## The epi-distance topology: Continuity and stability results with applications to convex optimization problems.(English)Zbl 0767.49011

Let $$C(X)$$, $$\Gamma(X)$$ denote the closed nonempty convex subsets of a Banach space $$X$$ and the proper lower semicontinuous convex functions on $$X$$. $$C(X)$$ is equipped with the topology $$\tau$$ of uniform convergence of distance functions on bounded subsets of $$X$$ and on $$\Gamma(X)$$ the related topology is defined: $$\lim f_ n=f$$ iff $$\lim(\text{epi} f_ n)=\text{epi} f$$ in $$(C(X),\tau)$$. It is proved that under standard regularity assumptions operations of addition and restriction are continuous on $$(\Gamma(X),\tau)$$. These results are applied to convex well-posed optimization problems $\min\{f(x)| x\in A\}\quad (=v(f,A));\;f\in\Gamma(X),\;A\in C(X). \tag{1}$ Well-posedness of (1) means that relations $$\lim_{n\to\infty}f(x_ n)=v(f| A)$$, $$x_ n\in A$$ imply $$\lim_{n\to\infty}x_ n=x_ *$$ where $$\{x_ *\}=\text{Argmin}\{f(x)| x\in A\}$$.
The following theorem is proved: Let $$\{(f_ n,A_ n)\}$$ be a sequence in $$(\Gamma(X),\tau)\times(C(X),\tau)$$ convergent to $$(f,A)$$. Suppose the problem (1) is well-posed and either $$f$$ is continuous at some point of $$A$$ or $$\text{dom} f\cap\text{int} A\neq\emptyset$$. Then (a) $$v(f| A)=\lim_{n\to\infty}v(f_ n| A_ n)$$; (b) if $$f_ n(x_ n)<v(f_ n| A_ n)+\varepsilon_ n$$, $$x_ n\in A_ n$$, $$\varepsilon_ n>0$$, $$\lim_{n\to\infty}\varepsilon_ n=0$$ then $$\lim_{n\to\infty}x_ n=x_ *$$. It is also shown that in a natural sense for most $$(f,A)$$ in $$(\Gamma(X),\tau)\times(C(X),\tau)$$ the function $$f$$ is continuous and whenever $$\limsup_{n\to\infty}f(x_ n)\leq v(f,A)$$, $$\lim_{n\to\infty}d(x_ n,A)=0$$ then $$\lim_{n\to\infty}x_ n=x_ *$$.

### MSC:

 49J52 Nonsmooth analysis 90C25 Convex programming
Full Text: