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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
##### Keywords:
global optimization problem; well-posedness criteria
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##### References:
 [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 Fréchet 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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