65 research outputs found
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
Computing Economic Equilibria by a Homotopy Method
In this paper the possibility of computing equilibrium in pure exchange and
production economies by a homotopy method is investigated. The performance of
the algorithm is tested on examples with known equilibria taken from the
literature on general equilibrium models and numerical results are presented.
In computing equilibria, economy will be specified by excess demand function.Comment: 12th IEEE International Symposium on Computational Intelligence and
Informatic
A Semismooth Newton Method for Tensor Eigenvalue Complementarity Problem
In this paper, we consider the tensor eigenvalue complementarity problem
which is closely related to the optimality conditions for polynomial
optimization, as well as a class of differential inclusions with nonconvex
processes. By introducing an NCP-function, we reformulate the tensor eigenvalue
complementarity problem as a system of nonlinear equations. We show that this
function is strongly semismooth but not differentiable, in which case the
classical smoothing methods cannot apply. Furthermore, we propose a damped
semismooth Newton method for tensor eigenvalue complementarity problem. A new
procedure to evaluate an element of the generalized Jocobian is given, which
turns out to be an element of the B-subdifferential under mild assumptions. As
a result, the convergence of the damped semismooth Newton method is guaranteed
by existing results. The numerical experiments also show that our method is
efficient and promising
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Using EPECs to model bilevel games in restructured electricity markets with locational prices
CWPE0619 (EPRG0602) Xinmin Hu and Daniel Ralph (Feb 2006) Using EPECs to model bilevel games in restructured electricity markets with locational prices We study a bilevel noncooperative game-theoretic model of electricity markets with locational marginal prices. Each player faces a bilevel optimization problem that we remodel as a mathematical program with equilibrium constraints, MPEC. This gives an EPEC, equilibrium problem with equilibrium constraints. We establish sufficient conditions for existence of pure strategy Nash equilibria for this class of bilevel games and give some applications. We show by examples the effect of network transmission limits, i.e. congestion, on existence of equilibria. Then we study, for more general EPECs, the weaker pure strategy concepts of local Nash and Nash stationary equilibria. We model the latter via complementarity problems, CPs. Finally, we present numerical examples of methods that attempt to find local Nash or Nash stationary equilibria of randomly generated electricity market games. The CP solver PATH is found to be rather effective in this context
An Incremental Gradient Method for Optimization Problems with Variational Inequality Constraints
We consider minimizing a sum of agent-specific nondifferentiable merely
convex functions over the solution set of a variational inequality (VI) problem
in that each agent is associated with a local monotone mapping. This problem
finds an application in computation of the best equilibrium in nonlinear
complementarity problems arising in transportation networks. We develop an
iteratively regularized incremental gradient method where at each iteration,
agents communicate over a cycle graph to update their solution iterates using
their local information about the objective and the mapping. The proposed
method is single-timescale in the sense that it does not involve any excessive
hard-to-project computation per iteration. We derive non-asymptotic agent-wise
convergence rates for the suboptimality of the global objective function and
infeasibility of the VI constraints measured by a suitably defined dual gap
function. The proposed method appears to be the first fully iterative scheme
equipped with iteration complexity that can address distributed optimization
problems with VI constraints over cycle graphs. Preliminary numerical
experiments for a transportation network problem and a support vector machine
model are presented
Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions
The aims of this paper are twofold. First, it is shown, for the first time, which types of nonsmooth functions are characterized by all vector critical points as being efficient or weakly efficient solutions of vector optimization problems in constrained and unconstrained scenarios on Hadamard manifolds. This implies the need to extend different concepts, such as the Karush-Kuhn-Tucker vector critical points and generalized invexity functions, to Hadamard manifolds. The relationships between these quantities are clarified through a great number of explanatory examples. Second, we present an economic application proving that Nash's critical and equilibrium points coincide in the case of invex payoff functions. This is done on Hadamard manifolds, a particular case of noncompact Riemannian symmetric spaces
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