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Published in G. Leitmann, F.E. Udwadia, A.V. Kryazhimskii, Dynamics and Control, 2020
In this local bimatrix game, strategy 1 is repeating pk for player 1 and repeating qk for player 2, and strategy 2 is switching from pk to p1 for player 1 and switching from qk to q2 for player 2. To identify (pk+1,qk+1), the players find a pure strategy Nash equilibrium, NE, in the local bimatrix game. One can prove that this game always has a pure strategy NE; moreover, four situations for NE, and, respectively, (pk+1,qk+1), may occur. Namely, the next cases are the only admissible ones: (1, 2) is a single NE, and (pk+1,qk+1) = (pk,q2),(2, 1) is a single NE, and (pk+1,qk+1) = (p1,qk),(2, 2) is a single NE, and (pk+1,qk+1) = (p1,q2),(1, 2) and (2, 1) are two NE, and (pk+1,qk+1) is either (pk,q2), or (p1,qk), with some (possibly equal) probabilities.
An improved generalized flexibility matrix approach for structural damage detection
Published in Inverse Problems in Science and Engineering, 2020
The LCP is to calculate a vector such that where , , and the notation ‘’ represents the componentwise defined partial ordering between two vectors. In the following, problem (22) and its solution are denoted by LCP () and , respectively. Many problems in scientific computing and engineering applications need to solve the LCP. For example, the market equilibrium problem, the contact problem and the Nash equilibrium point problem of a bimatrix game are eventually transformed into LCPs [39,40]. In addition, the LCP also constitutes the Karush-Kuhn-Tucker optimality condition of the quadratic programming problem. Let’s consider the following the quadratic programming problem: where , is a symmetric matrix. Then, we have the following result.
Γ-robust linear complementarity problems
Published in Optimization Methods and Software, 2022
Another classical example is the bimatrix game. Consider two players 1 and 2 with m and n pure strategies, respectively. The cost incurred for player 1 if she plays strategy and if player 2 plays strategy is given as the entry of the non-negative matrix . The analogous costs for player 2 are given in the non-negative matrix . A mixed strategy for player 1 is a non-negative vector with . A mixed strategy for the other player is defined in the same way. The expected costs of the players thus are and , respectively, and a pair of mixed strategies is called a Nash equilibrium [28,29] if It can be shown that computing a Nash equilibrium of a bimatrix game is equivalent to solving the LCP with data see [22] for an early study of this relation. This LCP is of rather special type since q does not depend on the problem's data but only contains 's and also M has a special structure. This example shows that for some LCPs, the consideration of uncertain q is not reasonable. Here, perturbations in q would yield an LCP that has no connection anymore to the original bimatrix game. As a consequence, only M can be reasonably considered uncertain, which corresponds to uncertain payoffs of the players; cf., e.g. [18].
Interior point methods for solving Pareto eigenvalue complementarity problems
Published in Optimization Methods and Software, 2023
Samir Adly, Mounir Haddou, Manh Hung Le
The area of complementarity problems (CP) has received great attention over the last few decades due to their various applications in engineering, economics and sciences. Since the pioneering work by Lemke and Howson, who showed that computing a Nash equilibrium point of a bimatrix game can be modelled as a linear complementarity problem [17], the theory of CP has become a useful and effective tool for studying a wide class of problems in numerical optimization. As a result, a variety of algorithms have been proposed and analysed to deal efficiently with these problems, see the thorough survey [7] and references therein. On the other hand, Eigenvalue Complementarity Problems (EiCP) (also known as cone-constrained eigenvalue problems) form a particular subclass of complementarity problems that extend the classical (linear algebra) eigenvalue problems. Solving classical eigenvalue problems is also a topic of great interest and finds its various applications in physics and engineering, see [9,34]. EiCP appeared for the first time in the study of static equilibrium states of finite dimensional mechanical systems with unilateral frictional contact [27], and since then it has been widely studied both theoretically and numerically. On this subject, we refer to [1–3,8,11–14,18,19,26,31] and references therein. Applications of EiCP were found in many fields such as the dynamic analysis of structural mechanical systems, vibro-acoustic systems, electrical circuit simulation, signal processing, fluid dynamics, contact problems in mechanics (see for instance [21–24,28]). Mathematically speaking, solving EiCP consists in finding a real number and a corresponding nonzero vector such that the following condition holds: where K is a closed convex cone in , ⊥ indicates the orthogonality, and stands for its positive dual cone, which is defined by In (1) is a given matrix (not necessarily symmetric).