v2010.10.26 - Convex Optimization

3.8. QUASICONVEX 269exponential always resides interior to the cone of positive semidefinitematrices in the symmetric matrix subspace; e A ≻0 ∀A∈ S M (1867). Thenfor any matrix Y of compatible dimension, Y T e A Y is positive semidefinite.(A.3.1.0.5)The subspace of circulant symmetric matrices contains all diagonalmatrices. The matrix exponential of any diagonal matrix e Λ exponentiateseach individual entry on the main diagonal. [251,5.3] So, changingthe function domain to the subspace of real diagonal matrices reduces thematrix exponential to a vector-valued function in an isometrically isomorphicsubspace R M ; known convex (3.1.1) from the real-valued function case[61,3.1.5].There are, of course, multifarious methods to determine functionconvexity, [61] [42] [132] each of them efficient when appropriate.3.7.3.0.8 Exercise. log det function.Matrix determinant is neither a convex or concave function, in general,but its inverse is convex when domain is restricted to interior of a positivesemidefinite cone. [34, p.149] Show by three different methods: On interior ofthe positive semidefinite cone, log detX = − log detX −1 is concave. 3.8 QuasiconvexQuasiconvex functions [61,3.4] [199] [330] [371] [244,2] are useful inpractical problem solving because they are unimodal (by definition whennonmonotonic); a global minimum is guaranteed to exist over any convex setin the function domain; e.g., Figure 79.3.8.0.0.1 Definition. Quasiconvex function.f(X) : R p×k →R is a quasiconvex function of matrix X iff domf is a convexset and for each and every Y,Z ∈domf , 0≤µ≤1f(µY + (1 − µ)Z) ≤ max{f(Y ), f(Z)} (642)A quasiconcave function is determined:f(µY + (1 − µ)Z) ≥ min{f(Y ), f(Z)} (643)△