Market Equilibrium

People have very different interpretations of the notion of an equilibrium, reflected in different formulations of the problem. Some consider it as rather a dynamic process, so formulated it as fixed point, which is more common in growth problems (also in the taxation dynamics). It is not clear yet how this is related to using duality in linear programming. According to Takayama, Arrow formulated it as games. Takayama's advocated formulation is actually by recognizing a resemblance between Pareto Optimum and the vector maximization in nonlinear programming, and as a result, the use of nonlinear programming to solve for competitive equilibrium ensures direct satisfaction of Pareto Optimum, and the existence conditions of such an equilibrium is already implicit in the requirements of nonlinear programming (such as concavity, local non-satiation, etc).

It seems that to understand the equilibrium problem is somewhat equivalent to understanding the relationship between the following few concepts:

fixed-point
duality (envelope)
nonlinear programming (saddle point)
intermediate value theorem (related to taylor expansion and asymptotic properties)
convergence of sequences
mean value theorem
central limit theorem