Abstract : This minisymposium focuses on recent advances in mathematical optimization with versatile subjects such as optimal control, variational analysis, dynamical systems, nonlinear functional analysis, network systems, fixed point theory, and so forth. Application topics discussed here mainly lie in mathematical economics and engineering, in particular, optimal economic growth, general equilibrium analysis, utility theory, and Marxian economics as well as generative adversarial networks, but possible applications are not necessarily restricted to such problems. The minisymposium serves for a communication with applied mathematics in different areas.
Organizer(s) : Nobusumi Sagara, Alexander Zaslavski, Yuhki Hosoya
Abstract : We study a method of calculating the utility function from a candidate of a demand function that is not differentiable but is locally Lipschitz. Using this method, we obtain two new necessary and sufficient conditions for a candidate of a demand function to be a demand function. The first is conditions for the Slutsky matrix, and the second is the existence of a concave solution to a partial differential equation. Moreover, we show that the upper semi-continuous weak order that corresponds to the demand function is unique, and this weak order is represented by our calculated utility function. We provide applications of these results to econometric theory. First, we show that, under several requirements, if a sequence of demand functions converges to some function with respect to the metric of compact convergence, then the limit is also a demand function. Second, the space of demand functions that have uniform Lipschitz constants on any compact set is complete under the above metric. Third, the mapping from a demand function to the calculated utility function becomes continuous. This implies that a consistent estimation method for the demand function immediately defines a consistent estimation method for the utility function using our calculation method.
[01446] Theoretical analysis of two time-scale update rule for training GANs
Format : Talk at Waseda University
Author(s) :
Naoki Sato
Hideaki Iiduka (Meiji University)
Abstract : A theoretical analysis of a two time-scale update rule $(\text{TTUR})$ for training generative adversarial networks $(\text{GANs})$ has been given using decaying learning rates. In this talk, we give a theoretical analysis of TTUR using constant learning rates and show that, for TTUR using constant learning rates, the number of steps needed to train GAN decreases as the batch size increases. We also provide numerical results to support our theoretical analyses.
[01450] Production Prices and Walrasian Intertemporal Competitive Equilibrium Prices in a Generalized Neoclassical Production Economy
Format : Talk at Waseda University
Author(s) :
Naoki Yoshihara (University of Massachusetts Amherst)
Abstract : We examine a general correspondence between production prices in classical and Marxian economics and the Walrasian competitive equilibrium prices in the standard general equilibrium theory by considering a standard intertemporal economy with a discounted lifetime utility function and a set of general neo-classical production technologies. This work resembles Duménil and Levy (1985) and Dana et. al (1989), but unlike these, a path of intertemporal Walrasian equilibrium prices is characterized by the Euler equation, derived from the economic model in this paper. In addition, equilibrium factor prices are endogenously determined and can vary across periods. Therefore, our intertemporal Walrasian equilibrium is much closer to the standard neoclassical type, compared to the intertemporal competitive equilibrium defined by Dana et. al (1989). However, we will show that any intertemporal Walrasian equilibrium prices converge to a system of production prices in the long term.
[01982] Optimal Growth in the Two-Sector Robinson-Shinkai-Leontief Model
Format : Talk at Waseda University
Author(s) :
Minako Fujio (Yokohama National University)
Ali M. Khan (The Johns Hopkins University)
Liuchun Deng (Yale-NUS College)
Abstract : In this talk we synthesize the findings on the two-sector Robinson-Shinkai-Leontief model of optimal growth with and without discounting and demonstrate a variety of optimal dynamics. We provide a taxonomy of the optimal policy and the dynamics it yields for the entire parameter space of the model. At the same time, we shall focus on the two approaches we rely on to delineate those results, the value-loss minimization and the dynamic programming.
[02153] On the approximate purification of mixed strategies in games with infinite action sets
Format : Talk at Waseda University
Author(s) :
Chaowen Yu (Rissho University)
Yuhki Hosoya (Chuo University)
Abstract : We consider a game in which the action set of each player is uncountable, and show that, from weak assumptions on the common prior, any mixed strategy has an approximately equivalent pure strategy. The assumption of this result can be further weakened if we consider the purification of a Nash equilibrium. Combined with the existence theorem for a Nash equilibrium, we derive an existence theorem for a pure strategy approximated Nash equilibrium under sufficiently weak assumptions. All of the pure strategies we derive in this paper can take a finite number of possible actions.
[03646] Numerical aspects of finding nonlinear production – consumption equilibrium
Format : Talk at Waseda University
Author(s) :
Igor Griva (George Mason University)
Roman A Polyak (George Mason University)
Abstract : We present and analyze numerical results obtained by using extra pseudo-gradient (EPG) method on a set of randomly generated nonlinear production – consumption equilibrium (NPCE) problems. The obtained results show that the number of EPG steps required for finding NPCE grows linearly with the number of products of a given economy. The number of arithmetic operations or time required for finding NPCE grows as a cube of the number of products.
[04476] Envelope theorems in Optimization
Format : Talk at Waseda University
Author(s) :
Joël Blot (Université PParis 1 Panthéon-SorbonneUniversi)
Abstract : We present recent results on the Envelope theorems in three chapters
of Optimization. First in Static Optimization, secondly in Calculus of Variations
and thirdly in Optimal Control Theory. To do that, we provide new results on
the continuous dependence of coordiates in a moving frame, on the Hadamard
di erentiability of functionals under the integral form, and on the proof of the
Pontryagin principle.
Date: November 13, 2022.
