Convex Optimization via Feedbacks

A method to approach a solution to a finite-dimensional convex optimization problem via trajectories of a control system is suggested. The feedbacks exploit the idea of extremal shifting control from the theory of closed-loop differential games. Under these feedbacks, system's velocities are formed through current relaxations of the initial problem. In relaxed problems, the initial equality constraint is replaced by a scalar equality or a scalar inequality showing, respectively, directions to keep or non-increase a current value of the discrepancy. The first (alpha-shifting) feedback minimizes Lagrangians for current relaxed problems, and results in a dynamical implementation of the penalty method. The second (half-space shifting) feedback solves relaxed problems directly. The first feedback is simpler but less accurate (accuracy bounds are pointed out). The sought solutions are approximated by state-over-time ratios. Discrete and continuous control patterns are considered. Asymptotical convergence with time growing to infinity is proved, and "immediate solution" trajectories having proper asymptotics with time shrinking to zero are designed

Minimizing the distance to one evader while chasing another

AbstractTo approach a simple game Δ2 of P and E = {E1, E2} with no a priori evaders' role assignment and the payoff equal to the distance to one evader at an instant of catching another, we introduce a concept of casting and study the games Δ1,2 and Δ2,1 for preassigned and Δp2 for open-loop casting procedures. Since Δp2 is reduced to Δ1,2 or Δ2,1 which, in turn, are distinguished only by their notations, we focus attention mainly on Δ1,2. According to the tenet of transition, Δ1,2 is divided into a concatenation of Δ1,2b (basic) and Δ1,2a (auxiliary) games that model the problem before and after the first instant of E1 capture. The games Δ1,2a, Δ1,2b, Δ1,2 are studied one after another with use of the Isaacs' approach extended by Berkowitz, Breakwell, Bernhard et al

Markov-Perfect Rent Dissipation in Rights-Based Fisheries

We present a general dynamic model of within-season harvesting competition in a fishery managed with individual transferable quotas. Markov-perfect equilibrium (MPE) harvesting and quota purchase strategies are derived using numerical collocation methods. We identify rent loss caused by a heterogeneous-in-value fish stock, congestion on the fishing ground, revenue competition, and stock uncertainty. Our results show that biological, technological, and market conditions under which rents will be dissipated in a standard individual transferable quota program are fairly special. We offer new insights for designing rights-based programs capable of generating resource rent in marine fisheries

Behavioral Equilibria for a 2x2 "Seller-Buyer" Game Evolutionary Model

Equilibric behaviors typical for differential and multi-step games are defined for a 2 by 2 evolutionary game (two populations of players, two strategies for each player) roughly modeling interactions between sellers and buyers. It is shown that currently optimal behaviors of individuals form long-run equilibric dynamics at both individual and population levels