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Well-posedness criteria in optimization with application to the calculus of variations. (English) Zbl 0841.49005
In this paper, for the global optimization problem \((X, J)\), to minimize the proper extended real-valued function \(J: X\to (- \infty, \infty)\) over the given subset \(X\) of a normed linear space equipped with the strong convergence, well-posedness criteria are derived. The given problem is embedded into a smoothly parametrized family \((X, I(., p))\) of minimization problems, where \(p\) is a parameter belonging to a given Banach space \(L\), and \(p^*\) is the parameter value to which the given unperturbed problem corresponds, i.e., \(I(x, p^*)= J(x)\) \(\forall x\in X\). Defining the value function \(V(p)= \inf\{I(x, p)\mid x\in X\}\) the author gives the following definition of well-posedness.
\((X, J)\) is well-posed with respect to the embedding iff \(V(p)> -\infty\), \(\forall p\in L\), and there exists a unique \(x^*= \arg\min(X, J)\) and for every sequence \(p_n\to p^*\) and every sequence \(x_n\in X\) such that \(I(x_n, p_n)- V(p_n)\to 0\) as \(n\to \infty\) we have \(x_n\to x^*\) in \(X\).
This definition is stronger than the Tikhonov well-posedness. In the following, the defined well-posedness is related under suitable conditions to the differentiability properties of \(V\) at \(p^*\). These abstract results are applied to one-dimensional problems of the calculus of variations.

MSC:
49J27 Existence theories for problems in abstract spaces
49K99 Optimality conditions
90C99 Mathematical programming
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[1] Asplund, E.; Rockafellar, R.T., Gradients of convex functions, Trans. am. math. soc., 139, 443-467, (1969) · Zbl 0181.41901
[2] Fitzpatrick, S., Metric projections and the differentiability of distance functions, Bull. aust. math. soc., 22, 291-312, (1980) · Zbl 0437.46012
[3] Fleming, W.H., The Cauchy problem for a nonlinear first order partial differential equation, J. diff. eqns, 5, 515-530, (1969) · Zbl 0172.13901
[4] Fleming, W.H.; Rishel, R., ()
[5] Clarke, F.H.; Loewen, P.D., The value function in optimal control: sensitivity, controllability and time-optimality, SIAM J. control optim., 24, 243-263, (1986) · Zbl 0601.49020
[6] Clarke, F.H., ()
[7] Ekeland, I.; Lebourg, G., Generic FrĂ©chet differentiability and perturbed optimization problems in Banach spaces, Trans. am. math. soc., 224, 193-216, (1976) · Zbl 0313.46017
[8] Georgiev, P.G., Well posed optimization problems depending on a parameter and differentiability of the optimal value function, ()
[9] Henry, D., Nonuniqueness of solutions in the calculus of variations: a geometric approach, SIAM J. control. optim., 18, 627-639, (1980) · Zbl 0454.49008
[10] Fleming, W.H.; Soner, H.M., ()
[11] Dontchev, A.L.; Zolezzi, T., ()
[12] Kuratowski, C., ()
[13] Ekeland, I., Nonconvex minimization problems, Bull. am. math. soc., 1, 443-474, (1979) · Zbl 0441.49011
[14] Kutznetzov, N.N.; Siskin, A.A., On a many dimensional problem in the theory of quasilinear equations, Zh. vychisl. mat. fiz., 4, 192-205, (1964)
[15] Cesari, L., ()
[16] Marcellini, P., (), 16-57
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