Journal of Optimization Theory and Applications

Some existence results for generalized variational inequalities and generalized complementarity problems involving quasimonotone and pseudomonotone set-valued mappings in reflexive Banach spaces are proved. In particular, some known results for nonlinear variational inequalities and complementarity problems in finite-dimensional and infinite-dimensional Hilbert spaces are generalized to quasimonotone and pseudomonotone set-valued mappings and reflexive Banach spaces. Application to a class of generalized nonlinear complementarity problems studied as mathematical models for mechanical problems is given.

In this paper, we consider a class of minimax decision problems which arise in the transmission of a Gaussian vector message over a vector channel with partially unknown statistical description. The statistically unknown part of the channel is modelled as one which is controlled by a jammer who can corrupt the transmitted message by sending noise which may be correlated with the original message under a given power constraint. Under two types of structural assumptions on the transmitter (encoder), the problem is posed as one in which the optimum decision rules at the encoder and the decoder jointly minimize a square distortion measure at the output, under worst possible choices for the jamming noise. It is shown that a saddle-point solution exists when the linear encoder structure is of the mixed type, whereas it does not exist when it is restricted to be deterministic. In the former case, explicit expressions for the saddle-point solution have been presented, whereas in the latter case minimax and maximin solutions have been obtained. An important feature of the saddle-point solution is that it depends on two integer-valued parameters, one of which determines (in a new rotated coordinate system) the number of components of the message vector to be transmitted through the channel, and the second one determines the number of channels that the jammer actually jams. Some worked out numerical examples complement the theoretical results.

The multidimensional assignment problem (MAP) is a NP-hard combinatorial optimization problem, occurring in many applications, such as data association. In this paper, we prove two conjectures made in Ref. 1 and based on data from computational experiments on MAPs. We show that the mean optimal objective function cost of random instances of the MAP goes to zero as the problem size increases, when assignment costs are independent exponentially or uniformly distributed random variables. We prove also that the mean optimal solution goes to negative infinity when assignment costs are independent normally distributed random variables.

It is generally assumed that any set of players can form a feasible coalition for classical cooperative games. But, in fact, some players may withdraw from the current game and form a union, if this makes them better paid than proposed. Based on the principle of coalition split, this paper presents an endogenous procedure of coalition formation by levels and bargaining for payoffs simultaneously, where the unions formed in the previous step continue to negotiate with others in the next step as “individuals,” looking for maximum share of surplus by organizing themselves as a partition. The structural stability of the induced payoff configuration is discussed, using two stability criteria of core notion for cooperative games and strong equilibrium notion for noncooperative games.

The paper contains a version of a minimax theorem with weakened convexity, extending a minimax theorem of Fan. The main result is obtained with the use of a generalized Gordan theorem, which is proved using a separation theorem. An example is also discussed.

The paper addresses the problem of recovering a pseudoconvex function from the normal cones to its level sets that we call the convex level sets integration problem. An important application is the revealed preference problem. Our main result can be described as integrating a maximally cyclically pseudoconvex multivalued map that sends vectors or “bundles” of a Euclidean space to convex sets in that space. That is, we are seeking a pseudoconvex (real) function such that the normal cone at each boundary point of each of its lower level sets contains the set value of the multivalued map at the same point. This raises the question of uniqueness of that function up to rescaling. Even after normalizing the function long an orienting direction, we give a counterexample to its uniqueness. We are, however, able to show uniqueness under a condition motivated by the classical theory of ordinary differential equations.

This paper presents a method for discrete-time control and estimation of flexible structures in the presence of actuator and sensor noise. The approach consists of complete decoupling of the modal equations and estimator dynamics based on the independent modal-space control technique and modal spatial filtering of the system output. The solution for the Kalman filter gains reduces to that of independent second-order modal estimators, thus permitting real-time digital control of distributed-parameter systems in a noisy environment. The method can be used to control and estimate any number of modes without computational restraints and is theoretically free of observation spillover. Two examples, the first using nonlinear, quantized control and the second using linear, state feedback control are presented.