v2009.01.01 - Convex Optimization

3.3. QUASICONVEX 239
exponential always resides interior to the cone of positive semidefinite
matrices in the symmetric matrix subspace; e A ≻0 ∀A∈ S M (1743). Then
for 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 diagonal
matrices. The matrix exponential of any diagonal matrix e Λ exponentiates
each individual entry on the main diagonal. [216,5.3] So, changing
the function domain to the subspace of real diagonal matrices reduces the
matrix exponential to a vector-valued function in an isometrically isomorphic
subspace R M ; known convex (3.1.1) from the real-valued function case
[53,3.1.5].
There are, of course, multifarious methods to determine function
convexity, [53] [37] [110] each of them efficient when appropriate.
3.2.3.0.7 Exercise. log det function.
Show by two different methods: log detX is concave on the interior of the
positive semidefinite cone.
3.3 Quasiconvex
Quasiconvex functions [53,3.4] [173] [286] [324] [210,2] are useful in
practical problem solving because they are unimodal (by definition when
nonmonotonic); a global minimum is guaranteed to exist over any convex set
in the function domain; e.g., Figure 69.
3.3.0.0.1 Definition. Quasiconvex function.
f(X) : R p×k →R is a quasiconvex function of matrix X iff domf is a convex
set and for each and every Y,Z ∈domf , 0≤µ≤1
f(µY + (1 − µ)Z) ≤ max{f(Y ), f(Z)} (577)
A quasiconcave function is determined:
f(µY + (1 − µ)Z) ≥ min{f(Y ), f(Z)} (578)
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