6.3.5 (c-Ascent Method (BeM71), BeM73]) www Consider the unconstrained maximization of a concave function : RR, and for > 0, consider the e-subdifferential at a vector u deg() = {919() 59(41) + g'( 4) + , V R"}. (a) Use the line of proof of Prop. B.24 (parts (a) and (b)] to show that deq(u) is nonempty, convex, and compact, and that glu + sd) --q(u)- sup inf d'o. 9E89) 80 S (b) Show that 0 acq() if and only if u is an e-optimal solution, i.e., sup qi) - 904). PERT (c) Show that if a direction d is such that inf gedeg() d'g > 0, then sup (u + sd) >9(x) + . (d) Show that if o acq(), then the direction 9H = arg min lgll 989) satisfies 9,9 > 0 for all ge deglu). (e) Consider the following procedure (first given in (Lem74]) that parallels the one of Exercise 6.3.4. Given w, let gi be some e-subgradient of q at H. For k = 2,3,..., let w* be the vector of minimum norm in the convex hull of g'...,9 = arg min 1911. 9conv{9?....k-"} If w* = 0, stop; we have 0 2.9(u), so superr 9) - 0}, determine whether there exists a scalar 3 such that g(x + zw*) > 9() - . If such a s can be found, stop and replace u with u + gut (the dual value has been improved by at least e). Otherwise let gk be an element of acq(u) such that min g'w*, gec() wa [note that from part (a), we have gk c*