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We consider the convergence rate of the alternating projection method for the nontransversal intersection of a semialgebraic set and a linear subspace. For such an intersection, the convergence rate is known as sublinear in the worst case. We study the exact convergence rate for a given semialgebraic set and an initial point, and investigate when the convergence rate is linear or sublinear. As a consequence, we show that the exact rates are expressed by multiplicities of the defining polynomials of the semialgebraic set, or related power series in the case that the linear subspace is a line, and we also decide the convergence rate for given data by using elimination theory. Our methods are also applied to give upper bounds for the case that the linear subspace has the dimension more than one. The upper bounds are shown to be tight by obtaining exact convergence rates for a specific semialgebraic set, which depend on the initial points., 22 pages, 1 figure

Flat Panel Displays (FPDs) are manufactured using many pieces of different processing equipment arranged sequentially in a line. Although the constant inter-arrival time (i.e., the tact time) of glass substrates in the line should be kept as short as possible, the collision probability between glass substrates increases as tact time decreases. Since the glass substrate is expensive and fragile, collisions should be avoided. In this paper, we derive a closed form formula of the approximate collision probability for a model, in which the processing time on each piece of equipment is assumed to follow Erlang distribution. We also compare some numerical results of the closed form and computer simulation results of the collision probability., Article, IEICE TRANSACTIONS ON INFORMATION AND SYSTEMS. E96D(3): 400-407 (2013)

An exact estimate on the modulus of metric regularity for linear systems is given. By applying the estimate, we obtain explicit forms of the modulus for linear conical systems and differentiable nonlinear systems on the space of continuous functions.

Consider a trader who exchanges one dollar into yen and assume that the exchange rate fluctuates within the interval [m, M]. The game ends without advance notice, then the trader is forced to exchange all the remaining dollars at the minimum rate m. El-Yaniv et al. presented the optimal worst-case threat-based strategy for this game (El-Yaniv et al. 2001). In this paper, under the assumption that the distribution of the maximum exchange rate is known, we provide average-case analyses using all the reasonable optimization measures and derive different optimal strategies for each of them. Remarkable differences in behavior are as follows: Unlike other strategies, the average-case threat-based strategy that minimizes E[OPT/ALG] exchanges little by little. The maximization of E[ALG/OPT] and the minimization of E[OPT]/E[ALG] lead to similar strategies in that both exchange all at once. However, their timing is different. We also prove minimax theorems with respect to each objective function., Article, JOURNAL OF COMBINATORIAL OPTIMIZATION. 21(1): 83-107 (2011)

We consider sensitivity of a semidefinite program under perturbations in the case that the primal problem is strictly feasible and the dual problem is weakly feasible. When the coefficient matrices are perturbed, the optimal values can change discontinuously as explained in concrete examples. We show that the optimal value of such a semidefinite program changes continuously under conditions involving the behavior of the minimal faces of the perturbed dual problems. In addition, we determine what kinds of perturbations keep the minimal faces invariant, by using the reducing certificates, which are produced in facial reduction. Our results allow us to classify the behavior of the minimal face of a semidefinite program obtained from a control problem., Comment: 19 pages

Polynomial optimization appears various areas of mathematics. Although it is a fully nonlinear nonconvex optimization problems, there are numerical algorithms to approximate the global optimal value by generating sequences of semidefinite programming relaxations. In this paper, we study how real radicals of ideals have roles in duality theory and finite convergence property. Especially, duality theory is considered in the case that the truncated quadratic module is not necessarily closed. We will also try to explain the results by giving concrete examples.

The main result of the paper contains an exact formula for the rate of regularity of a set-valued mapping with a convex graph. As a consequence we find an exact expression for the rate of regularity of set-valued mappings associated with so-called constraint systems. It turns out that the rate is equal to the upper bound of Robinson-type estimates over all norms in the graph space of the homogenized mapping majorizing the norm of the underlying space. We further introduce a concept of a perfectly regular mapping and find some criteria for perfect regularity.

Nojima waterway is a typical closed water in Tokyo Bay. To improve its water quality and remove the sediment odor, the waterway was opened to outer sea and the sediment was dredged away. As a result, salinity in the waterway increased due to outer sea water flowing in, and successions of the biotic community occurred. Goby fishing is getting vigorously around this area.

We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algorithm to calculate the generators, which is based on some techniques of the cylindrical algebraic decomposition. By applying these, polynomial optimization problems with polynomial equality constraints can be modified equivalently so that the associated semidefinite programming relaxation problems have no duality gap. Elementary proofs for some criteria on reality of ideals are also given., Comment: 15 pages, 1 figure

Flat panel displays (FPDs) are manufactured through many different processing equipments arranged sequentially in a line. Although the constant inter-arrival time (i.e., the tact time) of glass substrates in the line should be kept as short as possible, the collision probability between glass substrates increases as the time becomes shorter. Since the glass substrate is expensive and fragile, the collision should be avoided. In this paper, we derive a closed form formula of the collision probability for a model, in which the processing time on each equipment is assumed to follow the Erlang distribution. We also show some numerical results and computer simulation results of the collision probability.

Nondestructive burnup measurement techniques have been extensively developed and demonstrated for spent fuel assemblies at various facilities. Japanese commercial spent fuel reprocessing plant, which will be built by Japan Nuclear Fuel Service Co., will be equipped with nondestructive burnup monitors to verify shipper's data at a spent fuel receiving-storage pool. Motivated by these facts, this paper reviews the present status of nondestructive measurement techniques of burnups and plutonium contents in spent light-water-reactor fuel assemblies, focusing on γ-spectrometry, passive- and active-neutron measurement techniques from the view point of inline burnup monitoring in a reprocessing plant.