| # Optimality condition for a convex, linearly constrained problem Theorem: optimality condition for convex, linearly constrained problems Consider the linearly constrained optimization problem where is convex and differentiable, and , define the equality constraints. A point is optimal for the above problem if and only if it satisfies for some vector . Proof: Let us re-formulate the optimality condition is where the feasible set is now the affine set We can write the above as: and Since we can always flip the sign of vectors with , we can express the above as This means that should be orthogonal to the nullspace of . From the Fundamental theorem of linear algebra, this in turn says that should belong to the range of the transpose matrix, . In other words, the above condition is equivalent to We conclude that the optimality conditions for to be optimal are that there exists a vector such that This ends our proof. Example: