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We study continuity, and lack thereof, of thermodynamical properties for one-dimensional dynamical systems. Under quite general hypotheses, the free energy is shown to be almost upper-semicontinuous: some normalised component of a limit measure will have free energy at least that of the limit of the free energies. From this, we deduce results concerning existence and continuity of equilibrium states (including statistical stability). Metric entropy, not semicontinuous as a general multimodal map varies, is shown to be upper semicontinuous under an appropriate hypothesis on critical orbits. Equilibrium states vary continuously, under mild hypotheses, as one varies the parameter and the map. We give a general method for constructing induced maps which automatically give strong exponential tail estimates. This also allows us to recover, and further generalise, recent results concerning statistical properties (decay of correlations, etc.). Counterexamples to statistical stability are given which also show sharpness of the main results.

Lately, there has been a lot of interest in game-theoretic approaches to the trajectory planning of autonomous vehicles (AVs). But most methods solve the game independently for each AV while lacking coordination mechanisms, and hence result in redundant computation and fail to converge to the same equilibrium, which presents challenges in computational efficiency and safety. Moreover, most studies rely on the strong assumption of knowing the intentions of all other AVs. This paper designs a novel autonomous vehicle trajectory planning approach to resolve the computational efficiency and safety problems in uncoordinated trajectory planning by exploiting vehicle-to-everything (V2X) technology. Firstly, the trajectory planning for connected and autonomous vehicles (CAVs) is formulated as a game with coupled safety constraints. We then define the interaction fairness of the planned trajectories and prove that interaction-fair trajectories correspond to the variational equilibrium (VE) of this game. Subsequently, we propose a semi-decentralized planner for the vehicles to seek VE-based fair trajectories, in which each CAV optimizes its individual trajectory based on neighboring CAVs’ information shared through V2X, and the roadside unit takes the role of updating multipliers for collision avoidance constraints. The approach can significantly improve computational efficiency through parallel computing among CAVs, and enhance the safety of planned trajectories by ensuring equilibrium concordance among CAVs. Finally, we conduct Monte Carlo experiments in multiple situations at an intersection, where the empirical results show the advantages of SVEP, including the fast computation speed, a small communication payload, high scalability, equilibrium concordance, and safety, making it a promising solution for trajectory planning in connected traffic scenarios. To the best of our knowledge, this is the first study to achieve semi-distributed solving of a game with coupled constraints in a CAV trajectory planning problem.

In this paper, we first introduced systems of generalized vector quasi-variational equilibrium problems as well as systems of vector quasi-variational equilibrium problems as their special cases in abstract convex spaces. Next, we established some existence theorems of solutions for systems of generalized vector quasi-variational equilibrium problems and systems of vector quasi-variational equilibrium problems in non-compact abstract convex spaces. Furthermore, we applied the obtained existence theorem of solutions for systems of vector quasi-variational equilibrium problems to solve the existence problem of Nash equilibria for noncooperative games. Then, as applications of the existence result of Nash equilibria for noncooperative games, we studied the existence of weighted Nash equilibria and Pareto Nash equilibria for multi-objective games. The results derived in this paper extended and unified the primary findings presented by some authors in the literature.

In this monograph we consider the general setting of conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We deal with two classes of such systems: attracting and parabolic. The latter being treated by means of the former. We prove fairly complete asymptotic counting results for multipliers and diameters associated with preimages or periodic orbits ordered by a natural geometric weighting. We also prove the corresponding Central Limit Theorems describing the further features of the distribution of their weights. These results have direct applications to a wide variety of examples, including the case of Apollonian Circle Packings, Apollonian Triangle, expanding and parabolic rational functions, Farey maps, continued fractions, Mannenville-Pomeau maps, Schottky groups, Fuchsian groups, and many more. This gives a unified approach which both recovers known results and proves new results. Our new approach is founded on spectral properties of complexified Ruelle–Perron–Frobenius operators and Tauberian theorems as used in classical problems of prime number theory.

The electricity distribution network is experiencing a profound transformation with the concept of the smart grid, providing possibilities for selfish consumers to interact with the distribution system operator (DSO) and to maximize their individual energy consumption utilities. However, this profit-seeking behavior among consumers may violate the network constraints, such as line flows, transformer capacity and bus voltage magnitude limits. Therefore, a network-constrained energy consumption (NCEC) game among active load aggregators (ALAs) is proposed to guarantee the safety of the distribution network. The temporal and spatial constraints of an ALA are both considered, which leads the formulated model to a generalized Nash equilibrium problem (GNEP). By resorting to a well-developed variational inequality (VI) theory, we study the existence of solutions to the NCEC game problem. Subsequently, a two-level distributed algorithm is proposed to find the variational equilibrium (VE), a fair and stable solution to the formulated game model. Finally, the effectiveness of the proposed game model and the efficiency of the distributed algorithm are tested on an IEEE-33 bus system.

We consider a class of decomposition methods for variational inequalities, which is related to the classical Dantzig–Wolfe decomposition of linear programs. Our approach is rather general, in that it can be used with certain types of set-valued or nonmonotone operators, as well as with various kinds of approximations in the subproblems of the functions and derivatives in the single-valued case. Also, subproblems may be solved approximately. Convergence is established under reasonable assumptions. We also report numerical experiments for computing variational equilibria of the game-theoretic models of electricity markets. Our numerical results illustrate that the decomposition approach allows to solve large-scale problem instances otherwise intractable if the widely used PATH solver is applied directly, without decomposition.

In this paper, we consider a class of Stackelberg quasi-equilibrium problem with two players in finite dimensional spaces. Existence and location of the Stackelberg quasi-equilibrium is discussed by employing the quasi-variational inequality techniques and the fixed point arguments. The results presented in this paper generalize some corresponding ones due to Nagy [Nagy, S., Stackelberg equilibia via variational inequalities and projections, J. Global Optim., 57 (2013), 821–828].

We provide Completely Positive and Copositive Optimization formulations for the Constrained Fractional Quadratic Problem (CFQP) and Standard Fractional Quadratic Problem (StFQP). Based on these formulations, Semidefinite Programming relaxations are derived for finding good lower bounds to these fractional programs, which can be used in a global optimization branch-and-bound approach. Applications of the CFQP and StFQP, related with the correction of infeasible linear systems and eigenvalue complementarity problems are also discussed.

This paper addresses a problem that is typical of multi-period capacity expansion equilibrium models: plants or sectors have different risk exposures that may warrant different costs of capital. The paper examines modifications of a capacity expansion model interpreted in equilibrium terms to account for asset-specific costs of capital.

This paper examines the convergence of no-regret learning in games with continuous action sets. For concreteness, we focus on learning via “dual averaging”, a widely used class of no-regret learning schemes where players take small steps along their individual payoff gradients and then “mirror” the output back to their action sets. In terms of feedback, we assume that players can only estimate their payoff gradients up to a zero-mean error with bounded variance. To study the convergence of the induced sequence of play, we introduce the notion of variational stability, and we show that stable equilibria are locally attracting with high probability whereas globally stable equilibria are globally attracting with probability 1. We also discuss some applications to mixed-strategy learning in finite games, and we provide explicit estimates of the method’s convergence speed.