Nonmonotone Equilibrium Research Articles

A Novel Method for Solving Nonmonotone Equilibrium Problems

A Novel Method for Solving Nonmonotone Equilibrium Problems

A modified viscosity type inertial subgradient extragradient algorithm for nonmonotone equilibrium problems and application to cardiovascular disease detection

A modified viscosity type inertial subgradient extragradient algorithm for nonmonotone equilibrium problems and application to cardiovascular disease detection

Extragradient Algorithms with Linesearches for Solving Nonmonotone Equilibrium Problems in Banach Spaces

Extragradient Algorithms with Linesearches for Solving Nonmonotone Equilibrium Problems in Banach Spaces

The global convergence properties of an adaptive QP-free method without a penalty function or a filter for minimax optimization.

In this paper, we proposed an adaptive QP-free method without a penalty function or a filter for minimax optimization. In each iteration, solved two linear systems of equations constructed from Lagrange multipliers and KKT-conditioned NCP functions. Based on the work set, the computational scale is further reduced. Instead of the filter structure, we adopt a nonmonotonic equilibrium mechanism with an adaptive parameter adjusted according to the result of each iteration. Feasibility of the algorithm are given, and the convergence under some assumptions is demonstrated. Numerical results and practical application are reported at the end.

An extragradient method for non-monotone equilibrium problems on Hadamard manifolds with applications

In the present paper, we consider a non-monotone equilibrium problem on Hadamard manifolds and define Armijo's type extragradient algorithm for the proposed equilibrium problem. The introduced algorithm does not require objective bifunction's monotonicity and the solution set's nonemptiness. A convergence result of our algorithm under very mild assumptions is presented. Moreover, we investigate the applications of the established results to the non-monotone set-valued variational inequalities and the generalized Nash equilibrium problems from the considered method.

Reflected Iterative Method for Non-Monotone Equilibrium Problems with Applications to Nash-Cournot Equilibrium Models

Reflected Iterative Method for Non-Monotone Equilibrium Problems with Applications to Nash-Cournot Equilibrium Models

Solving non-monotone equilibrium problems via a DIRECT-type approach

A global optimization approach for solving non-monotone equilibrium problems (EPs) is proposed. The class of (regularized) gap functions is used to reformulate any EP as a constrained global optimization program and some bounds on the Lipschitz constant of such functions are provided. The proposed global optimization approach is a combination of an improved version of the DIRECT algorithm, which exploits local bounds of the Lipschitz constant of the objective function, with local minimizations. Unlike most existing solution methods for EPs, no monotonicity-type condition is assumed in this paper. Preliminary numerical results on several classes of EPs show the effectiveness of the approach.

Image restorations using an inertial parallel hybrid algorithm with Armijo linesearch for nonmonotone equilibrium problems

In this work, we modify the inertial type parallel monotone hybrid algorithm with Armijo linesearch to solve common nonmonotone equilibrium problems. The weak convergence theorem is established under some suitable conditions in Hilbert spaces. Finally, the numerical example is shown to support our main result. The efficiencies of the proposed algorithm are presented by comparing with previously parallel algorithms in novel image restoration. It’s illustrated that the proposed algorithm can increase deblurring quality when the subproblems are increasing.

Solving common nonmonotone equilibrium problems using an inertial parallel hybrid algorithm with Armijo line search with applications to image recovery

In this work, we modify the inertial hybrid algorithm with Armijo line search using a parallel method to approximate a common solution of nonmonotone equilibrium problems in Hilbert spaces. A weak convergence theorem is proved under some continuity and convexity assumptions on the bifunction and the nonemptiness of the common solution set of Minty equilibrium problems. Furthermore, we demonstrate the quality of our inertial parallel hybrid algorithm by using image restoration, as well as its superior efficiency when compared with previously considered parallel algorithms.

All-pay auctions with private signals about opponents’ values

We study all-pay auctions where each player observes her private value as well as a noisy private signal about the opponent’s value, following Fang and Morris’s (J Econ Theory 126(1):1–30, 2006) analysis of winner-pay auctions with multidimensional private signals. A unique symmetric monotonic equilibrium exists if the signal is not informative enough. When the signal is sufficiently informative, there exists a symmetric non-monotonic equilibrium in which all types of players randomize in overlapping supports. The revenue is lower than that in the standard independent private value setting. Fixing the signal’s informativeness, the all-pay auction raises lower revenue than the second-price auction, whereas the revenue ranking between the all-pay and the first-price auction is ambiguous.

A New Linesearch Algorithm for Split Equilibrium Problems

In this paper, we propose a new algorithm for solving a split equilibrium problem involving nonmonotone and monotone equilibrium bifunctions in real Hilbert spaces by using a shrinking projection method with a general Armijo line search rule on the e-subdifferential. We obtain a strong convergence theorem for the new algorithm.

The Last Step to the Throne, the Conflict between Monarchs and Crown Princes

I model the dynamic between ruler and successor. The ruler wants to cultivate a successor for a smooth power transition but fears being ousted by him, while the successor fears being removed by the ruler; these mutual fears may induce ruler--successor conflict. Each party follows a non-monotonic equilibrium strategy. The successor accumulates power while not threatening the ruler, and he prolongs their relationship by maintaining a low profile. The ruler gradually becomes more intolerant of the successor's growing power but, as her life nears its end, has less incentive to replace him. Thus conflict is most probable in the middle of their relationship. Although an institutionalized procedure may render conflict less likely, a predetermined succession order could increase its likelihood by restricting the ruler's optimal time to select a successor. If there are two candidates then a ruler prefers the weaker one only if their capabilities are similar.

On the Existence of Equilibrium in Bayesian Games Without Complementarities

This paper presents new results on the existence of pure-strategy Bayesian equilibria in specified functional forms. These results broaden the scope of methods developed by Reny (2011) well beyond monotone pure strategies. Applications include natural models of first-price and all-pay auctions not covered by previous existence results. To illustrate the scope of our results, we provide an analysis of three auctions: (i) a first-price auction of objects that are heterogeneous and imperfect substitutes; (ii) a first-price auction in which bidders’ payoffs have a very general interdependence structure; and (iii) an all-pay auction with non-monotone equilibrium.

Voltage-actuated snap-through in bistable piezoelectric thin films: a computational study

Bistable piezoelectric structures exhibit snap-through motions in response to applied voltage and play important roles in achieving functions such as fast actuation and structural morphing. However, modeling the nonlinear snap-through behaviors in such structures remains a challenge. In this paper, we develop a theoretical framework to model the voltage-actuated snap-through in bistable piezoelectric composites. Based on a revised continuum theory capable of characterizing the coupled finite deformation and electric field in piezoelectric materials, we establish a universal nonlinear finite element framework where the unknowns are the displacement and a scalar factor characterizing the magnitude of the applied voltage. By using the traditional Riks method, a supplementary arc-length equation related to the increment of the displacement and the scalar factor is constructed to complete the linearized incremental equation. A general solution scheme for obtaining the increments of all unknowns is developed, which enables automatic tracing of the nonmonotonic equilibrium path evolution. The feasibility and efficiency of this numerical method were demonstrated by the voltage-actuated snap-through phenomena for several bistable piezoelectric structures, including a simply-supported bilayer beam, a 3D square bilayer plate with free ends and a 3D constrained circular bilayer plate. This method will benefit the numerical design of high-performance bistable piezoelectric structures.

On the Existence of Equilibrium in Bayesian Games Without Complementarities

This paper presents new results on the existence of pure-strategy Bayesian equilibria in specified functional forms. These results broaden the scope of methods developed by Reny (2011) well beyond monotone pure strategies. Applications include natural models of first-price and all-pay auctions not covered by previous existence results. To illustrate the scope of our results, we provide an analysis of three auctions: (i) a first-price auction of objects that are heterogeneous and imperfect substitutes; (ii) a first-price auction in which bidders’ payoffs have a very general interdependence structure; and (iii) an all-pay auction with non-monotone equilibrium.

An inertial type iterative method with Armijo linesearch for nonmonotone equilibrium problems

This paper studies an iterative method with inertial term extrapolation step for solving an equilibrium problem of nonmonotone bifunctions in real Hilbert spaces. The inertia term extrapolation step is introduced to speed up the rate of convergence of the iteration process. We obtain convergence result under some continuity and convexity assumptions on the bifunction and the condition that the solution set of the associated Minty equilibrium problem is nonempty. Numerical comparisons of our proposed method with some other related method in the literature are given.

An extragradient-type method for solving nonmonotone quasi-equilibrium problems

In this paper, a quasi-equilibrium problem with a nonmonotone bifunction is considered in a finite-dimensional space. The primary difficulty with this problem is related to the fact that one must simultaneously solve a nonmonotone equilibrium problem and calculate a fixed point of a multivalued mapping. An extragradient-type method is presented and analysed for its solution. The convergence of the method is proved under the assumption that the solution set of an associated dual equilibrium problem is nonempty. Finally, some numerical experiments are reported.

Monotone and bounded interval equilibria in a coordination game with information aggregation

We analyze how private learning in a class of games with common stochastic payoffs affects the form of equilibria, and how properties such as player welfare and the extent of strategic miscoordination relate across monotone and non-monotone equilibria. Researchers typically focus on monotone equilibria. We provide conditions under which non-monotone equilibria also exist, where players attempt to coordinate to obtain the stochastic payoff whenever signals are in a bounded interval. In bounded interval equilibria (BIE), an endogenous fear of miscoordination discourages players from coordinating to obtain the stochastic payoff when their signals suggest coordination is most beneficial. In contrast to monotone equilibria, expected payoffs from successful coordination in BIE are lower than the ex-ante expected payoff from ignoring signals and always trying to coordinate to obtain the stochastic payoff. We show that BIE only exist when, absent private information, the game would be a coordination game.

Projection algorithms for solving nonmonotone equilibrium problems in Hilbert space

We propose two projection algorithms for solving an equilibrium problem where the bifunction is not required to be satisfied any monotone property. Under assumptions on the continuity, convexity of the bifunction and the nonemptyness of the solution set of the Minty equilibrium problem, we show that the sequences generated by the proposed algorithms converge weakly and strongly to a solution of the primal equilibrium problem respectively.

A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces

A new class of extragradient-type methods is introduced for solving an equilibrium problem in a real Hilbert space without any monotonicity assumption on the equilibrium function. The strategy is to replace the second projection step in the classical extragradient method by a projection onto shrinking convex subsets of the feasible set. Furthermore, to ensure a sufficient decrease on the equilibrium function, a general Armijo-type condition is imposed. This condition is shown to be satisfied for four different linesearches used in the literature. Then, the weak and strong convergence of the resulting algorithms is obtained under non-monotonicity assumptions. Finally, some numerical experiments are reported.