Technical Note - Direct Proof of the Existence Theorem for Quadratic Programming

T HE EXISTENCE theorem of quadratic programming states that a quadratic (not necessarily convex) function bounded from below on a nonempty polyhedral set R 5 6Rn assumes a minimum on R. The theorem was stated and proved for the first time by FRANK AND WOLFE. The original proof is very short and elegant; however, according to our experience, it is not well suited for expository purposes, since it is geometrical in nature and depends on a decomposition theorem for convex polyhedra often not available at the beginning of, say, an LP/QPcourse. For the convex case, a direct proof has been given by COLLATZ AND WETTERLING,[2] using the theory of Lagrange multipliers. The analytical proof we offer below-also for the nonconvex case-is completely elementary, using only some simple facts about limits and quadratic forms. With this proof it should be possible to start any systematic exposition of quadratic programming with what we think should logically stand at the beginning of such an exposition-the existence theorem. THEOREM. With xE(Rn let Q(x) =pTx+xTCx (CT= C) be a quadratic function (not necessarily convex, possibly degenerate). Let lj(x) =ajTx+bj(jEJ) be afinitefamily of linearfunctions. Let R= {x/lj(x) O, the sets R, = Rn{x/IxI so. Since Rp is compact, there exists at least one tERp such that Q(t) =q(p). Since, on the nonvoid compact set { t/tER, Q(t) = q(p) }, the continuous function 141 assumes its minimal value, we conclude that, for any p>O, there exists at least one point xp