## Theory of Convex Programming |

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arbitrary point assume assumptions of Theorem bounded linear functional bounded set compact set concave function continuous function convex programming problem convex programming theory convex set convex subsets corresponding dual problem duality relation duality theorems exists a point exists a vector feasible solutions finite number fixed function defined function f(x,y gramming problem holds hyperplanes inequality infF(x,y infimum initial problem interiority condition ip(x Lagrangian function Lemma Let f(x,y linear programming problem lower semicontinuous marginal values modified Slater condition Neumann's theorem nonempty and bounded obtain Obviously optimality criterion optimizing sequence pair of dual piecewise linear point of f(x,y problem 9 problem dual problems 64 proof of Theorem saddle point satisfy the Slater separation theorem set G set of feasible sets X(t solution of problem sup f(x sup F(x,y sup inf/(x,y supremum taking into account Theorem 13 upper semicontinuous variable von Neumann's theorem xux2 yEY xEX