The popular matching problem introduced by Abraham, Irving, Kavitha, and Mehlhorn is one of bipartite matching problems with one-sided preference lists. In this paper, we first propose a matroid generalization of the weighted variant of popular matchings introduced by Mestre. Then we give a characterization of weighted popular matchings in bipartite graphs with matroid constraints and one-sided preference lists containing no ties. This characterization is based on the characterization of weighted popular matchings proved by Mestre. Lastly we prove that we can decide whether a given matching is a weighted popular matching under matroid constraints in polynomial time by using our characterization.

A game with an incumbent and an entrant is considered. Each of them decides a timing of an investment into a new project. The profit flows of both firms involve two uncertain factors: (i) demand for the market observed only by the incumbent, and (2) fluctuation of the profit flow described by a geometric Brownian motion observed by both firms. The optimal timing of the investment of the incumbent under high demand is earlier than that under low demand, if the entrant's timing is fixed. However, the earlier investment of the incumbent may reveal information of high demand to the entrant, so that the early investment of the incumbent would accelerate the timing of the investment of the entrant. This earlier investment of the entrant reduces monopolistic profit of the incumbent before the investment of the entrant. Therefore, the incumbent under high demand may delay the timing of the investment strategically in order to hide the information. In the present paper, equilibria of this signaling game are characterized and sufficient conditions for the strategic behavior of the incumbent are obtained.

The values of both firms in equilibria are compared with the case of complete information and the welfare loss is investigated. I observe that the marginal profit in the duopoly for the high-demand incumbent is small the incumbent invests strategically, whereas the incumbent invests truthfully if marginal profit in the duopoly is sufficiently large. Strategic behavior of the incumbent is also observed when the investment cost or volatility of the incumbent is large or the investment cost of the entrant is small.

In some localities, a large-scale chain retailer competes against a small-scale local independent retailer that specializes in, for instance, vegetables, fruits, and flowers produced locally for local consumption. The former usually attracts consumers by emphasizing its width and depth of products variety, whereas the latter seeks to overcome its limited products assortment by offering lower prices for them than the chain store. This is possible for the local store partly because of lower labor costs and for various other reasons.

This study employs the Hotelling unit interval to examine price competition in a duopoly featuring one large-scale chain retailer and one local retailer. To express differences in their product assortments, we assume that the large-scale retailer denoted by A sells two types of product, G₁ and G₂, whereas the local retailer denoted by B sells only G₁. Moreover, we assume that all the consumers purchase G₁ at A or B after comparing prices and buy G₂ at A on an as-needed basis. We examine both Nash and Stackelberg equilibrium to indicate that the local retailer can survive competition with the large-scale chain retailer even if all the consumers purchase both G₁ and G₂. We also reveal that a monopolistic market structure, not duopoly, can optimize the social welfare if consumers always purchase both G₁ and G₂.

Quasi-Newton methods are widely used for solving unconstrained optimization problems. However, it is difficult to apply quasi-Newton methods directly to large-scale unconstrained optimization problems, because they need the storage of memories for matrices. In order to overcome this difficulty, memoryless quasi-Newton methods were proposed. Shanno (1978) derived the memoryless BFGS method. Recently, several researchers studied the memoryless quasi-Newton method based on the symmetric rank-one formula. However existing memoryless symmetric rank-one methods do not necessarily satisfy the sufficient descent condition. In this paper, we focus on the symmetric rank-one formula based on the spectral scaling secant condition and derive a memoryless quasi-Newton method based on the formula. Moreover we show that the method always satisfies the sufficient descent condition and converges globally for general objective functions. Finally, preliminary numerical results are shown.

In this paper, we study classes of discrete convex functions: submodular functions on modular semilattices and L-convex functions on oriented modular graphs. They were introduced by the author in complexity classification of minimum 0-extension problems. We clarify the relationship to other discrete convex functions, such as k-submodular functions, skew-bisubmodular functions, L^♮-convex functions, tree submodular functions, and UJ-convex functions. We show that they actually can be viewed as submodular/L-convex functions in our sense. We also prove a sharp iteration bound of the steepest descent algorithm for minimizing our L-convex functions. The underlying structures, modular semilattices and oriented modular graphs, have rich connections to projective and polar spaces, Euclidean building, and metric spaces of global nonpositive curvature (CAT(0) spaces). By utilizing these connections, we formulate an analogue of the Lovász extension, introduce well-behaved subclasses of submodular/L-convex functions, and show that these classes can be characterized by the convexity of the Lovász extension. We demonstrate applications of our theory to combinatorial optimization problems that include multicommodity flow, multiway cut, and related labeling problems: these problems had been outside the scope of discrete convex analysis so far.

A continuous-time barrier option model is developed for valuing executive stock options (ESOs), in which early exercise takes place whenever the underlying stock price reaches a certain upper barrier after vesting. We analyze the ESO value and the ESO exercise time to obtain their solutions in simple forms, which are consistent with principal features of early exercise, delayed vesting and random exit. For the perpetual case, these solutions are given in explicit forms and shown to be exact in the Black-Scholes-Merton formulation. Using an endogenous approximation for the barrier level, we numerically compare our approximation for the ESO value with a benchmark result generated by a binomial-tree model and the quadratic approximation previously established. From numerical comparisons for some particular cases, we see that our approximations always underestimate the benchmark results and the absolute values of the relative percentage errors are less than 1% for all cases, whereas the quadratic approximations overestimate the benchmarks and the relative percentage errors are less than about 2%.

The two-dimensional strip packing problem arises in wide variety of industrial applications. In this paper, we focus on the bitmap shape packing problem in which a set of arbitrarily shaped objects represented in bitmap format should be packed into a larger rectangular container without overlap. The complex geometry of bitmap shapes and the large amount of data to be processed make it difficult to check for overlaps. In this paper, we propose an efficient method for checking for overlaps and design efficient implementations of two construction algorithms, which are based on the bottom-left strategy. In this strategy, starting from an empty layout, items are packed into the container one by one. Each item is placed in the lowest position where there is no overlap relative to the current layout. We consider two algorithms, the bottom-left and the best-fit algorithm, which adopt this strategy. The computational results for a series of instances that are generated from well-known benchmark instances show that the proposed algorithms obtain good solutions in remarkably short time and are especially effective for large-scale instances.

This paper discusses the problem of determining whether a given plane graph is a Delaunay graph, i.e., whether it is topologically equivalent to a Delaunay triangulation. There exist theorems which characterize Delaunay graphs and yield polynomial time algorithms for the problem only by solving some linear inequality systems. A polynomial time algorithm proposed by Hodgson, Rivin and Smith solves a linear inequality system given by Rivin, which is based on sophisticated arguments about hyperbolic geometry. Independently, Hiroshima, Miyamoto and Sugihara gave another linear inequality system and a polynomial time algorithm. Although their discussion is based on primitive arguments on Euclidean geometry, their proofs are long and intricate, unfortunately. In this paper, we give a simple proof of the theorem shown by Hiroshima et al. by employing the fixed point theorem.