Evolution Solutions of Equilibrium Problems: A Computational Approach

Abstract

This paper proposes a computational method to describe evolution solutions of known classes of time-dependent equilibrium problems (such as time-dependent traffic network, market equilibrium or oligopoly problems, and dynamic noncooperative games). Equilibrium solutions for these classes have been studied extensively from both a theoretical (regularity, stability behaviour) and a computational point of view. In this paper we highlight a method to further study the solution set of such problems from a dynamical systems perspective, namely we study their behaviour when they are not in an (market, traffic, financial, etc.) equilibrium state. To this end, we define what is meant by an evolution solution for a time-dependent equilibrium problem and we introduce a computational method for tracking and visualizing evolution solutions using a projected dynamical system defined on a carefully chosen L ²-space. We strengthen our results with various examples.

Appendix to Example 3

Recalling the mapping

$$\displaystyle{ F(x) = \left (\begin{array}{l} 3x_{1} - x_{2} + x_{3}\\ \\ \frac{3} {2}x_{1} - 4x_{2} + x_{3}\\ \\ 2x_{1} - 7x_{2} + x_{3} \end{array} \right ), }$$

with the constraint set

$$\displaystyle{ \mathbb{K}(t)\,=\,\{x\in L^{2}([0,90], \mathbb{R}^{3})\mid 0\leq x(t),x_{ 1}(t)+x_{2}(t)+x_{3}(t)=\rho (t),x_{1}(t)+\frac{1} {2}x_{2}(t)\,=\,\psi (t)\}, }$$

We show below that F is monotone, but not strictly so. In what follows we make use of the following identities which can be derived from $\mathbb{K}$: for all $x,y \in \mathbb{K}$ we have that

$$\displaystyle{ x_{3} - y_{3} = -(x_{1} - y_{1}) - (x_{2} - y_{2}), }$$

(7)

and

$$\displaystyle{ x_{2} - y_{2} = -2(x_{1} - y_{1}). }$$

(8)

We evaluate now $\langle F(y) - F(x),y - x\rangle =\sum _{ i=1}^{3}F_{ i}(x)(x_{i} - y_{i})$ with (7), (8) yielding

$$\displaystyle{ 2(x_{1} - y_{1})^{2} \geq 0,\,\forall x\neq y \in \mathbb{K}(t). }$$