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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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