ACTA issues

Explicit solutions of a special class of linear programming problems in Banach spaces

S. H. Kulkarni, K. C. Sivakumar

Acta Sci. Math. (Szeged) 62:3-4(1997), 457-465

Abstract. Let $X_1$ and $X_2$ be real Banach spaces. Let $K$ be a weakly compact subset in $X_2$, $A\colon X_1\to X_2$ be a closed linear map and $\phi $ be a bounded linear functional on $X_1$. We consider the following linear programming problem: $$\hbox{Maximize }\phi(x) \hbox{ subject to }Ax\in K.$$ Conditions under which explicit solutions to the above problem can be found are studied. The solutions are represented in terms of generalized inverses of $A$ and an optimal solution of a linear program in the dual space $X_2$. These results are then applied to linear programs with equality constraints and explicitly constrained feasible sets.

AMS Subject Classification (1991): 46B, 90C

Keyword(s): Banach spaces, Linear programming problems, generalised inverses, explicit solutions

Received April 12, 1995 and in revised form December 13, 1995. (Registered under 6128/2009.)