Infinite Dimensional Duality Theory Applied to Investment Strategies in Environmental Policy

Abstract

In this paper, we develop an infinite dimensional Lagrangian duality framework for modeling and analyzing the evolutionary pollution control problem. Specifically, we examine the situation in which different countries aim at determining the optimal investment allocation in environmental projects and the tolerable pollutant emissions, so as to maximize their welfare. We state the equilibrium conditions underlying the model, and provide an equivalent formulation in terms of an evolutionary variational inequality. Moreover, by means of infinite dimensional duality tools, we prove the existence of Lagrange multipliers that play a fundamental role in order to describe countries' decision-making processes.

The infinite dimensional Lagrange multiplier rule for convex optimization problems

“…Since many infinite dimensional equilibrium problems have ordering cone empty, these usual conditions cannot be used to guarantee the strong duality. This is the case of all optimization problems or variational inequalities connected with network equilibrium problems, the obstacle problems, the elastic plastic torsion problems (see [1,[5][6][7][8][9][11][12][13][14]16,19,20,23]) which use positive cones of L p (Ω) or Sobolev spaces. Recently, in [9,8,10,18] the authors overcome this important difficulty by introducing a condition called Assumption S which ensures the strong duality.…”