barrier function methods for minimization. Numerische Mathematik, 68:17--33, 1994.

....to a Newton iteration on the KKT conditions of the barrier problem (2.1) which are given by #f(x) A h (x)# h A g (x)# g = 0 (3.3) S 1 e # g = 0 (3.4) h(x) 0 (3.5) g(x) s = 0. 3.6) Several authors, including Jarre and S. Wright [30] M. Wright [41] and Conn, Gould and Toint [16] have given arguments suggesting that the primal search direction will often cause the slack variables to become negative, and can be ine#cient. Although those papers consider a di#erent formulation of the problem, it is easy to see [29] that the arguments apply in our case. Research in linear ....

A.R. Conn, N.I.M. Gould and Ph.L. Toint. A note on using alternative second-order models for the subproblems arising in barrier function methods for minimization, Numer. Math. 68, pp 17--33, 1994.

....convergence of this approach is obtained by decreasing k superlinearly to zero (that is, lim k 1 k 1 = k = 0) while taking no more than a fixed number of Newton steps at each value of k . In the case of LICQ, rapid convergence of this type has been investigated by Conn, Gould, and Toint [7], Benchakroun, Dussault, and Mansouri [4] Wright and Jarre [29] and Wright [26] We anticipate that similar results will continue to hold when LICQ is replaced by MFCQ, because the central path continues to be smooth and the convergence domain (46) for Newton s method is similar in both cases. A ....

A. R. Conn, N. I. M. Gould, and Ph. L. Toint. A note on using alternative secondorder models for the subproblems arising in barrier function methods for minimization. Numerische Mathematik, 68:17--33, 1994.

....convergence of this approach is obtained by decreasing k superlinearly to zero (that is, lim k 1 k 1 = k = 0) while taking no more than a fixed number of Newton steps at each value of k . In the case of LICQ, rapid convergence of this type has been investigated by Conn, Gould, and Toint [7], Benchakroun, Dussault, 28 Stephen J. Wright, Dominique Orban and Mansouri [4] Wright and Jarre [29] and Wright [26] We anticipate that similar results will continue to hold when LICQ is replaced by MFCQ, because the central path continues to be smooth and the convergence domain (46) for ....

A. R. Conn, N. I. M. Gould, and Ph. L. Toint. A note on using alternative secondorder models for the subproblems arising in barrier function methods for minimization. Numerische Mathematik, 68:17--33, 1994.

....3.2, observe that Newton s method automatically imposes the choice y k = h(x k ) and in particular for k = 0. 4 The Homotopic Approach The discussion displayed in this section was motivated on comments by N. I. M. Gould, who also called the attention of the authors to the references Refs. [10], 11] 12] and [13] The iteration (5) applied to the solution of (4) can be written in the following way: r 2 (x k ) m X k=1 y k i r 2 h i (x k ) 1 h 0 (x k ) T h 0 (x k ) x k 1 Gamma x k ) GammarP (x k ) 31) where y k = h(x k ) 32) ....

....are chosen conveniently, only one modified Newton iteration is necessary for each value of the penalty parameter and overall convergence of the algorithm to the solution of the original problem is global and two step quadratic. Related results, for different homotopies, were obtained in Refs. [10], 11] and [13] The predictor corrector idea underlies most theoretically justified versions of penalty methods. In the predictor phase, a tangent step to the homotopy curve is taken, which is able to provide a good initial point 12 for Newton s method. The corrector phase consists of the ....

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Conn, A. R., Gould, N. I. M., and Toint, Ph. L., A Note on Using Alternative Second-Order Models for the Subproblems Arising in Barrier Function Methods for Minimization, Numerische Mathematik, Vol. 68, pp. 17--33, 1994.

.... , where x is a local minimizer of (1) After x( has been approximated for a particular value of 0, an obvious way to proceed is to decrease to some new value and then take a Newton step for minimizing the barrier function P (x; Unfortunately, as shown in Conn, Gould, and Toint [1] and Wright [6] for example, the Newton direction may be a poor search direction. Newton s method usually takes This research was supported by Obermann Fellowships in the Center for Advanced Studies at the University of Iowa and by the Mathematics, Information, and Computational Sciences ....

A. R. Conn, N. I. M. Gould, and P. L. Toint, A note on using alternative second-order models for the subproblems arising in barrier function methods for minimization, Numerische Mathematik, 68 (1994), pp. 17--33.

....of In solving the equation (11) there are a number of potential dangers. For example r xx Gamma x k ; w k ; s k Delta may be badly conditioned or x k Deltax may violate some of the shifted constraints. Searching for an alternative Newton model we follow an idea described in [Conn et al. 1994]. Suppose x and (corresponding to def = u(x; w k ; s k ) being independent variables and have a look at the equations Cx d Gamma A = 0 Gamma Gamma A T x Gamma b Delta i s k Delta Delta i Gamma w k i = 0 i = 1; m; 27) Note that (27) and (10) have identical ....

....i (A T x k Gamma b) i s k Gamma1 8 i = 1; m may be a potentially better choice. By (6) we see that the first term of the right hand side of (28) becomes small for sufficiently large k and the second term becomes small too if the shift s k and the weights fw k i g converge. See [Conn et al. 1994] for details) Moreover, if after a change in the shift and weights the maximum element in D A i (A T x Gamma b) i s k 1 7 NUMERICAL RESULTS 18 exceeds a given limit ( 10 4 ) we reject the shift s k , the weights w k and the current iterate x k Gamma1 . Continuing with s ....

Conn, A.R. , Gould N. , Toint Ph.L. : A note on using alternative second-order models for the subproblems arising in barrier functions methods for minimization ; Numer. Math. 68 : 17-33 (1994)

....exact quadratic augmented Lagrangian of Spellucci, 23] We compare the numerical results with those obtained by tranforming the problem according theorem 3. 1 followed by a subsequent solution using a shifted logarithmic barrier function method based on work of Polyak [22] Conn, Gould and Toint [4], which is fully described in [7] We compile the properties of these approaches here and refer the reader to the original papers for the proofs. 4 THE METHOD OF FRIEDLANDER, MART INEZ AND SANTOS The essence of this approach is given by Theorem 4.1 In addition to the assumptions of section 3 ....

....an inequality constrained QP problem of the form f(x) 1 2 x T Bx b T x = min subject to c(x) C T x c 0 0 : 9.1) Problems of this type may be solved efficiently by interior point methods. We decided to use an implementation of a shifted log barrier approach (see [22] [4], 7] Here the function (x; fl; s) def = f(x) Gamma r X i=1 fl i log(c i (x) s i ) 9.2) is minimized with respect to x for given parameter vectors fl 0 and s 0. r is the number of components of inequalities c, r = n 2m 1 here. A sequence of weights fl k and shifts s k has ....

Conn, A.R.; Gould, N.; Toint, Ph.L.: A note on using alternative second-order models for the subproblems arising in barrier function methods for minimzation. Num. Math. 68, (1994), 17--33 .

....and sparse Cholesky factorizations. Very briefly, the former approach tries to assemble the required entries in a piecemeal manner. Once a complete column and row are assembled one can do the corresponding elimination, thus building up the corresponding elements of L and U. For details see Conn et al. 1993a) Duff et al. 1986, Chapter 10) Duff et al. 1988) Duff and Reid (1982) Duff and Reid (1983) and Duff and Reid (1993) By contrast, the sparse Cholesky factorization primarily tries to order the rows and columns of B whilst maintaining reasonable stability by including the possibility of ....

....using a limited memory BFGS method. It is intended for large scale problems. ffl VE09 of Gould (1991) This package obtains local solutions to general, non convex quadratic programming problems, using an active set method, and is intended to be suitable for large scale problems. ffl VE14 of Conn et al. 1993g) This package solves bound constrained quadratic programming problems using a barrier function method and is again intended to be suitable for large scale problems. ffl VF13 of Powell (1982) This package solves general nonlinearly constrained problems using a sequential quadratic programming ....

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A. R. Conn, Nick Gould, and Ph. L. Toint. A note on using alternative second-order models for the subproblems arising in barrier function methods for minimization. Research Report RC18898, IBM T. J. Watson Research Center, Yorktown Heights, USA, 1993.

.... for a barrier subproblem is likely not to be accepted as a rst step if the minimization process is started from x k 1 , even if x k 1 is close to a solution of NLP [17] The determination of better initial points in interior methods has recently been examined by several authors, both for primal [3, 12] and primal dual barrier methods [1, 22, 23] as well as for exterior penalty methods (see [10] whose results were the inspiration for [5] In particular, Dussault [5] expands the primal central path about the current iterate. In this paper, we apply a similar analysis in the more general ....

.... we intend to determine conditions under which, asymptotically, a single Newton step is strictly feasible (in contrast with the purely primal case) and results in a point that satis es suitable barrier subproblem termination rules, after every reduction of the barrier parameter (see for instance [1, 3, 4] for previous work on the subject) This is shown to imply a componentwise Q superlinear rate of convergence, a stronger result than simply Q superlinear convergence of the vector of variables and Lagrange multipliers. Furthermore, this rate of convergence may be made arbitrarily close to ....

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A. R. Conn, N. I. M. Gould, and Ph. L. Toint. A note on using alternative second-order models for the subproblems arising in barrier function methods for minimization. Numerische Mathematik, 68:17-33, 1994.

....using a limited memory BFGS method. It is intended for large scale problems. ffl VE09 of Gould (1991) This package obtains local solutions to general, non convex quadratic programming problems, using an active set method, and is intended to be suitable for large scale problems. ffl VE14 of Conn et al. 1994) This package solves bound constrained quadratic programming problems using a barrier function method and is again intended to be suitable for large scale problems. ffl VF13 of Powell (1982) This package solves general nonlinearly constrained problems using a sequential quadratic programming ....

A. R. Conn, Nick Gould, and Ph. L. Toint. A note on using alternative second-order models for the subproblems arising in barrier function methods for minimization. Numerische Mathematik, 68(1):17--33, 1994. 38

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A.R. Conn, N.I.M. Gould and Ph.L. Toint. A note on using alternative second-order models for the subproblems arising in barrier function methods for minimization, Numer. Math. 68, pp 17--33, 1994.

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