... Fur-thermore, the optimal valuef∗is finite and thereexists a vector¯x ∈ Xsuch thatgj(¯x) < 0, ∀ j =1, ,r.•Strong Duality Theorem: There exists at leastone geometric multiplier and there ... 0.DUALITY THEOREM FOR INEQUALITIES•Assume thatXis convex and the functionsf : n→,gj: n→are convex overX. Fur-thermore, the optimal valuef∗is finite and thereexists ... dual optimal solution and therefore there isno geometric multiplier. (Even though there is noduality gap.)•Assumptions are violated (the feasible set and the relative interior ofXhave no common...