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.
4 (GP) : minimize f (x) s.t. x ∈ n, where f (x): n → is a function. We often design algorithms for GP by building a local quadratic model of f (·)atagivenpointx =¯x.We form the gradient ∇f (¯x) (the vector of partial derivatives) and the Hessian H(¯x) (the matrix of second partial derivatives), and approximate GP by the following problem which uses the Taylor expansion of f (x)atx ...
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