Solution concepts in two-person multicriteria games

In this paper, we propose new solution concepts for multicriteria games and compare them with existing ones. The general setting is that of two-person finite games in normal form (matrix games) with pure and mixed strategy sets for the players. The notions of efficiency (Pareto optimality), security levels, and response strategies have all been used in defining solutions ranging from equilibrium points to Pareto saddle points. Methods for obtaining strategies that yield Pareto security levels to the players or Pareto saddle points to the game, when they exist, are presented. Finally, we study games with more than two qualitative outcomes such as combat games. Using the notion of guaranteed outcomes, we obtain saddle-point solutions in mixed strategies for a number of cases. Examples illustrating the concepts, methods, and solutions are included.

Multicriterion differential games with applications to combat problems

In this paper a theory for two-person zero sum multicriterion differential games is presented. Various solution concepts based upon the notions of Pareto optimality (efficiency), security and equilibrium are defined. These are shown to have interesting applications in the formulation and analysis of two target or combat differential games. The methods for obtaining outcome regions in the state space, feedback strategies for the players and the mode of play has been discussed in the framework of bicriterion zero sum differential games. The treatment is conceptual rather than rigorous.

Solution concepts in continuous-kernel multicriteria games

This paper considers nonzero-sum multicriteria games with continuous kernels. Solution concepts based on the notions of Pareto optimality, equilibrium, and security are extended to these games. Separate necessary and sufficient conditions and existence results are presented for equilibrium, Pareto-optimal response, and Pareto-optimal security strategies of the players.

On optimum stratification with proportional allocation for a class of pareto distributions

Cum ./LSTA_A_8828879_O_XML_IMAGES/LSTA_A_8828879_O_ILM0001.gif rule [Singh (1975)] has been suggested in the literature for finding approximately optimum strata boundaries for proportional allocation, when the stratification is done on the study variable. This paper shows that for the class of density functions arising from the Wang and Aggarwal (1984) representation of the Lorenz Curve (or DBV curves in case of inventory theory), the cum ./LSTA_A_8828879_O_XML_IMAGES/LSTA_A_8828879_O_ILM0002.gif rule in place of giving approximately optimum strata boundaries, yields exactly optimum boundaries. It is also shown that the conjecture of Mahalanobis (1952) “. . .an optimum or nearly optimum solutions will be obtained when the expected contribution of each stratum to the total aggregate value of Y is made equal for all strata” yields exactly optimum strata boundaries for the case considered in the paper.

Uplink Power Control and Base Station Association in Multichannel Cellular Networks

A combined base station association and power control problem is studied for the uplink of multichannel multicell cellular networks, in which each channel is used by exactly one cell (i.e., base station). A distributed association and power update algorithm is proposed and shown to converge to a Nash equilibrium of a noncooperative game. We consider network models with discrete mobiles (yielding an atomic congestion game), as well as a continuum of mobiles (yielding a population game). We find that the equilibria need not be Pareto efficient, nor need they be system optimal. To address the lack of system optimality, we propose pricing mechanisms. It is shown that these mechanisms can be implemented in a distributed fashion.

Pareto-optimal solutions for multi-objective optimization of fed-batch bioreactors using nondominated sorting genetic algorithm

Many optimal control problems are characterized by their multiple performance measures that are often noncommensurable and competing with each other. The presence of multiple objectives in a problem usually give rise to a set of optimal solutions, largely known as Pareto-optimal solutions. Evolutionary algorithms have been recognized to be well suited for multi-objective optimization because of their capability to evolve a set of nondominated solutions distributed along the Pareto front. This has led to the development of many evolutionary multi-objective optimization algorithms among which Nondominated Sorting Genetic Algorithm (NSGA and its enhanced version NSGA-II) has been found effective in solving a wide variety of problems. Recently, we reported a genetic algorithm based technique for solving dynamic single-objective optimization problems, with single as well as multiple control variables, that appear in fed-batch bioreactor applications. The purpose of this study is to extend this methodology for solution of multi-objective optimal control problems under the framework of NSGA-II. The applicability of the technique is illustrated by solving two optimal control problems, taken from literature, which have usually been solved by several methods as single-objective dynamic optimization problems. (C) 2004 Elsevier Ltd. All rights reserved.

On the optimality of exhaustive service policies in multiclass queueing systems with modulated arrivals and switchovers

Consider a single-server multiclass queueing system with K classes where the individual queues are fed by K-correlated interrupted Poisson streams generated in the states of a K-state stationary modulating Markov chain. The service times for all the classes are drawn independently from the same distribution. There is a setup time (and/or a setup cost) incurred whenever the server switches from one queue to another. It is required to minimize the sum of discounted inventory and setup costs over an infinite horizon. We provide sufficient conditions under which exhaustive service policies are optimal. We then present some simulation results for a two-class queueing system to show that exhaustive, threshold policies outperform non-exhaustive policies.

Optimality of orthogonal sequences with multi-dimensional signaling

The sum capacity on a symbol-synchronous CDMA system having processing gain N and supporting K power constrained users is achieved by employing any set of N orthogonal sequences if a few users are allowed to signal along multiple dimensions. Analogously, the minimum received power (energy-per-chip) on the symbolsynchronous CDMA system supporting K users that demand specified data rates is attained by employing any set of N orthogonal sequences. At most (N - 1) users need to be split and if there are no oversized users, these split users need to signal only in two dimensions each. These results show that sum capacity or minimum sum power can be achieved with minimal downlink signaling.

Enhanced DMT-optimality criterion for STBC schemes for asymmetric MIMO systems

For any n(t) transmit, n(r) receive antenna (n(t) x n(r)) multiple-input multiple-output (MIMO) system in a quasi-static Rayleigh fading environment, it was shown by Elia et al. that linear space-time block code schemes (LSTBC schemes) that have the nonvanishing determinant (NVD) property are diversity-multiplexing gain tradeoff (DMT)-optimal for arbitrary values of n(r) if they have a code rate of n(t) complex dimensions per channel use. However, for asymmetric MIMO systems (where n(r) < n(t)), with the exception of a few LSTBC schemes, it is unknown whether general LSTBC schemes with NVD and a code rate of n(r) complex dimensions per channel use are DMT optimal. In this paper, an enhanced sufficient criterion for any STBC scheme to be DMT optimal is obtained, and using this criterion, it is established that any LSTBC scheme with NVD and a code rate of min {n(t), n(r)} complex dimensions per channel use is DMT optimal. This result settles the DMT optimality of several well-known, low-ML-decoding-complexity LSTBC schemes for certain asymmetric MIMO systems.

OPTIMALITY OF THE PRODUCT-MATRIX CONSTRUCTION FOR SECURE MSR REGENERATING CODES

In this paper, we consider the security of exact-repair regenerating codes operating at the minimum-storage-regenerating (MSR) point. The security requirement (introduced in Shah et. al.) is that no information about the stored data file must be leaked in the presence of an eavesdropper who has access to the contents of l(1) nodes as well as all the repair traffic entering a second disjoint set of l(2) nodes. We derive an upper bound on the size of a data file that can be securely stored that holds whenever l(2) <= d - k +1. This upper bound proves the optimality of the product-matrix-based construction of secure MSR regenerating codes by Shah et. al.

Optimality Conditions and Duality for Semi-Infinite Mathematical Programming Problem with Equilibrium Constraints

This article considers a semi-infinite mathematical programming problem with equilibrium constraints (SIMPEC) defined as a semi-infinite mathematical programming problem with complementarity constraints. We establish necessary and sufficient optimality conditions for the (SIMPEC). We also formulate Wolfe- and Mond-Weir-type dual models for (SIMPEC) and establish weak, strong and strict converse duality theorems for (SIMPEC) and the corresponding dual problems under invexity assumptions.

Wireless scheduling with partial channel state information: large deviations and optimality

We consider a server serving a time-slotted queued system of multiple packet-based flows, where not more than one flow can be serviced in a single time slot. The flows have exogenous packet arrivals and time-varying service rates. At each time, the server can observe instantaneous service rates for only a subset of flows ( selected from a fixed collection of observable subsets) before scheduling a flow in the subset for service. We are interested in queue length aware scheduling to keep the queues short. The limited availability of instantaneous service rate information requires the scheduler to make a careful choice of which subset of service rates to sample. We develop scheduling algorithms that use only partial service rate information from subsets of channels, and that minimize the likelihood of queue overflow in the system. Specifically, we present a new joint subset-sampling and scheduling algorithm called Max-Exp that uses only the current queue lengths to pick a subset of flows, and subsequently schedules a flow using the Exponential rule. When the collection of observable subsets is disjoint, we show that Max-Exp achieves the best exponential decay rate, among all scheduling algorithms that base their decision on the current ( or any finite past history of) system state, of the tail of the longest queue. To accomplish this, we employ novel analytical techniques for studying the performance of scheduling algorithms using partial state, which may be of independent interest. These include new sample-path large deviations results for processes obtained by non-random, predictable sampling of sequences of independent and identically distributed random variables. A consequence of these results is that scheduling with partial state information yields a rate function significantly different from scheduling with full channel information. In the special case when the observable subsets are singleton flows, i.e., when there is effectively no a priori channel state information, Max-Exp reduces to simply serving the flow with the longest queue; thus, our results show that to always serve the longest queue in the absence of any channel state information is large deviations optimal.

Necessary and sufficient conditions for optimality in constrained general sum stochastic games

In this paper we first derive a necessary and sufficient condition for a stationary strategy to be the Nash equilibrium of discounted constrained stochastic game under certain assumptions. In this process we also develop a nonlinear (non-convex) optimization problem for a discounted constrained stochastic game. We use the linear best response functions of every player and complementary slackness theorem for linear programs to derive both the optimization problem and the equivalent condition. We then extend this result to average reward constrained stochastic games. Finally, we present a heuristic algorithm motivated by our necessary and sufficient conditions for a discounted cost constrained stochastic game. We numerically observe the convergence of this algorithm to Nash equilibrium. (C) 2015 Elsevier B.V. All rights reserved.

Aircraft pursuit-evasion problems with variable speeds

Aircraft pursuit-evasion encounters in a plane with variable speeds are analysed as a differential game. An engagement-dependent coordinate system confers open-loop optimality on the game. Each aircraft's optimal motion can be represented by extremel trajectory maps which are independent of role, adversary and capture radius. These maps are used in two different ways to construct the feedback solution. Some examples are given to illustrate these features. The paper draws on earlier results and surveys several existing papers on the subject.

A Hierarchical System of Learning Automata

A learning automaton operating in a random environment updates its action probabilities on the basis of the reactions of the environment, so that asymptotically it chooses the optimal action. When the number of actions is large the automaton becomes slow because there are too many updatings to be made at each instant. A hierarchical system of such automata with assured c-optimality is suggested to overcome that problem.The learning algorithm for the hierarchical system turns out to be a simple modification of the absolutely expedient algorithm known in the literature. The parameters of the algorithm at each level in the hierarchy depend only on the parameters and the action probabilities of the previous level. It follows that to minimize the number of updatings per cycle each automaton in the hierarchy need have only two or three actions.