We consider multi-criteria sequential decision making problems where the vector-valued evaluations are compared by a given, fixed total ordering. Conditions for the optimality of stationary policies and the Bellman optimality equation are given. The analysis requires special care as the topology introduced by pointwise convergence and the order-topology introduced by the preference order are in general incompatible. Reinforcement learning algorithms are proposed and analyzed. Preliminary computer experiments confirm the validity of the derived algorithms. It is observed that in the medium-term multicriteria RL often converges to better solutions (measured by the first criterion) than their single-criterion counterparts. These type of multicriteria problems are most useful when there are several optimal solutions to a problem and one wants to choose the one among these which is optimal according to another fixed criterion. Example applications include alternating games, when in addition...

The solution to an instance of the standard Shortest Path problem is a single shortest route in a directed graph. Suppose, however, that each arc has both a distance and a cost, and that one would like to find a route that is both short and inexpensive. In general, no single route will be both shortest and cheapest; rather, the solution to an instance of this multi-criteria problem will be a set of efficient or Pareto optimal routes. The (distance, cost) pairs associated with the efficient routes define an efficient frontier or tradeoff curve. An efficient set for a multi-criteria problem can be exponentially large, even when the underlying singlecriterion;oblem is in P. This work therefore considers approximate solutions to rlulti-criteria discrete optimization problems and investigates when they can be found quickly. This requires generalizing the notion of a fully polynomial time approximatiofi scheme to multi-criteria problems. In this paper, necessary and sufficient conditions are developed for the existence of such a fast approximation scheme for a problem. Although the focus is multi-criteria problems, the conditions are of interest even in the single criterion case. In addition, an appropriate form of problem reduction is introduced to facilitate the application of these conditions to a variety of problems. A companion paper uses the results of this paper to study the existence of fast approximation schemes for several interesting network flow, knapsack, and

We consider multi-criteria sequential decision making problems where the vector-valued evaluations are compared by a given, fixed total ordering. Conditions for the optimality of stationary policies and the Bellman optimality equation are given. The analysis requires special care as the topology introduced by pointwise convergence and the order-topology introduced by the preference order are in general incompatible. Reinforcement learning algorithms are proposed and analyzed. Preliminary computer experiments confirm the validity of the derived algorithms. It is observed that in the medium-term multicriteria RL often converges to better solutions (measured by the first criterion) than their single-criterion counterparts. These type of multicriteria problems are most useful when there are several optimal solutions to a problem and one wants to choose the one among these which is optimal according to another fixed criterion. Example applications include alternating games, when in addition...

this article we take another starting point and that is to consider perfect dynamics. We say that a recurrent ANN admits perfect dynamics if the dynamical system given by the update operator of the network has an attractor whose basin of attraction covers the set of all possible initial solution candidates. One may wonder whether neural networks that admit perfect dynamics can be interesting in applications. In this article we show that there exist a family of such networks (or dynamics). We introduce

\Ve cOllf:iider multi-criteria f:iequent,ial decision making problems where the vcctor-"valucd evaluations arc compared by a given, fixed total ordering. Condit.ions for the opt.irnalit�y of statiOIl<-l,r} ' p()lich�s;-weI the Bellman optimalit,y equation are given for a. special, but. important class of problems ''v hell the evaluation of policies can be computed for the criteria, independently of each other. The anal)'sis requires special cafC as t.he t.opolag,Y int.roduced by polnL\visc convergence a.ncl the or<1cr-topology introduced by the preference order arc in general incompatible. Reinforcement. learning algorithms are proposed and analY7,ed. Prelimina,ry computer experiments confirm t,he val idit.y of the derived algorithms. These type of multi-criteria problem� are most useful,,,,hen t.here are several optimal solutions to a. problem and one 'Vl-lIlt.S to choose the one among these,vhich is optimal according Lo another fixed criLerion. Possible application in robot.ics and repeat.ed ga.mes are outlined.