A. R. Conn, N. I. M. Gould, and Ph. L. Toint. A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Technical Report 92/07, Department of Mathematics, FUNDP, Namur, Belgium, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....been studying this basic algorithm. While their algorithm differs from ours in many particulars, the general approach is similar. For related work on primal dual interior point methods see [10] 2] 13] 5] An alternative barrier approach to inequality constrained problems is discussed in [1]. AN INTERIOR POINT ALGORITHM FOR NONCONVEX NONLINEAR PROGRAMMING 5 2. ALGORITHM MODIFICATIONS FOR CONVEX OPTIMIZATION As mentioned in the previous section, the search directions given by (8) together with a steplength selected simply to ensure that the vectors w and y remain component wise ....

A.R.Conn, N. Gould, and Ph.L. Toint. A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Math. of Computation, 66:261--288, 1997.

....shifted penalty functions, including twice continuously differentiable ones. We add that the continuing interest in nonquadratic modified Lagrangians stems from the fact that, in contrast with the quadratic one, they are twice continuously differentiable, and this facilitates their minimization [Ber82, BTYZ92, BrS93, BrS94, CGT92, CGT94, GoT89, IST94, JeP94, Kiw96, NPS94, Pol92, PoT94, Teb92, TsB93]. By the way, our convergence results seem stronger than ones in [IST94, PoT94] for modified barrier functions, resulting from a dual application of (1.3) with D k h (x; x k ) replaced by an entropy like OE divergence. The paper is organized as follows. In x2 we recall the definitions of ....

....1 c k m X i=1 ln[1= k i Gamma c k g i (x) k 1 i = k i = 1 Gamma c k k i g i (x k 1 ) i = 1: m; i.e. to an inexact shifted logarithm barrier method (which was also derived heuristically in [Cha94, Ex. 4. 2] This method is related, but not indentical, to ones in [CGT92, GMSW88]; cf. CGT94] Example 7.12. If (t) Gammat ff =ff, ff 2 (0; 1) cf. Ex. 2.7.2) 7.20) reduces to x k 1 2 Arg min x ( f(x) Gamma 1 fic k m X i=1 [ k i ) 1= fi Gamma1) Gamma c k g i (x) fi ) k 1 i = k i ) 1= fi Gamma1) Gamma c k g i (x k 1 ) fi ....

A. R. Conn, N. I. M. Gould and Ph. L. Toint, A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds, Report 92/07, D'epartement de Math'ematique, Facult'es Universitaires de Namur, Namur, 1992.

.... 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 adding appropriate quantities to the diagonals of B, if necessary, Chapter 3 of Conn et al. 1992b, Gill and Murray, 1974, Gill et al. 1992, Schlick, 1993 and Schnabel and Eskow, 1991) For example, Schnabel and Eskow use Gerschgorin bounds to determine the amount to add to the diagonal. They choose diagonal pivots and change the diagonal as little as is 5 reasonable in order to maintain ....

....provided one does at least as well as the generalized Cauchy point. One obtains better convergence, and ultimately a satisfactory asymptotic convergence rate, by further reducing the model function. This is the trust region basis for the kernel algorithm SBMIN (Conn et al. 1988a) of LANCELOT (Conn et al. 1992b) It can be summarized as follows: ffl Find the generalized Cauchy point based upon a local (quadratic) model. ffl Fix activities to those at the generalized Cauchy point. ffl (Approximately) solve the resulting reduced problem whilst maintaining account of the trust region and bounds. ffl ....

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A. R. Conn, N. I. M. Gould, and Ph. L. Toint. A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Research Report RC 18049, IBM T. J. Watson Research Center, Yorktown Heights, USA, 1992.

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Conn, A. R., Gould, N. I. M. and Toint, P. L. (1992a). A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds, Technical Report 92/07, Department of Mathematics, FUNDP, Namur, Belgium.

....amongst others. Variations on the theme include the modified (unshifted) barrier function of Jittorntrum and Osborne (1980) the shifted barrier functions of Gill et al. 1988) and Freund (1991) the modified (shifted) barrier function of Polyak (1992) and the Lagrangian barrier function of Conn et al. 1992a) A typical barrier function method attempts to solve (1.1) by (approximately) minimizing a sequence of barrier functions 9(x; w (k) s (k) for appropriate sequences of weights fw (k) g and shifts fs (k) g. The approximate minimizer x (k) of 9(x; w (k) s (k) is generally ....

....small. We need to be cautious here as there is no guarantee that x (k) is feasible for the shifted constraints once the updates (4.30) 8 have been applied. It may then be necessary to find an alternative starting point for the k 1 st inner iteration. Suitable methods are given by Conn et al. 1992a) If the asymptotic phase of the algorithm is reached, the penalty parameter (k) remains fixed at some value 3 0 and the Lagrange multiplier estimates (k 1) are defined by (k 1) i = w (k) i c i (x (k) s (k) i ; for i = 1; m (4:32) c.f. 3.9) Here, the shifts ....

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A. R. Conn, N. I. M. Gould, and Ph. L. Toint. A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Technical Report 92/07, Department of Mathematics, FUNDP, Namur, Belgium, 1992.

....values of and ) one normally constructs a quadratic model of the ALF and (approximately) minimizes this within the region defined by the simple bounds, and, perhaps, a trust region, on x and y. A simple minded approach to this in fact the approach taken within the LANCELOT code SBMIN (see Conn et al. 1992b) is to treat all variables in the same way. Thus slack variables are not treated differently from the problem variables x. If there are many slack variables relative to the number of problem variables for instance, as would be the case for problems where a parameterized (or semi infinite) ....

.... along the piecewise linear path obtained by projecting the arc (x ffp x ; y ffp y ) ff 0, back into the feasible region (see Bertsekas, 1982 or Conn et al. 1988) Furthermore, additional Newton steps may be performed using reduced sets of free variables if so required (see, for instance, Conn et al. 1992b, Section 3.2.3) 3.2 Iterative methods If we wish to use an iterative method, we merely need to find a vector (p x ) Ix ] for which (p x ) T [Ix ] g [Ay ] x) A [Ay ;Ix ] x) T [Ay ] 0: 3:10) We might achieve this by, for instance, applying a CG truncated Newton method (see, for ....

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A. R. Conn, N. I. M. Gould, and Ph. L. Toint. A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Technical Report 92/07, Department of Mathematics, FUNDP, Namur, Belgium, 1992. 5

....Introduced by Griewank and Toint (1982a) this particular structure and its generalization to group partial separability have shown to be very useful in the design of algorithms for large scale optimization problems, both constrained and unconstrained. For instance, the LANCELOT package (see Conn et al. 1992b) is based on this structural concept. In this context, we will consider that the partially separable structure of a function is given, and will then try to improve it with a very specific goal in mind: we aim at reducing the amount of computational time spent in the calculation of a step of a ....

....structures In order to motivate our approach in a simple framework, we consider the unconstrained optimization problem of minimizing f(x) m X i=1 f i (x) 2:1) a partially separable function of the n dimensional real vector x. We refer the reader to Conn et al. 1990) or Chapter 2 of Conn et al. 1992b) for a detailed introduction of partial separability and group partial separability. Assume furthermore, for the moment, that f(x) is convex and twice continuously differentiable. Suppose finally that the problem is to be solved on a sequential computer by applying Newton s method, and that the ....

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A. R. Conn, N. I. M. Gould, and Ph. L. Toint. A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Technical Report 92-067, Rutherford Appleton Laboratory, Chilton, England, 1992. 9

.... (ff) ff 0p for any p 0. The shifts are all nonnegative, the weights for infinite bounds are zero while those corresponding to finite bounds are strictly positive. A variety of barrier functions have been proposed and the reader is referred to Fiacco and McCormick (1968) Wright (1992) or Conn et al. 1992) for details. In Section 2 of this paper, we discuss general issues of convergence for schemes for solving (1.1) 1.2) and lay the foundations for the linear algebraic processes we shall employ; we formally state the aims of the paper in Section 2.3. In Section 3, we consider how convergence may ....

A. R. Conn, N. I. M. Gould, and Ph. L. Toint. A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Technical Report 92-067, Rutherford Appleton Laboratory, Chilton, England, 1992.

....algorithm for optimization with general nonlinear inequality constraints and simple bounds by A. R. Conn 1 , Nick Gould 2 , and Ph. L. Toint 3 RAL 92 068 November 16, 1992 Abstract. This paper considers the number of inner iterations required per outer iteration for the algorithm proposed by Conn et al. 1992a) We show that asymptotically, under suitable reasonable assumptions, a single inner iteration suffices. 1 Mathematical Sciences Department, IBM T.J. Watson Research Center, PO Box 218, Yorktown Heights, NY 10598, USA 2 Central Computing Department, Rutherford Appleton Laboratory, Chilton, ....

....of a globally convergent algorithm for optimization with general nonlinear inequality constraints and simple bounds A. R. Conn, Nick Gould and Ph. L. Toint November 16, 1992 Abstract This paper considers the number of inner iterations required per outer iteration for the algorithm proposed by Conn et al. 1992a) We show that asymptotically, under suitable reasonable assumptions, a single inner iteration suffices. 1 Introduction In this paper, we consider the nonlinear programming problem minimize x2 n f(x) 1:1) subject to the general constraints c i (x) 0; i = 1; m; 1:2) and the ....

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A. R. Conn, N. I. M. Gould, and Ph. L. Toint. A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Technical Report 92/07, Department of Mathematics, FUNDP, Namur, Belgium, 1992.

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A. R. Conn, N. I. M. Gould, and Ph. L. Toint. A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Technical Report 92/07, Department of Mathematics, FUNDP, Namur, Belgium, 1992.

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Conn, A.R. , Gould N. , Toint Ph.L. : A globally convergent Lagrangian barrier algorithm for optimization with general constraints and simple bounds ; Tech. Report 92/07(2nd rev.) FUNDP , Namur , Belgium (1995)

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