Inverse optimization problems with multiple weight functions

We introduce a new class of inverse optimization problems in which an input solution is given together with k linear weight functions, and the goal to modify weights by same deviation vector p so that becomes optimal respect each them, while minimizing ‖p‖1. In particular, we concentrate on three multiple functions: shortest s−t path, bipartite perfect matching, arborescence problems. Using LP duality, give min–max characterizations for ℓ1-norm vector. Furthermore, show not necessarily integral even when functions are so, therefore computing significantly more difficult than single-weighted case. also necessary sufficient condition existence changes values only elements solution, thus giving unified understanding previous results arborescences matchings.