Let P be an open set in E('n), and C be an arbitrary cone in E('n). LetF and g be functions from P to E('l) and E('n) respectively;Consider the following minimization problem, Problem P, find anx('0) (epsilon) E('n), if it exists, such that;F(x('0)) = min F(x); x (epsilon) X; x('0) (epsilon) X;where,;X = x : x (epsilon) P (L-HOOK) E('n), g(x) (epsilon) C (L-HOOK) E('m);and F is differentiable at x('0);Associated with the minimization problem is a modified Kuhn-Tucker stationary point problem over cone domains. Find anx('0) (epsilon) P (L-HOOK) E('n) and u('0) (epsilon) -C* (L-HOOK) E('n) (C* is the polar cone of C) such that;(DEL)'F(x('0)) + u('0)'(DEL)g(x('0)) = 0; u('0)'g(x('0)) = 0; g(x('0)) (epsilon) C;Necessary and sufficient optimality conditions are established between Problem P and the modified Kuhn-Tucker stationary point problem for a certain class of nonlinear programming problems over arbitrary cone domains;These results are used to prove a modified Farkas Lemma over degenerate and nondegenerate cone domains which uses only a "partial" linear duality theorem;A quadratic programming problem is considered. Its dual problem is constructed in a natural way over degenerate and nondegenerate cone domains and quadratic duality is established between the two problems.