Note that the feasible region of points satisfying constraints in the LPP is defined by the intersection of m half-planes, and so it must be a convex subset of Rn (more specifically, a convex polytope, a generalization of a polygon to higher dimensions). It is also easy to see that the optimum for the problem must lie on the boundary of this feasible region. For consider f(x)=cTx, our objective function which we are trying to minimize. Thenrf(x)= c, a constant vector. To minimizef(x) we should move the direction of rf(x)=c, and so the furthest feasible point in the c direction would be lying on the boundary of our polytope (in fact, on a vertex of the polytope). Many algorithms take advantage this fact (notably, the simplex algorithm).
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