6,022 research outputs found
Mixed Strategy Constraints in Continuous Games
Equilibrium problems representing interaction in physical environments
typically require continuous strategies which satisfy opponent-dependent
constraints, such as those modeling collision avoidance. However, as with
finite games, mixed strategies are often desired, both from an equilibrium
existence perspective as well as a competitive perspective. To that end, this
work investigates a chance-constraint-based approach to coupled constraints in
generalized Nash equilibrium problems which are solved over pure strategies and
mixing weights simultaneously. We motivate these constraints in a discrete
setting, placing them on tensor games (-player bimatrix games) as a
justifiable approach to handling the probabilistic nature of mixing. Then, we
describe a numerical solution method for these chance constrained tensor games
with simultaneous pure strategy optimization. Finally, using a modified
pursuit-evasion game as a motivating examples, we demonstrate the actual
behavior of this solution method in terms of its fidelity, parameter
sensitivity, and efficiency
Deflation for semismooth equations
Variational inequalities can in general support distinct solutions. In this
paper we study an algorithm for computing distinct solutions of a variational
inequality, without varying the initial guess supplied to the solver. The
central idea is the combination of a semismooth Newton method with a deflation
operator that eliminates known solutions from consideration. Given one root of
a semismooth residual, deflation constructs a new problem for which a
semismooth Newton method will not converge to the known root, even from the
same initial guess. This enables the discovery of other roots. We prove the
effectiveness of the deflation technique under the same assumptions that
guarantee locally superlinear convergence of a semismooth Newton method. We
demonstrate its utility on various finite- and infinite-dimensional examples
drawn from constrained optimization, game theory, economics and solid
mechanics.Comment: 24 pages, 3 figure
Scenario-Game ADMM: A Parallelized Scenario-Based Solver for Stochastic Noncooperative Games
Decision making in multi-agent games can be extremely challenging,
particularly under uncertainty. In this work, we propose a new sample-based
approximation to a class of stochastic, general-sum, pure Nash games, where
each player has an expected-value objective and a set of chance constraints.
This new approximation scheme inherits the accuracy of objective approximation
from the established sample average approximation (SAA) method and enjoys a
feasibility guarantee derived from the scenario optimization literature. We
characterize the sample complexity of this new game-theoretic approximation
scheme, and observe that high accuracy usually requires a large number of
samples, which results in a large number of sampled constraints. To accommodate
this, we decompose the approximated game into a set of smaller games with few
constraints for each sampled scenario, and propose a decentralized, consensus
ADMM algorithm to efficiently compute a generalized Nash equilibrium of the
approximated game. We prove the convergence of our algorithm and empirically
demonstrate superior performance relative to a recent baseline
SOLVING TWO-LEVEL OPTIMIZATION PROBLEMS WITH APPLICATIONS TO ROBUST DESIGN AND ENERGY MARKETS
This dissertation provides efficient techniques to solve two-level optimization problems. Three specific types of problems are considered. The first problem is robust optimization, which has direct applications to engineering design. Traditionally robust optimization problems have been solved using an inner-outer structure, which can be computationally expensive. This dissertation provides a method to decompose and solve this two-level structure using a modified Benders
decomposition. This gradient-based technique is applicable to robust optimization problems with quasiconvex constraints and provides approximate solutions to problems with nonlinear constraints. The second types of two-level problems considered are mathematical and equilibrium programs with equilibrium constraints. Their two-level structure is simplified using Schur's decomposition and reformulation
schemes for absolute value functions. The resulting formulations are applicable to game theory problems in operations research and economics. The third type of two-level problem studied is discretely-constrained mixed linear complementarity problems. These are first formulated into a two-level mathematical program with equilibrium constraints and then solved using the aforementioned technique for mathematical and equilibrium programs with equilibrium constraints. The techniques for all three problems help simplify the two-level structure into one level, which helps gain numerical and application insights. The computational effort for solving these problems is greatly reduced using the techniques in this dissertation. Finally, a host of numerical examples are presented to verify the approaches. Diverse applications to economics, operations research, and engineering design motivate the relevance of the novel methods developed in this dissertation
Computation and analysis of evolutionary game dynamics
Biological processes are usually defined based on the principles of replication, mutation, competition, adaption, and evolution. In evolutionary game theory, such a process is modeled as a so-called evolutionary game, which not only provides an alternative interpretation of dynamical equilibrium in terms of the game nature of the process, but also bridges the stability of the biological process with the Nash equilibrium of the evolutionary game. Computationally, the evolutionary game models are described in terms of inverse and direct games, which are estimating the payoff matrix from data and computing the Nash equilibrium of a given payoff matrix respectively. We discuss the necessary and sufficient conditions for the Nash equilibrium states, and derive the methods for both inverse and direct games in this thesis. The inverse game is solved by a non-parametric smoothing method and penalized least squares method, while different schemes for the computation of the direct game are proposed including a specialized Snow-Shapley algorithm, a specialized Lemke-Howson algorithm, and an algorithm based on the solution of a complementarity problem on a simplex. Computation for the sparsest and densest Nash equilibria is investigated. We develop a new algorithm called dual method with better performance than the traditional Snow-Shapley method on the sparse and dense Nash equilibrium searching. Computational results are presented based on examples. The package incorporating all the schemes, the Toolbox of Evolution Dynamics Analysis (TEDA), is described
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