A Greedy Algorithm for Minimizing a Separable Convex Function Over an Integral Bisubmodular Polyhedron

A Greedy Algorithm for Minimizing a Separable Convex Function Over a Finite Jump System

“…Since -s E St(xs,:r'), it fbllows from (4.18) and the separable convexity of w that As in the proof of Theorem 4,l in [1], we can show…”

“…Recent･ly, we presented a greedy algorit-hm fbr minimizing a separable convex function over an integral bisubmodular polyhedron ( [1]). We show in this paper that the algorithm given in [1] also works over a finite jump system (E,f).…”

“…Recent･ly, we presented a greedy algorit-hm fbr minimizing a separable convex function over an integral bisubmodular polyhedron ( [1]). We show in this paper that the algorithm given in [1] also works over a finite jump system (E,f). Our algorithm starts with an arbitrary initial feasible point and repeat/s coordinate-wise augmentations andlor exchanges in a greedy wa"r, In our previous paper Il] we did not･ give an estimation of the number of the required transformations of feasible solutions but by examining the behaJvior of the greedy algorithm we will show t,hat the greedy algorithm for a finite jurnp system (E,f) terminates' after changing an initial feasible solnt,ion at most 2{u(e)-i(e)} (1,2) aEE times, where fbr each e E II U(e)= ip,{?g x(e), ICe) = I}},i,n, x(e).…”

Personal reminiscence: combinatorial and discrete optimization problems in which I have been interested

“…The subject on bisubmodular functions is one of them. I worked on bisubmodular functions with my then doctoral students Takeshi Naitoh, Toshio Nemoto, and Kazutoshi Ando ([2], [3], [5], [6], [7]) and with Sachin Patkar [59]. I also found a nice min-max theorem for bisubmodular functions [41], which gave a basis for bisubmodular function minimization (see joint work with Iwata [54] and with Tom McCormick [90]).…”

“…I also found a nice min-max theorem for bisubmodular functions [41], which gave a basis for bisubmodular function minimization (see joint work with Iwata [54] and with Tom McCormick [90]). Also an extension of [3] to jump systems appeared in [4].…”