Optimal flows in nonlinear gain networks

Abstract

Computation of globaZly optima2 fZms i n nonlinear gain networks i s generally quite d i f f i m Z t . cZasses of these networks for which optimal solutions are easi Zy obtained.

Concave Generalized Flows with Applications to Market Equilibria

“…The following extension has been proposed by Truemper [39] and Shigeno [35]. On each arc ij ∈ E, we are given lower and upper arc capacities ℓ, u : E → R and a monotone increasing concave function Γ ij : [ℓ ij , u ij ] → R ∪ {−∞}; we are also given node demands b : V → R. As for generalized flows, a pseudoflow is a function f : E → R with ℓ ≤ f ≤ u.…”

“…In this paper, we consider a nonlinear extension, concave generalized flows, studied by Truemper [39] in 1978, and by Shigeno [35] in 2006. For each arc e we are given a concave, monotone increasing function Γ e such that if α units enter e then Γ e (α) units leave it.…”

Spatial price equilibrium networks with flow-dependent arc multipliers

“…Interestingly, in the majority of generalized network models the arc multipliers are fixed [ 1 , 10 , 32 , 36 , 52 ]; that is, they are not flow-dependent. Truemper in [ 50 ] recognized that, to that date, most of the generalized networks that had been studied had fixed arc multipliers, which he noted correspond to linear, as opposed to, nonlinear, functions of flow on the arcs. He argued that it was important to consider nonlinear functions, since there are real-world problems, including some associated with chemical processes, that could not be modeled using, in effect, fixed arc multipliers for gains (see [ 53 ]).…”

“…Subsequently, Shigeno in [ 43 ] also studied such generalized networks, wherein, specifically, the flow leaving an arc is an increasing concave function of the flow entering it. More recently, Vegh in [ 51 ], building on the earlier work in [ 43 , 50 ], established that the resulting general convex programming model is relevant for several market equilibrium problems, including the linear Fisher market model and its various extensions. Vegh in [ 51 ] constructed a polynomial time combinatorial algorithm for solving corresponding flow maximization problems, which also yields a new algorithm for linear generalized flows.…”

A SURVEY OF COMBINATORIAL MAXIMUM FLOW ALGORITHMS ON A NETWORK WITH GAINS(<Special Issue>Network Design, Control and Optimization)

“…(312)il", which implies that Step 2 in the above framework will find a 1!2-optimal flew. [59] max-fiowalgo. }l (7(C,))if" = ('),geO(C,))if" ) (7"O(ak.+imi)1(cr"O(ak.+,) ･ .…”