Separable convex programs with ratio constraints over a directed tree
The authors develop a simple procedure for solving separable convex programs with ratio constraints in the form \[ \text{Min }\sum_{j\in N} f_ j(x_ j)\text{ s.t. } x_ i\geq r_{i_ j} x_ j,\;(i,j)\in A,\;\ell_ i\leq x_ i\leq u_ i,\;i\in N,\leqno(\text P) \] where the \(x_ i\)'s are the decision variables, \(\ell_ i\) and \(u_ i\) are the lower and upper bounds on each decision variable, \(N\) is the set of nodes and \(A\), called the arcs, is a subset of \(N\times N\). The functions \(f_ j(.)\) are assumed to be strictly convex and differentiable. The authors notice that this procedure is an extension of the technique developed by the second author [Manage. Sci. 36, No. 11, 1407-1412 (1990; Zbl 0727.90060)]. It is claimed that such problems arise in production planning, regression analysis and other related areas.
scientific article; zbMATH DE number 92009