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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
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