Transcription of Quadratic Functions, Optimization, and Quadratic Forms
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Quadratic Functions, Optimization, and Quadratic Forms
Quadratic Functions, Optimization, and Quadratic Fo r m s Robert M. Freund February, 2004 1 2004 Massachusetts Institute of 1 Quadratic Optimization A Quadratic optimization problem is an optimization problem of the form: T(QP) : minimize f (x):= 1 xT Qx + c x . Problems of the form QP are natural models that arise in a variety of settings. For example, consider the problem of approximately solving an over-determined linear system Ax = b, where A has more rows than columns. We might want to solve: (P1) : minimize Ax b x . Now notice that Ax b 2 = xT AT Ax 2bT Ax+bT b, and so this problem is equivalent to: (P1) : minimize xT AT Ax 2bT Ax + bT b x , which is in the format of QP. A symmetric matrix is a square matrix Q n n with the property that Qij = Qji for all i, j =1.
Notice in the general model QP that we can always presume that Q is a symmetric matrix, because: xT Qx = 1 x T (Q + QT)x 2 and so we could replace Q by the symmetric matrix Q¯ := 1 2 (Q + QT). Now suppose that f (x):= 1 x T Qx + c T x 2 where Q is symmetric. Then it is easy to see that: ∇f (x)=Qx + c and H(x)=Q.
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