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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.

90C05 Linear programming
massive set
Full Text: DOI
[1] V. L. Levin, Set-Valued Anal. 9, 383–390 (2001). · Zbl 1049.49031 · doi:10.1023/A:1012614520391
[2] R. R. Phelps, Convex Functions, Monotone Operators, and Differentiability (Springer-Verlag, Berlin, 1993). · Zbl 0921.46039
[3] R. Rockafellar, Convex Analysis (Princeton Univ. Press, Princeton, 1970; Mir, Moscow, 1973). · Zbl 0193.18401
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