## Generalized Lagrangian functions in mathematical programming |

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attains its minimum blem closed convex set compact set concave concave function continuous function convex combination convex function convex program convex set C C E convex subset COROLLARY cp x,y cp(x DEFINITION denoted differentiably quasi-convex directional derivative drietal dual problem exists a neighbourhood f is differentiable f x1 feasible fixed yGD follows from Theorem function f defined Hence hyperplane implies iMx,y iMx.y Kuhn-Tucker Lagrangian function Lagrangian triple Let f Let N(y lower semi-continuous Mangasarian mapping mathematical programming monotonic strictly quasi-convex neighbourhood N(x nonempty nonlinear program open set oplossen optimal solution optimal value partial derivatives primal problem problem 3.1 PROOF pseudo-concave relatively open saddle point satisfies section 4.2 sequential strict duality strictly quasi-concave unconstrained upper semi-continuous vector wordt x e F x e N(x x e Xy ye N(y yGD xGC