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The generalized Nash equilibrium problem (GNEP) is a kind of game to find strategies for a group of players such that each player's objective function is optimized. Solutions for GNEPs are called generalized Nash equilibria (GNEs). In this paper, we propose a numerical method for finding GNEs of GNEPs of polynomials based on the polyhedral homotopy continuation and the Moment-SOS hierarchy of semidefinite relaxations. We show that our method can find all GNEs if they exist, or detect the nonexistence of GNEs, under some genericity assumptions. Some numerical experiments are made to demonstrate the efficiency of our method., Comment: 25 pages, Version to appear in Journal of Scientific Computing

This paper concerns the generalized Nash equilibrium problem of polynomials (GNEPP). We apply the Gauss–Seidel method and Moment-SOS relaxations to solve GNEPPs. The convergence of the Gauss–Seidel method is known for some special GNEPPs, such as generalized potential games (GPGs). We give a sufficient condition for GPGs and propose a numerical certificate, based on Putinar’s Positivstellensatz. Numerical examples for both convex and nonconvex GNEPPs are given for demonstrating the efficiency of the proposed method.

This paper studies convex Generalized Nash Equilibrium Problems (GNEPs) that are given by polynomials. We use rational and parametric expressions for Lagrange multipliers to formulate efficient polynomial optimization for computing Generalized Nash Equilibria (GNEs). The Moment-SOS hierarchy of semidefinite relaxations are used to solve the polynomial optimization. Under some general assumptions, we prove the method can find a GNE if there exists one, or detect nonexistence of GNEs. Numerical experiments are presented to show the efficiency of the method., 29 pages

Background Heat collection rate and heat loss coefficient are crucial indicators for the evaluation of in service water-in-glass evacuated tube solar water heaters. However, the direct determination requires complex detection devices and a series of standard experiments, wasting too much time and manpower. Findings To address this problem, we previously used artificial neural networks and support vector machine to develop precise knowledge-based models for predicting the heat collection rates and heat loss coefficients of water-in-glass evacuated tube solar water heaters, setting the properties measured by “portable test instruments” as the independent variables. A robust software for determination was also developed. However, in previous results, the prediction accuracy of heat loss coefficients can still be improved compared to those of heat collection rates. Also, in practical applications, even a small reduction in root mean square errors (RMSEs) can sometimes significantly improve the evaluation and business processes. Conclusions As a further study, in this short report, we show that using a novel and fast machine learning algorithm—extreme learning machine can generate better predicted results for heat loss coefficient, which reduces the average RMSEs to 0.67 in testing.

To determine the effect of different denture cleaning methods on surface roughness in resin denture base.We established 20 wafer samples of fuser base resin (14 mm×14 mm×3 mm), and then randomly divided them into 4 groups: group A was the control group, which were placed in water, while group B, C and D were the experimental groups, whose cleaning methods were toothbrush and water, toothbrush and toothpaste, denture cleaning tablets, respectively. Each procedure in group B and C lasted for 3 seconds, but group D lasted 5 min and repeated for 1080 times. We cleaned twice a day to simulate the method of cleaning dentures. Surface roughness was checked after different procedures by laser scanning confocal microscopy.Significant difference on surface roughness was found between group B, C and A (P0.05), while no significant difference in the surface roughness between group A and D (P0.05).Significant surface roughness alterations have been observed in toothbrush and toothpaste group, but no change has been found in denture cleaning tablets group, which does not produce scratches on the surface of resin denture base.