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Chapter 3 Quadratic Programming

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Optimization I; Chapter 356Chapter 3 Quadratic Constrained Quadratic Programming problemsA special case of the NLP arises when the objective functionalfis quadraticand the constraintsh, gare linear inx lRn. Such an NLP is called a QuadraticProgramming (QP) problem. Its general form isminimizef(x) :=12xTBx xTb( )overx lRnsubject toA1x=c( )A2x d ,( )whereB lRn nis symmetric,A1 lRm n, A2 lRp n, andb lRn, c lRm, d we shall see in this Chapter , the QP ( )-( ) can be solved iterativelyby active set strategies or interior point methods where each iteration requiresthe solution of an equality constrained QP Equality constrained Quadratic programmingIf only equality constraints are imposed, the QP ( )-( ) reduces tominimizef(x) :=12xTBx xTb( )overx lRnsubject toAx=c ,( )whereA lRm n, m n.

Lemma 3.2 Existence and uniqueness Assume that A 2 lRm£n has full row rank m • n and that the reduced Hessian ZTBZ is positive deflnite. Then, the KKT matrix K is nonsingular. Hence, the KKT system (3.3) has a unique solution (x⁄;‚⁄). Proof: The proof is left as an exercise. †

  Existence, Uniqueness, Existence and uniqueness

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