Guddat, J., Guerra, F. and Jongen, H.Th., Parametric optimization: singularities, pathfollowing and jumps, B.G. Teubner, Stuttgart and John Wiley, Chichester, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....2 IR n j g i (x) 0; i 2 I; g k (x) 0; k 2 K g (2) and where f j ; j = 1 : L; g k ; k 2 I [ K are given functions , I = f1 : mg and K = f1 : pg n I; p m . We use the following well known notions of optimality for a multiobjective optimization problem. Definition 1.1 (c.f. e.g. [3]) A point x 2 IR n is called an efficient point if ( f(x) D ) f(M) 3) where f = f 1 ; fL ) and D = GammaIR L n f0g. The concepts of ffl efficient point and weakly efficient point are defined analogously writing D ffl = f y 2 IR L n f0g j dist(y; D) ffl jjyjj g and D ....

....[ f 1g; k 2 L and 0 2 GammaD d . 2 We denote by Psi 1 ( Psi 1loc ( the set of global and local optimal points of the problem P 1 ( The following relation is known. c.f. e.g. 5] M eff = 2 IR L Psi 1 ( M loceff = 2 IR L Psi 1loc ( 2nd.Parametrization: c.f. e.g. [3, 4]) s(f(x) max i 2 L 0 i ( f i (x) Gamma i ) ffi L X k=1 0 k ( f k (x) Gamma k ) where ffi 2 (0; 1) sufficiently small is fixed. The problem min s(x; x 2 M (5) has the following properties: c.f. e.g. 5] 2 IR L Psi 2 ( ae M eff ; 2 IR L Psi 2loc ( ae M ....

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Guddat J., Guerra F., Jongen H.T. : Parametric Optimization: singularities, pathfollowing and jumps. John Wiley, Chichester and Teubner, Stuttgart 1990. 21

....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 ....

.... Embedding 3 and corresponding (A) local minimizers x(t i ) of P (t i ) i = 1; N or (B) stationary points x(t i ) of P (t i ) i = 1; N or (C) generalized critical points (g:c: points) x(t i ) of P (t i ) i = 1; N Now, we recall some basic definitions (cf. e.g. [13], 18] A point x 2 IR n is called generalized critical point (g.c. point) for the problem (P) if x 2 M and the set fDf(x) Dh i (x) i 2 I; Dg j (x) j 2 J 0 (x)g is linearly dependent, where J 0 (x) fj 2 J jg j (x) 0g is the index set of active constraints and Df(x) denotes the row ....

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Guddat,J., Guerra,F. and Jongen,H.Th. Parametric Optimization: Singularities, Pathfollowing and Jumps, B.G.Teubner and John Wiley, Chichester, 1990.

....n such that: Dh i (x) 0; i 2 I; Dg j (x) 0; j 2 J 0 (x) Here and throughout the paper D k y denotes the partial derivatives of order k with respect to the variables y. Let us now recall some well known notions. First, the notion of generalized critical points. Definition 2 (see [12] or [7]) A point x 2 M is called a generalized critical point (shortly g. c. point) of the problem (P ) if the vectors fDf(x) Dh i (x) i 2 I; Dg j (x) j 2 J 0 (x)g are linearly dependent. 5 Thus, if x 2 M is a g.c.point of (P ) then there exist u 0 , i ; i 2 I and u j ; j 2 J 0 (x) such that u ....

....the problem (P ) if there exist i ; i 2 I and u j 0; j 2 J 0 (x) such that the relation (2) is satisfied. It is well known that the validity of the constraint qualifications LICQ or MFCQ at a local minimizer x of the problem (P ) implies that x is a stationary point. Definition 3 (see [12] or [7]) Let x 2 M be a g.c. point of (P ) It is called nondegenerated if the following conditions are fulfilled: ND1) LICQ holds at x. Then, there exist uniquely determined numbers i ; i 2 I and u j ; j 2 J 0 (x) such that relation (2) holds. ND2) u j 6= 0; j 2 J 0 (x) ND3) D 2 L(x)j T ....

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Guddat J., Guerra F. and Jongen, H.Th., Parametric Optimization: singularities, pathfollowing and jumps. John Wiley, Chichester and Teubner, Stuttgart 1990.

....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 the sequential approach and under the regularity of the embeddings obtained. This regularity ....

....composed from the components d i , i 2 K. For a differentiable function f : IR s IR, Df(x) denotes the row vector of partial derivatives. 2 Genericity result. We begin with recalling the definition of a JJT regular parametric optimization problem in a slightly modified form (see [14] or [10]) 4 Definition 1 Let us consider a general parametric optimization problem P (t) defined by the functions (f(y; t) H(y; t) G(y; t) 2 C 3 (IR n 1 1 ; IR n 2 1 ) Let S be a subset of IR n 1 1 . We call the parametric problem P (t) JJT regular on S if each generalized critical ....

Guddat J., Guerra F. and Jongen, H.Th., Parametric Optimization: singularities, pathfollowing and jumps. John Wiley, Chichester and Teubner, Stuttgart 1990.

.... authors have used bifurcation and singularity theory to investigate the local behavior and persistence of minima at the singular points of the above system, see, e.g. Bank, Guddat, Klatte, Kummer and Tammer (1983) Gfrerer, Guddat and Wacker (1983) Gfrerer, Guddat, Wacker and Zulehner (1985) Guddat, Guerra Vasquez and Jongen (1990), Guddat, Jongen, Kummer and Nozicka (1987) Jongen, Jonker and Twilt (1983, 1986) Jongen and Weber (1990) Kojima and Hirabayashi (1984) Poore and Tiahrt (1987, 1990) Rakowska, Haftka and Watson (1991) discuss algorithms for tracking paths of optimal solutions. Lundberg and Poore (1993) report ....

J. Guddat, F. Guerra Vasquez and H. T. Jongen (1990), Parametric Optimization: Singularities, Path Following, and Jumps, John Wiley and Sons, Chichester, England.

....different because we need to trace minimizers of h. However, in general it is not possible to define a continuous trajectory of minimizers, and thus we must be prepared to jump curves. For additional information on issues related to tracing minimizers, see Gudat, Guerra Vazquez, and Jongen [17]. 5.2 Smoothing The Gaussian transform is isotropic because if we view the function f in a different coordinate system via the function h : IR n 7 IR defined by h(x) f(P T x) then hhi (Px) hfi (x) for any orthogonal matrix P 2 IR n Thetan . If we wish to emphasize some directions, ....

J. Guddat, F. G. Vazquez, and H. T. Jongen, Parametric Optimization: Singularities, Pathfollowing and Jumps, John Wiley & Sons, 1990.

....The problem of determining a solution to a system of nonlinear inequalities is a fundamental problem in nonlinear optimization, which plays a major role both in global optimization and in constrained local optimization. In the general case, it is equivalent to a global optimization problem [10][8]. Indeed, the problem of determining x 2 IR n that satisfies a system of nonlinear inequalities g( x) 0, with g : IR n IR m can obviously be recast into the problem of determining whether there exists a solution to the problem glob min x2IR n kg (x)k = 0 where k Delta k is any ....

J. Guddat, F. Guerra Vasquez, and H. Th. Jongen. Parametric Optimization: Singularities, Pathfollowing and Jumps. John Wiley and Sons Ltd., Stuttgard, 1990.

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Guddat, J., Guerra, F. and Jongen, H.Th., Parametric optimization: singularities, pathfollowing and jumps, B.G. Teubner, Stuttgart and John Wiley, Chichester, 1990.

....; IR) k 2 K; j 2 J; q 2 f2; 3g We assume (A1) M 6= Let f k ; k 2 K, be the global minimum of f k (x) subject to M or, in the nonconvex case, a sufficiently good lower bound. We assume that f k ; k 2 K, to be known. In the following we describe a well known dialogue procedure (cf. e.g. 9] [12]) which constitutes the basis of our investigation. Let x i be the currently computed feasible point. Then we consider a telescreen picture as described in Table 1.1. The main information consists in estimating the objective values at the point x i in comparison to f k = infff k (x) j x 2 ....

....k ; k 2 K; p sufficiently large, f 0 k 0; k 2 K and g 0 j 0; j 2 J with f 0 k 1 6= f 0 k 2 ; k 1 ; k 2 2 K; k 1 6= k 2 and g 0 j 1 6= g 0 j 2 ; j 1 ; j 2 2 J; j 1 6= j 2 . 2 Theoretical Background and the Program Package PAFO. First, we present a very short version of 2.5, 2. 6 in [12]. We consider the general one parametric problem: P (t) minff(x; t)jx 2 M(t)g; t 2 IR; 2.1) J. Guddat, F. Guerra, D. Nowack 5 where M(t) fx 2 IR n j h i (x; t) 0; i 2 I; g j (x; t) 0; j 2 Jg, and f; h i ; g j 2 C q (IR n Theta IR; IR) i 2 I; j 2 J; q 2. Furthermore, we introduce ....

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Guddat, J., Guerra, F., Jongen, H.Th. (1990): Parametric Optimization: Singularities, Pathfollowing and Jumps. BG Teubner, Stuttgart and John Wiley, Chichester.

....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 ....

....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 ....

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Guddat, J., Guerra, F., Jongen, H.Th. (1990): Parametric optimization: Singularities, pathfollowing and jumps. BG Teubner, Stuttgart and John Wiley, Chichester.

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