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

On a Boundedly Rational Pareto-Optimal Trade in Emission Reduction

We consider the emission reduction process involving several countries, in which the countries negotiate, in steps, frequently enough, on small, local emission reductions and implement their decisions right away. In every step, the countries either find a mutually acceptable local emission reduction vector and use it as a local emission reduction plan, or terminate the emission reduction process. We prove that the process necessarily terminates in some step and the final total emission reduction vector lies in a small neighborhood of a certain Pareto maximum point in the underlying emission reduction game. We use examples to illustrate some features of the proposed decision making scheme and discuss a way to organize negotiations in every step of the emission reduction process

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

Long-term and short-term targets: Conflict and reconciliation

We address the issue of a tradeoff between long- and short-term interests in economic management. Starting with an obvious observation that the actions targeted to long- and short-term goals are generally in conflict, we pass on to a less obvious satement that it is not an exceptional situation that a smart decision maker can reduce or even eliminate the conflict. We illustrate the statement by an informal analysis of a stylized model of management of an enterprise. We show that if the managers put enough effort in identification of the current-value shadow prices of the enterprise, their current short-term-optimal actions become non-distinguishable from the long-term-optimal ones

Equilibrium and Guaranteeing Solutions in Evolutionary Nonzero Sum Games

Advanced methods of theory of optimal guaranteeing control and techniques of generalized (viscosity, minimax) solutions of Hamilton-Jacobi equations are applied to nonzero game interaction between two large groups (coalitions) of agents (participants) arising in economic and biological evolutionary models. Random contacts of agents from different groups happen according to a control dynamical process which can be interpreted as Kolmogorov's differential equations in which coefficients describing flows are not fixed a priori and can be chosen on the feedback principle. Payoffs of coalitions are determined by the functionals of different types on infinite horizon. The notion of a dynamical Nash equilibrium is introduced in the class of control feedbacks. A solution feedbacks maximizing with the guarantee the own payoffs (guaranteeing feedback) is proposed. Guaranteeing feedbacks are constructed in the framework of the the theory of generalized solutions of Hamilton-Jacobi equations. The analytical formulas are obtained for corresponding value functions. The equilibrium trajectory is generated by guaranteeing feedbacks and its properties are investigated. The considered approach provides new qualitative results for the equilibrium trajectory in evolutionary models. The first striking result consists in the fact that the designed equilibrium trajectory provides better (in some bimatrix games strictly better) index values for both coalitions than trajectories which converge to static Nash equilibria (as, for example, trajectories of classical models with the replicator dynamics). The second principle result implies both evolutionary properties of the equilibrium trajectory: evolution takes place in the characteristic domains of Hamilton-Jacobi equations and revolution at switching curves of guaranteeing feedbacks. The third specific feature of the proposed solution is "positive" nature of guaranteeing feedbacks which maximize the own payoff unlike the "negative" nature of punishing feedbacks which minimize the opponent payoff and lead to static Nash equilibrium. The fourth concept takes into account the foreseeing principle in constructing feedbacks due to the multiterminal character of payoffs in which future |states are also evaluated. The fifth idea deals with the venturous factor of the equilibrium trajectory and prescribes the risk barrier surrounding it. These results indicate promising applications of theory of guaranteeing control for constructing solutions in evolutionary models

Why Prognostic Systems Analysis Has To Change - Learning from the Past Tells How

Prognostic systems analysis is widely applied to generate "sharp" projections into the future. However, prognostic scenarios and "sharp" futures are a physical impossibility! The key questions arising are (1) whether it is possible to determine the Heisenberg-like relation of a model; and (2) whether it is even possible to determine the model's characteristic unsharpness regime by learning from the past

Towards Detection of Early Warning Signals on Financial Crises

The financial crises of 2001-2002 and 2008-2009 had a significant impact on the world economy. In this paper, we investigate whether early warning signals can be seen in financial time series preceding the crises. In our analysis, we use data on the Dow Jones Industrial Average and Federal Reserve Interest Rate. We construct a random process describing the occurrence of positive and negative signals in a time series preceding the financial crisis of 2001-2002. We use the constructed random process and a time series for the period 2001-2008 to assess the probability of a crisis to occur in 2008-2009. We show that the probability exhibits a steady growth and conclude that the proposed method demonstrates an ability to register early warning signals on the global financial crisis of 2008-2009

The Pontryagin Maximum Principle for Infinite-Horizon Optimal Controls

this paper (motivated by recent works on optimization of long-term economic growth) suggests some further developments in the theory of first-order necessary optimality conditions for problems of optimal control with infinite time horizons. We describe an approximation technique involving auxiliary finite-horizon optimal control problems and use it to prove new versions of the Pontryagin maximum principle. A special attention is paid to behavior of the adjoint variables and the Hamiltonian. Typical cases, in which standard transversality conditions hold at infinity, are described. Several significant earlier results are generalized

Central Path Dynamics and a Model of Competition, II

Growth -- the change in number or size -- and adaptation -- the change in quality or structure -- are key attributes of global processes in natural communities, society and economics (see, e.g. Hofbauer and Sigmund, 1988; Freedman, 1991; Young, 1993). In this paper we describe a model with explicit growth-adaptation feedbacks. We treat it in the form of an economic model of competition of two firms (with several departments) on the market. Their size is measured by their capital, and their quality by their productive power (production complexity). It is assumed that the production complexity of a department or firm is a simple function (that is more general than the one considered in Krazhimskii and Stoer, 1999) of its capital. The model works on both the firm level (competition among the departments) and the market level (competition among the firms). The model shows some empirically observable phenomena. Typically, one of the firms will finally cover the market. The winner is not necessarily the firm with the potentially higher maximum productivity. A long-term coexistence of firms may arise in exceptional situations occurring only when the maximum potential productivities (not the actual productivities) are equal. The analysis is also based on the concept of central paths from the interior point optimization theory (see Sonnevend, 1985; and e.g., Ye, 1997)

Input Reconstructibility for Linear Dynamics. Ordinary Differential Equations

The paper deals with the standard input-output observation scheme for a dynamic system governed by a linear ordinary differential equation. The initial problem is to reconstruct the actually working time-varying input, given a state observation result. Normally, the problem has no solution: observation is too poor to select the real input from the collection of "possible" ones. It is proposed to turn the problem as follows: what information of the real input is reconstructible precisely? The dual setting: what information of the real input is totally non-reconstructible? The question of aftereffect arises naturally: does accumulation of observation results lead to the informational jump -- from nonreconstructibility to complete reconstructibility -- in the past? Posing and answering these questions is the goal of the present study