Jongen, H. Th., Jonker, P. and Twilt, F., Critical sets in parametric optimization. Math. Programming, Vol. 34, pp. 333 -- 353, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= 0, 2.18) let the set of active points T (u) t # T : g(u, t) 0 consist of a finite number of elements. Then, for the PIP method, the relation lim i## max 1#s#s(i) m i,s T i = 0 is valid. Finiteness of T (u) can be considered as a generic property of semi infinite programs [10]. 2.3 Rate of convergence In order to prove linear convergence of the sequence # i,s , # i,s = dist 2 (u i,s , U # ) dist(u, A) inf v#A #u v#) we introduce the following Assumption 2 inf u#K U # J(u) J min dist 2 (u, U # ) # d 0, 2.19) is valid with J min = min ....

Jongen, H. Th., Jonker, P. and Twilt, F., Critical sets in parametric optimization. Math. Programming, Vol. 34, pp. 333 -- 353, 1986.

.... Background We consider the following general one parametric optimization problem: P (t) min f f(x; t) j x 2 M(t) g where M(t) f x 2 IR n j g i (x; t) 0; i 2 I; g k (x; t) 0; k 2 K g I and K are index sets defined as in (2) We first recall the class F of Jongen, Jonker and Twilt [8]. 4 Definition 2.1 (c.f. e.g. 3, 8] Sigma gc is the set of all (x; t) such that x 2 M(t) and the vectors D x f(x; t) D x g k (x; t) k 2 I [ K 0 (x; t) are linearly dependent, where K 0 (x; t) f j 2 K j g j (x; t) 0 g Here D x f denotes for the row vectors of first partial ....

....general one parametric optimization problem: P (t) min f f(x; t) j x 2 M(t) g where M(t) f x 2 IR n j g i (x; t) 0; i 2 I; g k (x; t) 0; k 2 K g I and K are index sets defined as in (2) We first recall the class F of Jongen, Jonker and Twilt [8] 4 Definition 2.1 (c.f. e.g. [3, 8]) Sigma gc is the set of all (x; t) such that x 2 M(t) and the vectors D x f(x; t) D x g k (x; t) k 2 I [ K 0 (x; t) are linearly dependent, where K 0 (x; t) f j 2 K j g j (x; t) 0 g Here D x f denotes for the row vectors of first partial derivatives with respect to x. If (x; t) 2 ....

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Jongen H.T., Jonker P., Twilt F. : Critical sets in parametric optimization. Mathematical programming 34, 1986.

.... 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 vector of ....

.... of X gc is completely described for (f; H;G) belonging to an open and dense subset F of (C 3 s (IR n Theta IR; IR) 1 m s with respect to the strong C 3 topology (Whitneytopology) We assume k 3 and (f; H;G) 2 F and give a short characterization of this class (for details see [18], 13] The points of X gc can be divided into 5 types: Type 1: A point z = x; t) 2 X gc is of Type 1 if the following 4 conditions are satisfied: D x f X i2I ff i D x h i X j2J 0 (z) fi j D x g j )j z = 0; 2.1) LICQ is satisfied at x 2 M( t) 2.2) fi j 6= 0; j 2 J 0 ....

Jongen,H.Th., Jonker,P. and Twilt,F., Critical sets in parametric optimization, Math.Progr. 34 (1986), 333-353. On Modifications of the Standard Embedding 45

....1 s s(i) ku i,s uk = 0, 65) let the set of active points T (u) t 2 T : g(u, t) 0 consist of a finite number of elements 2 . Then, for the PIP method, the relation lim i 1 max 1 s s(i) m i,s T i = 0 is valid. 2 cf. genericity property of semi infinite programms [4] Proximal Interior Point Approach for Convex SIP 23 Proof Let T (u) t 1 , t q (T (u) is not excluded) We fix a small # 0 and define the set T = q [ k=1 t 2 T : kt t k k # . Because of the continuity of g(u, on the compact set T T , the ....

H. Th. Jongen, P. Jonker, and F. Twilt. Critical sets in parametric optimization. Math. Programming, 34:333--353, 1986.

....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 Jonger, Jonker and Twilt (see [12]) or in the sense of Kojima and Hirabayashi (see [14] We use the regularity in the sense on Jongen, Jonker and Twilt (shortly JJTregularity) The success of a pathfollowing method (arriving t = 1) when being applied to a one parametric problem is not ensured even in the case of regularity. ....

....2 IR 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 ....

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Jongen, H.Th., Jonker, P. and Twilt, F., Critical sets in parametric optimization, Math. Program. 34 (1986), 333-353.

....present the necessary background for one parametric problems, of which P i (t) is a particular case. In Section 3 we study P i (t) and in section 4 we analyze under which perturbations on P i (t) we obtain a parametric problem PD (t) belonging to the generic class F of Jongen, Jonker and Twilt [15]. In Section 5 we present some applications. 2 Theoretical Background We consider the general one parametric problem: minff(x; t) j x 2 M (t)g; t 2 R; P (t) where M (t) fx 2 R n j h i (x; t) 0; i 2 I; g j (x; t) 0; j 2 Jg; I = f1; Delta Delta Delta ; mg; m n; J = f1; Delta ....

....; P stat : f(x; t) 2 R n Theta R j x is a stationary point of P (t)g ; P loc : f(x; t) 2 R n Theta R j x is a local minimizer of P (t)g ; H : h 1 ; Delta Delta Delta ; hm ) T ; G : g 1 ; Delta Delta Delta ; g s ) T : Note 1. For the denition of a g.c. point we refer to [15], see also [5] 2 Theoretical Background 5 x o M 1 g s 3 g s 2 P g s 1 g s 3 (x; 0) 0 (a) x o g s 3 (x; 0) 0 M 1 M 1 P g s 2 g s 1 g s 3 (b) x o g s 3 (x; 0) 0 M 2 P g s 1 (c) x o g s 3 (x; 0) 0 M 2 M 2 P g s 1 (d) Figure 2: possible structure of M 1 (t) and M 2 (t) ....

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Jongen, H. Th.; Jonker, P. and Twilt, F: Critical sets in parametric optimization. In: Math. Programming 34 (1986), 333 \Gamma 353

....Furthermore, it is shown that if 0 2 IR n m s is a regular value of H, then the set H Gamma1 (0) is a piecewise one dimensional C 1 manifold (briefly PC 1 manifold) Next, we cite our short characterization from 2. 5 in [12] of the class F introduced by Jongen, Jonker and Twilt ([15, 16]) In [16] the local structure of Sigma gc is completely 1 For the definition we refer to [16] see [12] too 2 We consider the gradient D x h i (x; t) as a row vector. 6 Multiobjective optimization: embeddings described if (f; H;G) belongs to a C 3 s open and dense subset F of C 3 ....

....it is shown that if 0 2 IR n m s is a regular value of H, then the set H Gamma1 (0) is a piecewise one dimensional C 1 manifold (briefly PC 1 manifold) Next, we cite our short characterization from 2. 5 in [12] of the class F introduced by Jongen, Jonker and Twilt ( 15, 16] In [16] the local structure of Sigma gc is completely 1 For the definition we refer to [16] see [12] too 2 We consider the gradient D x h i (x; t) as a row vector. 6 Multiobjective optimization: embeddings described if (f; H;G) belongs to a C 3 s open and dense subset F of C 3 (IR n ....

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Jongen, H.Th., Jonker, P., Twilt, F. (1986): Critical Sets in Parametric Optimization. Math. Programming 34, 333-353.

....sets is relatively simple to describe. One approach based on piecewise differentiable mappings is due to Kojima and Hirabayashi ( 18] and deals with the global structure of the Karush KuhnTucker set. The other approach, based on transversal intersection, is due to Jongen, Jonker and Twilt ([14], 15] In this latter approach the generalized critical points are classified 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 ....

....five types of generalized critical points defining regularity in the sense of Jongen, Jonker and Twilt. ffl The strong, or Whitney, topology in the space of all three time differentiable mappings. For a definition of most of the concepts (and even more) see e.g. the books [16, 17] and the paper [14]. The rest of the report is organized as follows. In the second section the generic property will be presented and its proof will be reduced to three claims. The proof of these claims is the purpose of the third section. Some conclusions and examples are presented in the last and fourth section. ....

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Jongen, H.Th., Jonker, P. and Twilt, F., Critical sets in parametric optimization, Math. Program. 34 (1986), 333-353.

....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 ....

....continuously differentiable functions. Furthermore, it is shown that if 0 2 IR n m s is a regular value of H, then the set H Gamma1 (0) is a piecewise one dimensional C 1 manifold. Next, we cite our short characterization from [2] of the class F introduced by Jongen, Jonker and Twilt ([11, 12]) In [12] the local structure of Sigma gc is completely described if (f; H;G) belongs to a C 3 s open and dense subset F of C 3 (IR n Theta IR; IR) 1 m s , where C 3 s denotes the strong (or Whitney ) C 3 topology. If (f; H;G) 2 F , then Sigma gc can be divided into 5 types. ....

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Jongen, H.Th., Jonker, P., Twilt, F. (1986): Critical sets in parametric optimization. Math. Programming 34, 333-353.

....(5) and (6) as FORM (Formulation for Optimal Realization of Models) The major mathematical difficulty in studying the model of Eq. 5) is that it undergoes large parametric deformations. Several key results have been recently obtained about such parametric models in the optimization literature [6, 9, 13, 17, 21, 26]. These results pertain to the theoretical behavior of the parametric models, and answer specific questions such as, what are the various continuity properties of maps X(p) f 1 (p) and X (p) and what is the nature of singularities (such as bifurcations and limit points) encountered ....

Jongen H. Th., Jonker P., and F. Twilt, 1986, "Critical Sets in Parametric Optimization," Mathematical Programming, Vol. 34, pp. 333-353.

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