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Epi/hypo-convergence: The slice topology and saddle points approximation. (English) Zbl 0880.49011
Consider \(F:X\times Y\rightarrow\overline{{\mathbb{R}}}\) a convex function and \(K:X\times Y^*\rightarrow\overline{{\mathbb{R}}}\) its Lagrangian, i.e., \(K(x,\cdot)=(F(x,\cdot))^*\). Taking \((F_n)\) a sequence of convex, proper and l.s.c. functions and \((K_n)\) the corresponding sequence of Lagrangians, the author obtains results concerning the convergence of \((K_n)\). For example, if \((F_n)\) slice converges to \(F\) then \((K_n)\) epi/hypo converges to \(K\) in the sense: \(\underline{\text{cl}}_s(e_s/h_{w^*}-\text{ls}\overline{K_n})\leq\underline K\), \(\underline{\text{cl}}_s(h_s/e_{w^*}-\text{li}\underline{K_n})\geq\overline K\). He also obtains the convergence of saddle points. The results are then applied to convex programming, internal approximation, optimal control and Chebyshev approximation.

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
90C25 Convex programming
49N10 Linear-quadratic optimal control problems
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
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