I see in pflug modeling and measuring risk book, CVaR is concave... But the other book definate cvar is convex... If assume cvar is concave, then cvar optimization problem give us a global optimal point?
CVaR is a convex function in the underlying portfolio (measured as for instance absolute value or profit). I won't get into proving anything so instead I am going to link the first result from Google search: https://pdfs.semanticscholar.org/a5df/128eed59668b525a743a4e7f3f0efe12f930.pdf
In fact, one of the reasons that we in general think of CVaR as a superior risk measure to VaR is the fact that CVaR is a coherent risk measure and VaR is not. Convexity needs to be satisfied in order for risk measure to be coherent.