Gfrerer,H., Guddat,J., Wacker,J. and Zulehner,W., Pathfollowing methods for KuhnTucker curves by an active index set strategy, in A. Bagchi, and H.Th.Jongen (eds), System and Optimization, Proc. Twente Workshop, Lect. Notes Control Inform. Sci. 66, Springer-Verlag, Berlin, 1985, pp.111-32.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(regular problems in the sense of Jongen, Jonker and Twilt) cf. Definition 2:4) For the analysis with respect to JJT regular problems, we will assume a higher degree of differentiability of the problemfunctions. Let us recall now the well known concept of embedding (cf. e.g. 1] 2] 4] 9] [10], 11] 13] 21] 22] 24] Construct a one parametric optimization problem P (t) minff(y; t)jy 2 M (t)g; t 2 [0; 1] where M(t) fy 2 IR n jh i (y; t) 0; i 2 I; g j (y; t) 0; j 2 Jg n n, J is a finite index set with J J , with at least the following properties: ....

....Constraint Qualification (shortly EnMFCQ) is satisfied for the problem (P) if for all q q and for all x 2 B q (0) there exists a vector j 2 IR n such that: g j (x) Dg j (x)j 0; j 2 J; with g j (x) 0; 2x T j 0; if kxk 2 = q: Remark 4. 6 In the literature (cf. e.g.[10], 13] 5] the EnMFCQ is known as a condition for the convergence of the penalty and exact penalty methods. Theorem 4.7 Assume that EnMFCQ is satisfied for the problem (P) Then the MFCQ is fulfilled for all x 2 M (t) and for all t 2 (0; 1] Furthermore if the set J 0 (x 0 ) fj 2 J jg j ....

Gfrerer,H., Guddat,J., Wacker,J. and Zulehner,W., Pathfollowing methods for KuhnTucker curves by an active index set strategy, in A. Bagchi, and H.Th.Jongen (eds), System and Optimization, Proc. Twente Workshop, Lect. Notes Control Inform. Sci. 66, Springer-Verlag, Berlin, 1985, pp.111-32.

.... have full dimension(cf. 13] M i = cl intM i i = 1; 2: C3) With the condition (C3) we want to exclude cases where M 1 ae (P 3 i=1 G i ) M 2 ae (P G 1 ) Let us recall now the well known concept of embedding (cf. e.g. Allgower [1] Guddat et al. 5] Dentcheva [3] Gfrerer et al. [10]) and propose the following embeddings in order to solve (P i ) i = 1; 2: Let x o 2 P be arbitrarily chosen and dene minff(x; t) j x 2 M 1 (t)g t 2 ( Gamma1; 1] P 1 (t) 2 Theoretical Background 4 where f(x; t) x Gamma x o ) T D(x Gamma x o ) M 1 (t) fx 2 R n j g j (x; ....

....at each x 2 M i (t) points of Type 4 are excluded. Hence, the case II in Section 2 may not appear. We ask for a condition on the original problems (P i ) i = 1; 2. We discuss the so called Enlarged Mangasarian Fromovitz Constraint Qualication (brieAEy EnMFCQ) introduced by Gfrerer et al. [10]. Denition 2 (EnMFCQ) The EnMFCQ is satised for P 1 if, for all x 2 P (polyhedron) there exists a vector 2 R n with g j (x) Dg j (x) 0; j 2 J (x) fj 2 J j g j (x) 0; j 6= s 2g D x g s 2 (x) 0; if g s 2 (x) 0: The EnMFCQ is satised for P 2 if, for all x 2 P, there exists a ....

Gfrerer, H.; Guddat, J.; Wacker, Hj. and Zulehner, W.: Pathfollowing methods for Kuhn-Tucker curves by an active index set strategy. In: Bagchi, A., Jongen, H.Th. (eds.) Systems and optimization. Lecture Notes in Control and Information Sciences 66, Springer-Verlag, Berlin, Heidelberg, New York (1985), 111 \Gamma 131

....all (b; c; d) 2 IR n Theta IR m Theta IR s , the problem P (b;c;d) t) minff(x; t) b T x fi fi fi fi fi fi fi h i (x; t) c i = 0; i 2 I; g j (x; t) d j 0; j 2 J 9 = is KH regular. 2 Now, we present two theorems that are essential for our analysis. Theorem 2. 5 (cf. [4]) We assume (C1) M(t) is non empty and there exists a compact set C with M(t) C for all t 2 [0; 1] C2) P (t) is KH regular with respect to [0; 1] C3) There exists a t 1 0 and a continuous function x : 0; t 1 ) IR n such that x(t) is the unique stationary point for P (t) for t 2 ....

.... and compact for all t 2 [0; 1) Finally, we discuss the assumption (A4) This is a condition to the parameter depending feasible set M 2 (t) for all t 2 [0; 1] First, we ask for a sufficient condition with respect to the set M( 1 ) E(p) which we will call, as in other papers (cf. e.g. [4] [11] 2] the Enlarged Mangasarian Fromovitz Constraint Qualification (briefly EnMFCQ) Let (F; G) 2 C 1 (IR n ; IR) s l . The EnMFCQ for M( 1 ) E(p) For all x 2 E(p) it holds: There exists a 2 IR n with (i) g j (x) Dg j (x) 0; j 2 fj 2 J j g j (x) 0g (ii) f k (x) Df k ....

Gfrerer, H., Guddat, J., Wacker, Hj., Zulehner, W. (1985): Pathfollowing Methods for Kuhn-Tucker Curves by an Active Index Set Strategy. In: Bagchi, A., Jongen, H.Th. (eds.) Systems and optimization. Lecture Notes in Control and Information Sciences 66, Springer-Verlag Berlin, Heidelberg, New York, 111-131.

....2 IR n j h i (x) 0; i 2 I; g j (x) 0; j 2 Jg; 1.0) I : f1; mg; m n; J : f1; sg; and f; h i ; g j 2 C 2 (IR n ; IR) i 2 I; j 2 J: For some of the results presented we need a higher degree of differentiability. We recall the well known concept of embedding (cf. e.g. [5, 8, 13]) Consider a one parametric optimization problem This research was supported by the Deutsche Forschungsgemeinschaft under Grants Gu304 1 4 P (t) minff(y; t) j y 2 M (t)g; t 2 [0; 1] 1.1) where M (t) fy 2 IR n j h i (y; t) 0; i 2 I; g j (y; t) 0; j 2 Jg n n, J ....

....PATH III the corresponding computer program is called PAFO (cf. 8, 7, 16] computes a numerical description of a compact connected component in Sigma gc and Sigma stat , respectively. In the last part of Chapter 2 we present two theorems that are essential for our analysis. Theorem 2. 5 ([5]) We assume (C1) M(t) is non empty and there exists a compact set C with M(t) C for all t 2 [0; 1] C2) P (t) is KH regular with respect to [0; 1] C3) There exists a t 1 0 and a continuous function x : 0; t 1 ) IR n such that x(t) is the unique stationary point for P (t) for t 2 [0; ....

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Gfrerer, H., Guddat, J., Wacker, Hj., Zulehner, W. (1985): Pathfollowing methods for Kuhn-Tucker curves by an active index set strategy. In: Bagchi, A., Jongen, H.Th. (eds.) Systems and optimization. Lecture Notes in Control and Information Sciences 66, Springer-Verlag Berlin, Heidelberg, New York, 111-131.

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