Guddat,J., Jongen,H.,Th. and Ruckmann,J., On stability and stationary points in nonlinear optimization, J. Aust. Math. Soc., Ser. B 28 (1986), 36-56.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....n Gamma m: In case when the MFCQ holds at each x 2 M(t) but not the LICQ, the local structure of M(t) is more complicated. The set M(t) is a manifold of dimension n Gamma m; but not a differentiable one. We only have a continuous change of coordinates, even if we assume k 1 (cf. 13] [15]) The following theorem is a direct consequence of Theorem B in [15] 6 R. Fandom Noubiap Theorem 2.3 Assume for all t 2 [0; 1] that M(t) is compact and the MFCQ is satisfied at all x 2 M(t) Then M(t 1 ) and M(t 2 ) are homeomorphic for all t i 2 [0; 1] i = 1; 2. Now we introduce the ....

....not the LICQ, the local structure of M(t) is more complicated. The set M(t) is a manifold of dimension n Gamma m; but not a differentiable one. We only have a continuous change of coordinates, even if we assume k 1 (cf. 13] 15] The following theorem is a direct consequence of Theorem B in [15]: 6 R. Fandom Noubiap Theorem 2.3 Assume for all t 2 [0; 1] that M(t) is compact and the MFCQ is satisfied at all x 2 M(t) Then M(t 1 ) and M(t 2 ) are homeomorphic for all t i 2 [0; 1] i = 1; 2. Now we introduce the following notations: X loc : f(x; t) 2 IR n Theta IR=x is a local ....

Guddat,J., Jongen,H.,Th. and Ruckmann,J., On stability and stationary points in nonlinear optimization, J. Aust. Math. Soc., Ser. B 28 (1986), 36-56.

....in particular it excludes that by changing x a (connected) component of Y (x) may disappear (or a new component may appear) Recall that a su#cient condition for the continuity of Y is the condition A MFCQ . We give an example. Example 2 Consider the problem P: min x 2 s.t. x # M = x # [ 2, 2] y x 1 # 0,y# Y (x) with Y (x) y # IR 0 # y, y # x . We find Y (x) x,0] x#0 # ,x 0 and M = 2, 1] # (0, 2] At x = 0 the condition MFCQ is not fulfilled for Y (x) 0 . Obviously, the mapping Y is not continuous at x =0,M is not closed and a solution of P does ....

....that a su#cient condition for the continuity of Y is the condition A MFCQ . We give an example. Example 2 Consider the problem P: min x 2 s.t. x # M = x # [ 2, 2] y x 1 # 0,y# Y (x) with Y (x) y # IR 0 # y, y # x . We find Y (x) x,0] x#0 # ,x 0 and M =[ 2, 1] # (0, 2] At x = 0 the condition MFCQ is not fulfilled for Y (x) 0 . Obviously, the mapping Y is not continuous at x =0,M is not closed and a solution of P does not exist. We now outline a possible way to construct a continuous discretization Y # (x)ofY (x)as 9 given in A3. In practice, this ....

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Guddat J., Jongen H.Th. and Ruckmann J., On stability and stationary points in nonlinear optimization, J. Australian Math. Soc., Series B 28, 36-56, (1986).

....of a smooth non degenerate function f : X R to the Betti numbers of X hold for X being a cornered manifold , provided critical point and index are suitably re defined. This paper was submitted for publication in 1990; however, the result was already known to Guddat, Jongen and Rueckmann [6] (1986) The proof presented here differs considerably from the proof of Guddat et al. Guddat et al. use a direct argument following the original approach of M. Morse. The proof presented here uses a logarithmic barrier function to approximate the constrained problem. Morse theory ( 7] ....

Guddat, J., Jongen, H. Th., and Rueckmann, J. On stability and stationary points in nonlinear optimization. J. Austral. Math. Soc. Ser. B, 28, 36--56, (1986).

....if (x; t) 2 K(x 0 ; 0) then (x; t) belongs to S 2f1;2;3;5g Sigma gc . 2 Remark 2.7 Assume (C4) Now we have a look at Fig. 2.2. Since the MFCQ is satisfied, points of Type 5 in (j) and (k) are excluded. 2 Finally we present a consequence of a general topological stability result given in [13]: Theorem 2.8 (cf. 13] We assume (C1) and (C4) Then M(t 1 ) is homeomorphic with M(t 2 ) for all t 1 ; t 2 2 [0; 1] 2 10 Multiobjective optimization: embeddings On the program package PAFO (this is a very short version of 4.5 and 5.2 in [12] PAFO (cf. 19] and [5] is based on a ....

....then (x; t) belongs to S 2f1;2;3;5g Sigma gc . 2 Remark 2.7 Assume (C4) Now we have a look at Fig. 2.2. Since the MFCQ is satisfied, points of Type 5 in (j) and (k) are excluded. 2 Finally we present a consequence of a general topological stability result given in [13] Theorem 2. 8 (cf. [13]) We assume (C1) and (C4) Then M(t 1 ) is homeomorphic with M(t 2 ) for all t 1 ; t 2 2 [0; 1] 2 10 Multiobjective optimization: embeddings On the program package PAFO (this is a very short version of 4.5 and 5.2 in [12] PAFO (cf. 19] and [5] is based on a pathfollowing method (called PATH ....

Guddat, J., Jongen, H.Th., Ruckmann, J.-J. (1986): On Stability and Stationary Points in Nonlinear Optimization. J. Austral. Math. Soc., Ser. B 28, 36-56.

....c(t) t 1 Gamma t tends to 1 if t tends to 1. This one parametric optimization problem has the following disadvantages: The problem is not defined for t = 1, the objective function is exactly once continuously differentiable (i.e. the results of parametric optimization presented in [8, 9, 10, 11, 12, 13, 15, 16, 7] a short summary is given in Chapter 2 are not applicable) we do not know any starting point for t = 0. It is easy to see that these disadvantages will not appear for P 1 (t) Moreover, there are further important properties of P 1 (t) cf. Theorem 1.1) The term (1 Gamma t) x Gamma x 0 ....

....(x ; 1) Furthermore (x; t) 2 K(x 0 ; 0) belongs to S 2f1;2;3;5g Sigma gc . 2 Remark 2.7 Now we have a look at Fig. 2.2. Since the MFCQ is satisfied, points of Type 5 in (j) and (k) are excluded. 2. Finally we present a consequence of a general structural stability result given in [9]: Theorem 2.8 ( 9] We assume (C1) and (C4) Then M(t 1 ) is homeomorphic with M(t 2 ) for all t 1 ; t 2 2 [0; 1] 2 3 On connecting curves Considering Theorem 2.5, Corollary 2.6 and Theorem 2.8 we first ask for a condition on the original problem (P ) ensuring that the MFCQ is satisfied for ....

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Guddat, J., Jongen, H.Th., Ruckmann, J.-J. (1986): On stability and stationary points in nonlinear optimization. J. Austral. Math. Soc., Ser. B 28, 36-56.

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Guddat, J.; Jongen, H. Th. and R#ckmann, J.: On stability and stationary points in nonlinear optimization. J. Aust. Math. Soc., Ser. B 28(1986); 36 \Gamma 56

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