Optimality condition for a convex, linearly constrained problemTheorem: optimality condition for convex, linearly constrained problems
Consider the linearly constrained optimization problem ![]() where A point ![]() for some vector Proof: Let us re-formulate the optimality condition is ![]() where the feasible set ![]() We can write the above as: ![]() Since we can always flip the sign of vectors ![]() This means that From the Fundamental theorem of linear algebra, this in turn says that ![]() We conclude that the optimality conditions for ![]() This ends our proof. Example: |