The multi-leader-follower game serves as an important model in game theory with many applications in economics, operations research and other fields. In this survey paper, we first recall some background materials in game theory and optimization. In particular, we present several extensions of Nash equilibrium problems including the multi-leader-follower game. We then give some applications as well as solution methods of multi-leader-follower games.

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Nonlinear semidefinite programming (SDP) problems have received a lot of attentions because of large variety of applications. In this paper, we survey numerical methods for solving nonlinear SDP problems. Three kinds of typical numerical methods are described; augmented Lagrangian methods, sequential SDP methods and primal-dual interior point methods. We describe their typical algorithmic forms and discuss their global and local convergence properties which include rate of convergence.

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Efficient allocation of indivisible goods is an important problem in mathematical economics and operations research, where the concept of Walrasian equilibrium plays a fundamental role. As a sufficient condition for the existence of a Walrasian equilibrium, the concept of gross substitutes condition for valuation functions is introduced by Kelso and Crawford (1982). Since then, several variants of gross substitutes condition as well as a discrete concavity concept, called M^♮-concavity, have been introduced to show the existence of an equilibrium in various models. In this paper, we survey the relationship among Kelso and Crawford's gross substitutes condition and its variants, and discuss the connection with M^♮-concavity. We also review various characterizations and properties of these concepts.

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Diffusion processes have been widely used for approximations in the queueing theory. There are different types of diffusion approximations. Among them, we are interested in those obtained through limits of a sequence of models which describe queueing networks. Such a limit is typically obtained by the weak convergence of either stochastic processes or stationary distributions. We already have nice reviews and text books for them. However, this area is still actively studied, and it seems getting hard to have a comprehensive overview because mathematical results are highly technical. We try to fill this gap presenting technical background. Although those diffusion approximations have been well developed, there remains a big problem, which is difficulty to get useful information from the limiting diffusion processes. Their state spaces are multidimensional, whose dimension corresponds to the number of nodes for a single-class case and the number of customer types for a multi-class case. We now have a better view for the two dimensional case, but still know very little about the higher dimensional case. This intractability is somehow against a spirit of diffusion approximation. This motivates us to reconsider diffusion approximation from scratch. For this, we highlight the stationary distributions, and make clear a mechanism to produce diffusion approximations.

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