Abstract: The theory of the proximal point algorithm for maximal monotone operators is applied to three algorithms for solving convex programs, one of which has not previously been formulated Rate-of-convergence results for the “method of multipliers,” of the strong sort already known, are derived in a generalized form relevant also to problems beyond the compass of the standard second-order conditions for oplimality The new algorithm, the “proximal method of multipliers,” is shown to have much the same convergence properties, but with some potential advantages

Abstract: We consider a mathematical programming problem on a Banach space, and we derive necessary conditions for optimality in Lagrange multiplier form. We prove further that “most mathematical programming problems are normal.” The novelty of our approach lies on the one hand in the absence of both differentiability and convexity hypotheses on the functions delimiting the problem, and on the other hand in the method of proof, which is new. The approach unifies the well-known smooth and convex cases besides treating a new general class of problems. The main analytical tool in this paper is an extension to infinite-dimensional spaces of the “generalized gradient” previously introduced by the author. The calculus of the generalized gradient is explored as a preliminary step.

Abstract: Multivalued functions with convex graphs are shown to exhibit certain desirable regularity properties when their ranges have internal points. These properties are applied to develop a perturbation theory for convex inequalities and to extend results on the continuity of convex functions.

Abstract: We study two person, zero sum stochastic games. We prove that limn→∞{Vn/n} = limr→0rVr, where Vn is the value of the n-stage game and Vr is the value of the infinite-stage game with payoffs discounted at interest rate r > 0. We also show that Vr may be expanded as a Laurent series in a fractional power of r. This expansion is valid for small positive r. A similar expansion exists for optimal strategies. Our main proof is an application of Tarski's principle for real closed fields.

Abstract: Consider a storage process X = {Xt, t ≥ 0} with compound Poisson input and a state-dependent release rule r· which is arbitrary except for the requirement that state zero be reachable in finite time from any positive starting state. We show that there exists a stationary distribution for X if and only if there is a limiting distribution independent of the initial state, in which case the stationary distribution is unique and coincides with the limiting distribution. A necessary and sufficient condition for the existence of a stationary distribution, as well as a general solution for the distribution when it exists, is given. We also give a general formula for Ux, the probability that level b is exceeded before level a is reached, starting from state x ∈ a, b]. Both the stationary distribution and Ux are expressed in terms of a certain positive kernel.

Abstract: Given a closed convex cone K in the n-dimensional Euclidean space Rn, its polar cone K+ and a point-to-set mapping f from K to subsets of Rn, the extension of the generalized complementarity problem we consider is to find a vector x in K such that there is a y in fx ∩ K+ with xTy = 0. We prove, using directly a fixed point theorem, with an appropriate condition on f, that the above problem has a solution. We also show that this condition is satisfied by a certain class of mappings f which are encountered in various applications. We further establish “the basic theorem of complementarity” for this problem, which thus extends the previous existence results to the case where K is a pointed cone.

Abstract: This paper examines necessary and sufficient conditions for the existence of voting equilibria over a multidimensional issue space, following the lead of the seminal work of Davis and Hinich Davis, O. A., M. J. Hinich. 1966. A mathematical model of policy formation in a democratic society. J. Bernard, ed. Mathematical Applications in Political Science II. Southern Methodist University Press, Dallas, 175--208. and Plott Plott, C. R. 1967. A notion of equilibrium and its possibility under majority rule. Amer. Econom. Rev.57 787--806.. We assume a finite number of voters who vote among alternatives located in an m-dimensional vector space and that the voters vote according to their preferences defined over the space. Here, we generalize and strengthen many of the existing theorems to deal with global rather than local equilibria and plurality as opposed to majority voting rules.

Abstract: Shapley introduced an index theory for bimatrix games that oriented the paths generated by the Lemke-Howson algorithm and thus partitioned equilibrium points into two sets. Here we develop a similar orientation theory for a generalized complementary pivot algorithm and apply our results to bimatrix games, the linear complementarity problem, and fixed point algorithms. We provide the weakest known general sufficiency conditions (based on those of Garcia) for Lemke's algorithm to process the linear complementarity problem. We also show that if the linear complementarity version of a quadratic programming problem is solved and gives a Kuhn-Tucker solution, then the hessian of the objective function restricted to the subspace of active constraints has positive determinant. Finally we show that fixed point algorithms will only approximate fixed points of f at which the determinant of the Jacobian of f minus the identity has the appropriate sign.

Abstract: A general multistage problem of stochastic optimization is studied. It is proved under some simple assumptions that the extremum in the problem is attained. The “Bellman functions” are constructed and a criterion of optimality in terms of these functions is given. The main tools used are measurable selection theorems. The paper generalizes the previous work of R. T. Rockafellar and R. J.-B. Wets devoted to the convex case.

Abstract: A generalization of the von Neumann-Morgenstern solution, called a subsolution, is introduced. Subsolulions exist for all games (in a nontrival way for games with a nonempty core), and can be interpreted as “standards of behavior.” A unique, distinguished subsolution called the supercore is also identified; it is the intersection of all subsolutions.

Abstract: The distribution of time to first system failure is considered for systems whose repairable components are separately maintained. It is shown that if repairable components have exponential failure law, repair distributions have decreasing repair rate, and nonrepairable components have increasing failure rate, then the distribution of time to first system failure is new better than used. Improved bounds are given for the exponential repair case.

Abstract: This paper considers algorithms that compute fixed points or more generally solve equations which are based on complementary pivoting. These algorithms, starting with a map f0 and its fixed point x0, deform ft to f∞ = f as t goes from 0 to ∞, and follow the path xt of fixed points of ft. In this paper we study these paths. In particular, we treat some special applications of these algorithms from a global point of view, and thus look at the behavior and properties of these paths sufficiently far from the solution. Our methods are motivated by methods of global analysis. We show that in many implementations of these algorithms the path can be specified. These include the cases when the mapping is linear and when it is smooth and monotone. The results of the linear analysis are used to study the local behavior of these paths i.e., properties and behavior sufficiently close to the solution for smooth mappings. An important implication of this study is that the paths can be modified and thus the work done by these algorithms can be controlled. We also show, for smooth mappings, how our results can be implemented to increase the efficiency of some standard algorithms.

Abstract: An algorithm is presented for computing the stationary distribution of a certain class of stochastic processes. The class comprises many known or potentially useful processes in the fields of operations research and applied probability.

Abstract: An arbitration model for cooperative two-person normal-form games is suggested in which each player's strength is measured by what he can obtain through committing himself to a course of action before his opponent does. This approach differs from earlier models in which threats were evaluated on the basis of their relative effects on the two players. Possibilities for extensions to the n-player case are included.

Abstract: Billera Billera, L. J. 1968. On cores and bargaining sets for N-person cooperative games without side payments. Ph.D. dissertation, The City University of New York, 1968; Billera, L. J. 1970. Existence of general bargaining sets for cooperative games without side payments. Bull. Amer. Math. Soc.76 375--379.] proposed a simple bargaining set for games without side payments, and announced it as an open question, whether his proposed bargaining set is never empty. An affirmative answer is given to Billera's question in this paper.

Abstract: Let X, X1, X2,... be independent and identically distributed random variables with E{X} 0 then a well-known result in random walk theory slates that M has a finite moment of order r if and only if the positive part of X has a finite moment of order r + 1. Utilizing the intimate relationship between random walks and single server queues, this paper presents a new and simple proof for this basic result. Identities involving the distribution functions and the integer moments of M and X are then derived by simple, probabilistic arguments. Bounds are then obtained for the expected values of some important variables associated with the random walk Sn, n ≥ 1. All of these random-walk results are applied to the stable queue GI/G/1, and some standard queueing results are subsequently obtained in an effortless manner.

Abstract: Self-penalties of the players in a two-person game are studied as a cooperative tool: by committing himself to a penalty in case he plays a particular strategy, a player may improve upon the equilibrium payoff of both players. A complete characterization of this phenomenon is given for mixed equilibria.

Abstract: Conditions are given which are sufficient to prove the existence of a unique unconstrained minimizer in a convex compact set. An exact formula is given for the amount by which a value of the function exceeds its global minimum.

Abstract: The ideas described in this paper arise from a particular investigation of the general question: when information relevant to a sequence of decisions is collected by statistical sampling, is it worth controlling the local rate of sampling to provide more accurate information at certain times? Here, the value of any sampling and decision procedure is measured by the long-term average of all information and decision costs. The model is concerned with the design of a control chart and it leads to a Markovian decision problem with three possible actions at each point of the state space. The determination of an optimal policy depends on the solution of a complicated free boundary problem. Although there is a well-established relation between the basic partial differential equation of this problem and Brownian motion, the investigation raises many questions, both analytical and probabilistic, which remain to be answered. However, some limited results are obtained by examining special formal solutions. In spite of serious gaps in the general theory, it is possible to establish useful bounds on the minimum average cost which can be attained.

Abstract: The minimum order (number of nodes) is determined for e-connected graphs of valence ≥c and diameter ≥d. The question arises naturally from reliability and defense considerations that may enter into the design of an armed communication network.

Abstract: The question of whether or not the diagonal property is a consequence of the axioms defining the axiomatic value is answered in the negative by means of a counter-example: a space and an axiomatic value on it are introduced not possessing the diagonal property.

Abstract: A simple procedure for the computation of a basis of the general solution of a finite system of linear inequalities is presented. It generalizes the well-known Burger's algorithm for homogeneous systems.

Abstract: The concept of infinite second guessing is known to play a role in the consideration of multi-person control problems. By considering the problem in its appropriate Hilbert space context we formalize this concept and show that it does in fact play a vital part in the structure of those solutions that are available. We give an explicit series solution to the constant coefficient quadratic cost team problem. This solution is valid for any distribution of noise, which is square integrable. The well-known linearity of optimal control in the Gaussian case is immediately apparent. An important special case is given further consideration and other applications are indicated. Finally a theorem characterising the limit behaviour of the series in terms of the information common to all players is presented.

Abstract: Ray Fulkerson, one of the founders of mathematical programming, was a symbol of high scholarly standards to many of us. We shall miss him. Compared with the sorrow that we feel, old cliches seem cold and impersonal. A permanent tribute to his memory lies in his fundamental contributions to mathematics. We are going to review some of those which are particularly relevant to operations research and to outline their impact on the development of this field.l Our survey will be brief; for a more detailed account of Ray Fulkerson's work, the reader is directed to a forthcoming article by Alan J. Hoffman in Mathematical Programming Studies. Furthermore, in the present issue of this journal, Louis J. Billera and William F. Lucas describe Ray Fulkerson's life and career.

Abstract: A new method designed to globally minimize concave functions over linear polyhedra is described. Properties of the method are discussed, an example problem is solved, and computational considerations are discussed.

Abstract: We associate with every real-valued function a number which measures its lack of convexity. This number is used to estimate the duality gap in optimization problems where the criterion and/or the constraints are nonconvex. It is shown that when the number of variables is very great with respect to the number of constraints, this duality gap is small in relative value. Approximating in this way problems where the criterion and constraints are given as integrals, we show that the duality gap vanishes.

Abstract: We show that there exists a Laurent series in a fractional power of n which approximates Vn up to log n, where Vn is the value of an n-stage two person zero sum stochastic game. We prove this result by showing that the Laurent series is an approximate solution of the dynamical programming equation for Vn, Vn+1 = fVn. It seems that our methods could be used to find approximate solutions to other difference equations. Our proof makes repeated use of Tarski's principle for real closed fields.

Abstract: To improve the computational efficiency of a branch-and-bound algorithm at the sacrifice of obtaining an optimal solution, the lower bound test is sometimes strengthened beyond its limit, i.e., a partial problem Pi is terminated if gPi ≥ z-ez instead of gPi ≥ z, where gPi is a lower bound of Pi, z is the current incumbent value and ez ≥0 specifies the allowance within which the value of an obtained solution can deviate from the optimal. This paper first shows that the efficiency as measured by the number of decomposed partial problems does not generally improve by introducing e or by increasing e, and then isolates subclasses of branch-and-bound algorithms for which a monotone increase in efficiency with respect to e is guaranteed.