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On generic uniqueness of optimal solution in an infinite dimensional linear programming problem. (English. Russian original) Zbl 1227.90023
Dokl. Math. 78, No. 1, 490-492 (2008); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 421, No. 1, 21-23 (2008).
Summary: In this paper, the class of infinite dimensional linear programming problems with a unique optimal solution for a massive set of maximized functionals is described.
Consider the following problem. Given a Banach space $$E$$, linear spaces $$F\subseteq E^*$$ and $$L$$, a convex cone $$L_+$$ in $$L$$, a linear mapping $$A:F\to L$$, and an element $$b\in L$$, we suppose that $$F$$ is weakly* dense in the dual space $$E^*$$. For $$e\in E$$, find the optimal value
$v(e)= \sup\big\{\langle e,x\rangle: x\in F,\;Ax\geq b\big\}, \tag{1}$
where the inequality $$y_1\geq y_2$$ for $$y_1,y_2\in L$$ means $$y_1-y_2\in L_+$$.
We are interested in the generic uniqueness problem, i.e., in the question as to when the set $$U$$ of functionals $$e$$ for which there is a unique optimal solution to problem (1) is massive.

##### MSC:
 90C05 Linear programming
massive set
Full Text:
##### References:
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