Gollmer,R.,Guddat,J.,Guerra,F.,Nowack,D. and Ruckmann,J., Pathfollowing methods in Nonlinear Optimization I: Penalty Embedding in [14], 163-214.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Jg 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: Humboldt Universitat, FB Mathematik, PSF 1297, D 10099 Berlin fandom mathematik.hu berlin.de 2 R. Fandom Noubiap We follow the concept of the investigations of the pathfollowing methods in [11] (Penalty Embedding) 3] Exact Penalty Embedding) and [4] Multiplier Embedding) In this paper we investigate why the so called standard embedding is not successful in some cases. Then we modify this embedding from several points of view and show some advantages of the modified embeddings. ....

....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: A1) A ....

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Gollmer,R.,Guddat,J.,Guerra,F.,Nowack,D. and Ruckmann,J., Pathfollowing methods in Nonlinear Optimization I: Penalty Embedding in [14], 163-214.

....i ; g j 2 C 2 (IR n ; IR) i 2 I; j 2 J: Humboldt Universitat, FB Mathematik, PSF 1297, D 10099 Berlin fandom mathematik.hu berlin.de 2 R. Fandom Noubiap We follow the concept of the investigations of the pathfollowing methods in [11] Penalty Embedding) 3] Exact Penalty Embedding) and [4] (Multiplier Embedding) In this paper we investigate why the so called standard embedding is not successful in some cases. Then we modify this embedding from several points of view and show some advantages of the modified embeddings. For our investigation we distinguish between two kinds of ....

....problems (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 ....

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D. Dentcheva, J. Guddat, J.-J. Ruckmann, K. Wendler, Pathfollowing Methods in Nonlinear Optimization III: Lagrange Multiplier Embedding in ZOR - Mathematical Methods of Operations Research (1995) 41: 127-152

.... n; J : f1; sg, and f; h i ; g j 2 C 2 (IR n ; IR) i 2 I; j 2 J: Humboldt Universitat, FB Mathematik, PSF 1297, D 10099 Berlin fandom mathematik.hu berlin.de 2 R. Fandom Noubiap We follow the concept of the investigations of the pathfollowing methods in [11] Penalty Embedding) [3] (Exact Penalty Embedding) and [4] Multiplier Embedding) In this paper we investigate why the so called standard embedding is not successful in some cases. Then we modify this embedding from several points of view and show some advantages of the modified embeddings. For our investigation we ....

....We observe that with both starting points the additional compactification constraint x 2 1000 is not active along the curve in X stat j [0;1] That means we can cancel this constraint. The standard embedding is successful while the penalty embedding ( 11] the exact penalty embedding ([3]) and the multiplier embedding ( 4] are not. In section 4; we will show that the so called Enlarged Mangasarian Fromovitz Constraint Qualification is not satisfied for this problem. t x Type 0.0999004 1.73801 4 0.998438 1.07132 4 0.998903 0.119176 4 0.998644 0.741932 4 0.999778 2.18657 4 ....

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D. Dentcheva, R. Gollmer, J. Guddat, J.-J. Ruckmann, Pathfollowing Methods in Nonlinear Optimization II: Exact Penalty Methods in [7]

....system, can be found in [15] In this case the parametrized system is also not differentiable. In this paper we use embeddings that are interpreted as one parametric optimization problems. Examples of embedings representing different methods of the nonlinear programming are studied in the papers [2, 3, 4, 6]. There are two theoretical conditions ensuring that the set of generalized critical points of a one parametric problem has a structure that is feasible for 2 the use of pathfollowing methods. These conditions are called the regularity condition of a one parametric problem in the sense of ....

....F is defined and open and dense. Naturally, there arises the question, how reasonable is to suppose that the obtained problem is JJT regular. With respect to this problem one usually presents a way to perturb each problem obtained with the specific embedding in order to get a regular problem (see [2, 3, 4, 6]) The density of the set of regular problems ensures for each one parametric problem the existence of a regular problem as close as wanted. The perturbation theorem presented in the paper [16] provides an explicit way to perturb an arbitrary one parametric problem in order to get a regular one. ....

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Gollmer, R., Guddat,J., Guerra, F., Nowack, D. and Ruckmann, J.-J.: Pathfollowing methods in nonlinear optimization I: penalty embedding. In: J. Guddat et al. (eds.), Parametric Optimization and Related Topics III, Peter Lang Verlag, Frankfurt a.M., Berlin, Bern, New York, Paris, Wien, (1993), 163-214

....system, can be found in [15] In this case the parametrized system is also not differentiable. In this paper we use embeddings that are interpreted as one parametric optimization problems. Examples of embedings representing different methods of the nonlinear programming are studied in the papers [2, 3, 4, 6]. There are two theoretical conditions ensuring that the set of generalized critical points of a one parametric problem has a structure that is feasible for 2 the use of pathfollowing methods. These conditions are called the regularity condition of a one parametric problem in the sense of ....

....F is defined and open and dense. Naturally, there arises the question, how reasonable is to suppose that the obtained problem is JJT regular. With respect to this problem one usually presents a way to perturb each problem obtained with the specific embedding in order to get a regular problem (see [2, 3, 4, 6]) The density of the set of regular problems ensures for each one parametric problem the existence of a regular problem as close as wanted. The perturbation theorem presented in the paper [16] provides an explicit way to perturb an arbitrary one parametric problem in order to get a regular one. ....

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Dentcheva, D., Guddat, J., Ruckmann, J.-J., andd Wendler, K., Pathfollowing methods in nonlinear optimization III: multiplier embedding, in ZOR 1994

....into five types, which are then analyzed in detail. This approach is of fundamental importance, since it treats the generic behaviour of one parametric optimization problems. With the tools developed by Jongen, Jonker and Twilt different sequential methods for constrained optimization ( 2] 3] [7], 9] have been studied. Results obtained by using pathfollowing procedures with jumps ( 10] 8] for different penalty, exact penalty and multiplier embeddings with the possibilities of their pure sequential versions were compared. Convergence analyses were stated under usual assumptions on ....

....optimization problems with a quadratic penalty term as in the following one parametric problem. P (t) minff(x) t (1 Gamma t) 2 2 4 X i2I h 2 i (x) X j2J (minfg j (x) 0g) 2 3 5 j x 2 IR n g In this case the values of the parameter are taken increasing to 1. In [7] the close connection of the solution of this problem with a oneparametric constrained one is stated. After some transformations in order to get better properties of the one parametric problem, the finally proposed embeddings are of the form: b P (t) minf f(x; v; w; t)j(x; v; w) 2 M (t)g; ....

Gollmer, R., Guddat,J., Guerra, F., Nowack, D. and Ruckmann, J.-J.: Pathfollowing methods in nonlinear optimization I: penalty embedding. In: J. Guddat et al. (eds.), Parametric Optimization and Related Topics III, Peter Lang Verlag, Frankfurt a.M., Berlin, Bern, New York, Paris, Wien, (1993), 163-214 24

....into five types, which are then analyzed in detail. This approach is of fundamental importance, since it treats the generic behaviour of one parametric optimization problems. With the tools developed by Jongen, Jonker and Twilt different sequential methods for constrained optimization ( 2] [3], 7] 9] have been studied. Results obtained by using pathfollowing procedures with jumps ( 10] 8] for different penalty, exact penalty and multiplier embeddings with the possibilities of their pure sequential versions were compared. Convergence analyses were stated under usual assumptions ....

Dentcheva, D., Guddat, J., Ruckmann, J.-J., and Wendler, K., Pathfollowing methods in nonlinear optimization III: Lagrange Multiplier Embedding, in ZOR 1994, 127-152.

.... stationary point of P (0) is known (and the corresponding Lagrange multipliers are known or easy to compute) A2) P (t) has a global minimizer for every t 2 [0; 1] A3) P (1) is equivalent to (P ) in a certain sense (to be specified below) In this paper we consider three embeddings (cf. [6, 2, 3]) motivated by penalty, exact penalty and Lagrange multiplier methods and we ask the following question: Are there conditions for finding a discretization of the interval [0; 1] 0 = t 0 Delta Delta Delta t i Delta Delta Delta t N = 1 and corresponding stationary points y(t i ) of ....

.... =f(x; v; w) 2 IR n Theta IR m Theta IR s j th i (x) 1 Gamma t) v i Gamma v 0 i ) 0; i 2 I; tg j (x) 1 Gamma t) w j Gamma w 1 j ) 0; j 2 J kx Gamma x 0 k 2 b T x Gamma p 0 kv Gamma v 0 k 2 kw Gamma w 0 k 2 Gamma q 0g: In distinction to P 4 (t) in [6] we consider two points w 0 and w 1 in a suitable manner. Then the starting point (x 0 ; v 0 ; w 0 ) has better properties (cf. Theorem 1.1 below) In [6] it is explained how we obtain such a kind of model for the penalty method. Let us consider this here only briefly. The model for the ....

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Gollmer, R., Guddat, J., Guerra, F., Nowack, D., Ruckmann, J.-J. (1993): Pathfollowing methods in nonlinear optimization I: Penalty embedding. In: Guddat, J. et al. (eds.) Parametric optimization and related topics III. In: Ser. approximation and optimization. Verlag Peter Lang, Frankfurt a.M., Berlin, Bern, New York, Paris, Wien.

.... stationary point of P (0) is known (and the corresponding Lagrange multipliers are known or easy to compute) A2) P (t) has a global minimizer for every t 2 [0; 1] A3) P (1) is equivalent to (P ) in a certain sense (to be specified below) In this paper we consider three embeddings (cf. [6, 2, 3]) motivated by penalty, exact penalty and Lagrange multiplier methods and we ask the following question: Are there conditions for finding a discretization of the interval [0; 1] 0 = t 0 Delta Delta Delta t i Delta Delta Delta t N = 1 and corresponding stationary points y(t i ) of ....

....; w 0 ) is the global minimizer and the only stationary point for P 2 (0) Furthermore, x 0 ; w 0 ) is a non degenerate stationary point. ii) Analogously to Theorem 1.1(ii) iii) Analogously to Theorem 1. 1(iii) 2 Thirdly, we introduce the following Lagrange multiplier embedding (cf. [3]) for the original method cf. e.g. 1] P 3 (t) minfF 3 (x; v; w; t) j (x; v; w; 2 M 3 (t)g; t 2 [0; 1] where F 3 (x; v; w; t) F (A;B;C;D;E;x 0 ; 0 ; 0 ;v 0 ;w 0 ) x; v; w; t[f(x) X i2I i h i (x; t) X j2J j g j (x; t) 1 Gamma t) x ....

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Dentcheva, D., Guddat, J., Ruckmann, J.-J., Wendler, K. (1995): Pathfollowing methods in nonlinear optimization III: Lagrange multiplier embedding. ZOR 41, 127-152.

.... stationary point of P (0) is known (and the corresponding Lagrange multipliers are known or easy to compute) A2) P (t) has a global minimizer for every t 2 [0; 1] A3) P (1) is equivalent to (P ) in a certain sense (to be specified below) In this paper we consider three embeddings (cf. [6, 2, 3]) motivated by penalty, exact penalty and Lagrange multiplier methods and we ask the following question: Are there conditions for finding a discretization of the interval [0; 1] 0 = t 0 Delta Delta Delta t i Delta Delta Delta t N = 1 and corresponding stationary points y(t i ) of ....

.... then there exist vectors v 2 IR m ; w 2 IR s , such that (x; v; w) is a stationary point for P 1 (1) 2 Secondly, we consider the so called exact penalty methods proposed e.g. in [14] for problems without equality constraints (i.e. I = We propose the following embedding (cf. M 5 (t) in [2]) P 2 (t) minfF 2 (x; w; t) j (x; w) 2 M 2 (t)g; t 2 [0; 1] where F 2 (x; w; t) F (A;B;x 0 ;w 0 ;w 1 ;d) x; w; t) tf(x) 1 Gamma t) X j2J j (x) w j Gamma w 0 j ) 1 Gamma t) x Gamma x 0 ) T A(x Gamma x 0 ) w Gamma w 0 ) T [diag(ff j Gamma g j (x) ....

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Dentcheva, D., Gollmer, R., Guddat, J., Ruckmann, J.-J. (1995): Pathfollowing methods in nonlinear optimization II: Exact penalty methods. In [4], 200-230.

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