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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.

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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