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Two approaches are applied to the set-valued optimization problem. The following problems have been examined by Corley, Luc and their colleagues: Take the union of all objective values and then search for (weakly, properly, etc.) minimal points in this union with respect to the vector ordering. This approach is called the vector approach to set optimization. The concept shifted when the set relations were popularized by Kuroiwa–Tanaka–Ha at the end of the twentieth century. They introduced six types of set relations on the power set of topological vector space using a convex ordering cone C with nonempty interior. Therefore, this approach is called the set relation approach to set optimization. For a given vector optimization problem, several approaches are applied to construct a dual problem. A difficulty lies in the fact that the minimal point in vector optimization problem is not necessarily a singleton, though it becomes a subset of the image space in general. In this paper, we first present new definitions of set-valued conjugate map based on comparison of sets (the set relation approach) followed by introducing some types of weak duality theorems. We also show convexity and continuity properties of conjugate relations. Lastly, we present some types of strong duality theorems using nonlinear scalarizing technique for set that is generalizations of Gerstewitz’s scalarizing function for the vector-valued case.

In this paper, we first introduce some types of set relations on the power set of n-dimensional Euclidean spaces which are proposed by Kuroiwa–Tanaka–Ha and Jahn–Ha. We also mention new types of cancellation laws of set relations. Second, we introduce a complete lattice-valued problem on the power set of n-dimensional Euclidean spaces proposed by Hamel et al. Applying nonlinear scalarizing technique in complete lattice, we present a new type of minimal element theorem and generalized Ekeland’s variational principles in complete lattice optimization problem. We also present an existence theorem of minimal solutions related to the famous Takahashi’s minimization theorem in complete lattice optimization problem.

In this paper, we prove Ekeland's type of variational principle for a vector equilibrium problem, and present a Caristi-Kirk type fixed point theorem and an existence result for vector equilibrium solution.

Abstract The Deci-hertz Interferometer Gravitational Wave Observatory (DECIGO) is a future Japanese space mission with a frequency band of 0.1 Hz to 10 Hz. DECIGO aims at the detection of primordial gravitational waves, which could have been produced during the inflationary period right after the birth of the Universe. There are many other scientific objectives of DECIGO, including the direct measurement of the acceleration of the expansion of the Universe, and reliable and accurate predictions of the timing and locations of neutron star/black hole binary coalescences. DECIGO consists of four clusters of observatories placed in heliocentric orbit. Each cluster consists of three spacecraft, which form three Fabry–Pérot Michelson interferometers with an arm length of 1000 km. Three DECIGO clusters will be placed far from each other, and the fourth will be placed in the same position as one of the other three to obtain correlation signals for the detection of primordial gravitational waves. We plan to launch B-DECIGO, which is a scientific pathfinder for DECIGO, before DECIGO in the 2030s to demonstrate the technologies required for DECIGO, as well as to obtain fruitful scientific results to further expand multi-messenger astronomy.

On January 15, 2022, at 04:14:45 (UTC), the Hunga Tonga-Funga Ha’apai, a submarine volcano in the Tongan archipelago in the southern Pacific Ocean, erupted and generated global seismic, shock, and electromagnetic waves, which also reached Japan, situated more than 8000 km away. KAGRA is a gravitational wave telescope located in an underground facility in Kamioka, Japan. It has a wide variety of auxiliary sensors to monitor environmental disturbances that obstruct the observation of gravitational waves. The effects of the volcanic eruption were observed by these environmental sensors both inside and outside of the underground facility. In particular, the shock waves made it possible to evaluate the transfer functions from the air pressure wave in the atmosphere to the underground environmental disturbances (air pressure and seismic motion).

Deci-hertz Interferometer Gravitational Wave Observatory (DECIGO) is the future Japanese space mission with a frequency band of 0.1 Hz to 10 Hz. DECIGO aims at the detection of primordial gravitational waves, which could be produced during the inflationary period right after the birth of the universe. There are many other scientific objectives of DECIGO, including the direct measurement of the acceleration of the expansion of the universe, and reliable and accurate predictions of the timing and locations of neutron star/black hole binary coalescences. DECIGO consists of four clusters of observatories placed in the heliocentric orbit. Each cluster consists of three spacecraft, which form three Fabry-Perot Michelson interferometers with an arm length of 1,000 km. Three clusters of DECIGO will be placed far from each other, and the fourth cluster will be placed in the same position as one of the three clusters to obtain the correlation signals for the detection of the primordial gravitational waves. We plan to launch B-DECIGO, which is a scientific pathfinder of DECIGO, before DECIGO in the 2030s to demonstrate the technologies required for DECIGO, as well as to obtain fruitful scientific results to further expand the multi-messenger astronomy.