Sjur Didrik Flåm

Equilibrium programming using proximal-like algorithms

We compute constrained equilibria satisfying an optimality condition. Important examples include convex programming, saddle problems, noncooperative games, and variational inequalities. Under a monotonicity hypothesis we show that equilibrium solutions can be found via iterative convex minimization. In the main algorithm each stage of computation requires two proximal steps, possibly using Bregman functions. One step serves to predict the next point; the other helps to correct the new prediction. To enhance practical applicability we tolerate numerical errors.

Relaxed outer projections, weighted averages and convex feasibility

A new algorithmic scheme is proposed for finding a common point of finitely many closed convex sets. The scheme uses weighted averages (convex combinations) of relaxed projections onto approximating halfspaces. By varying the weights we generalize Cimminos and Auslenders methods as well as more recent versions developed by Iusem & De Pierro and Aharoni & Censor. Our approach offers great computational flexibility and encompasses a wide variety of known algorithms as special instances. Also, since it is “block-iterative”, it lends itself to parallel processing.

Equilibrium, Evolutionary Stability And Gradient Dynamics

Considered here are equilibria, notably those that solve noncooperative games. Focus is on connections between evolutionary stability, concavity and monotonicity. It is shown that evolutionary stable points are local attractors under gradient dynamics. Such dynamics, while reflecting search for individual improvement, can incorporate myopia, imperfect knowledge and bounded rationality/competence.

A continuous approach to oligopolistic market equilibrium

We provide an algorithm for computing Cournot-Nash equilibria in a market that involves finitely many producers. The algorithm amounts to following a certain dynamical system all the way to its steady state, which happens to be a noncooperative equilibrium. The dynamics arise quite naturally as follows. Let each producer continuously adjust the planned production, if desired, as a response to the current aggregate supply. In doing so, the producer is completely guided by myopic profit considerations. We show, under broad hypothesis, that this adjustment process is globally, asymptotically convergent to a Nash equilibrium.

Approaches to economic equilibrium

Abstract We study mechanisms that help to explain why and how equilibrium may eventually be attained in many economic settings. Examples include competitive markets and many noncooperative games. The main object is an adaptive process, reflecting repeated adjustment of individual strategies, which under monotonicity conditions leads to equilibrium. Environmental uncertainty is accommodated, but agents need neither know the probability law, nor invoke any statistical learning theory, nor compute mean values. In fact, everyone can act — perhaps without precedent, maybe among strangers — within unidentified frames. Nonetheless, we show, under reasonable conditions, that individual optimality and system equilibrium obtains in the long run.

Lagrange multipliers in stochastic programming

A Fritz John multiplier rule is given for discrete time, finite horizon, stochastic programs. Particular emphasis is placed on constraint qualifications that allow for the application of this rule in normal Lagrange form.

On finite convergence and constraint identification of subgradient projection methods

This paper deals with a continuous time, subgradient projection algorithm, shown to generate trajectories that accumulate to the solution set. Under a strong convexity assumption we show that convergence is exponential in norm. A sharpness condition yields convergence in finite time, and the necessary lapse is estimated. Invoking a constraint qualification and a non-degeneracy assumption, we demonstrate that optimally active constraints are identified in finite time.

Oligopolistic Competition; from Stability to Chaos

This note deals with the classic Cournot (1838) model of oligopolistic competition, thoroughly reviewed in Tirole (1988). Under discussion is an industry composed of finitely many firms i ∈ I, all producing the same homogeneous good for one competitive market. Firm i furnishes the quantity qi ≥ 0 at cost ci(qi ), thus obtaining the profit

Solving convex programs by means of ordinary differential equations

{{\pi }_{i}} = p(Q){{q}_{i}} - {{c}_{i}}({{q}_{i}}).