Kojima,M. and Hirabayashi,R., Continuous deformation of nonlinear programs, Math.Program.Study 21 (1984), 150-198.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 generalized ....

....i2I ff i D x h i (x; t) P j2J fi j D x g j (x; t) Gammah i (x; t) i 2 I fi Gamma j Gamma g j (x; t) j 2 J 3 7 7 7 5 where fi j = maxffi j ; 0g and fi Gamma j = minffi j ; 0g: The mapping H is called Kojima mapping. Obviously H is piecewise continuously differentiable. In [21] the classical definition of a regular value of a continuously differentiable function is generalized for piecewise continuously differentiable functions. Furthermore, it is shown that if 0 2 R n m s is a regular value of H, then the set H Gamma1 (0) is a piecewise one dimensional C 1 ....

Kojima,M. and Hirabayashi,R., Continuous deformation of nonlinear programs, Math.Program.Study 21 (1984), 150-198.

.... P j2J y m j D x g j (x; t) Gammah i (x; t) i 2 I y Gamma m j Gamma g j (x; t) j 2 J 9 = 2.2) for ff 2 IR let ff = maxfff; 0g and ff Gamma = minfff; 0g) Obviously, the so called Kojima mapping H in (2.2) is piecewise continuously differentiable. In [17] the classical definition of a regular value of a continuously differentiable function is generalized for piecewise 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 ....

....gc ae 5 S =1 Sigma gc j . J. Guddat, F. Guerra, D. Nowack 9 (ii) The problem P (t) is called regular in the sense of Kojima Hirabayashi briefly KH regular (with respect to K) if 0 2 IR n m s is a regular value of H (Hj IR n ThetaIR m ThetaIR s ThetaK ) Theorem 2. 4 (cf. [17]) Let (f; H;G) 2 C 3 (IR n Theta IR; IR) 1 m s . Then, for almost 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 ....

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Kojima, M., Hirabayashi, R. (1984): Continuous Deformation of Nonlinear Programs. Math. Progr. Study 21, 150-198.

....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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Kojima, M., Hirabayashi, R. (1984): Continuous deformation of nonlinear programs. Math. Progr. Study 21, 150-198.

....(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 ....

Kojima M., and R. Hirabayashi, 1984, "Continuous Deformation of Nonlinear Programs," Mathematical Programming Study, Vol. 21, pp. 150-198.

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