Transcription of Chapter 12 Quadratic Optimization Problems
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Chapter 12 Quadratic Optimization Problems
Chapter 12 Quadratic Optimization Quadratic Optimization : The Positive DefiniteCaseIn this Chapter , we consider two classes of Quadratic opti-mization Problems that appear frequently in engineeringand in computer science (especially in computer vision) (x)=12x Ax+x bover allx Rn, (x)=12x Ax+x bover the unit 12. Quadratic Optimization PROBLEMSIn both cases,Ais a symmetric matrix. We also seeknecessary and sufficient conditions forfto have a Problems in physics and engineering can be statedas theminimization of some energy function,withorwithout , it is a fundamental principle of mechanics thatnature acts so as to minimize , if a physical system is in a stable state ofequilibrium, then the energy in that state should be simplest kind of energy function is a Quadratic Quadratic Optimization : THE POSITIVE DEFINITE CASE449 Such functions can be conveniently defined in the formP(x)=x Ax x b,whereAis a symmetricn nmatrix, andx, b,arevectorsinRn, , for reasons that will be clear shortly, it is prefer-able to put a factor12in front of the Quadratic term, sothatP(x)=12x Ax x question is, under what conditions (onA)doesP(x)have a global minimum, preferably unique?
12.1. QUADRATIC OPTIMIZATION: THE POSITIVE DEFINITE CASE 455 Thus, when the energy function P(x)ofasystemisgiven by a quadratic function P(x)= 1 2 xAx−xb, where A is symmetric positive definite, finding the global minimum of P(x) is equivalent to solving the linear system Ax = b. Sometimes, it is useful to recast a linear problem Ax = b
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