共查询到20条相似文献,搜索用时 828 毫秒
1.
In this paper we reformulate the generalized Nash equilibrium problem (GNEP) as a nonsmooth Nash equilibrium problem by means
of a partial penalization of the difficult coupling constraints. We then propose a suitable method for the solution of the
penalized problem and we study classes of GNEPs for which the penalty approach is guaranteed to converge to a solution. In
particular, we are able to prove convergence for an interesting class of GNEPs for which convergence results were previously
unknown. 相似文献
2.
Using a regularized Nikaido-Isoda function, we present a (nonsmooth) constrained optimization reformulation of the player convex generalized Nash equilibrium problem (GNEP). Further we give an unconstrained reformulation of a large subclass of player convex GNEPs which, in particular, includes the jointly convex GNEPs. Both approaches characterize all solutions of a GNEP as minima of optimization problems. The smoothness properties of these optimization problems are discussed in detail, and it is shown that the corresponding objective functions are continuous and piecewise continuously differentiable under mild assumptions. Some numerical results based on the unconstrained optimization reformulation being applied to player convex GNEPs are also included. 相似文献
3.
We define the concept of reproducible map and show that, whenever the constraint map defining the quasivariational inequality (QVI) is reproducible then one can characterize the whole solution set of the QVI as a union of solution sets of some variational inequalities (VI). By exploiting this property, we give sufficient conditions to compute any solution of a generalized Nash equilibrium problem (GNEP) by solving a suitable VI. Finally, we define the class of pseudo-Nash equilibrium problems, which are (not necessarily convex) GNEPs whose solutions can be computed by solving suitable Nash equilibrium problems. 相似文献
4.
Generalized Nash equilibrium problems (GNEPs) allow, in contrast to standard Nash equilibrium problems, a dependence of the
strategy space of one player from the decisions of the other players. In this paper, we consider jointly convex GNEPs which
form an important subclass of the general GNEPs. Based on a regularized Nikaido-Isoda function, we present two (nonsmooth)
reformulations of this class of GNEPs, one reformulation being a constrained optimization problem and the other one being
an unconstrained optimization problem. While most approaches in the literature compute only a so-called normalized Nash equilibrium,
which is a subset of all solutions, our two approaches have the property that their minima characterize the set of all solutions
of a GNEP. We also investigate the smoothness properties of our two optimization problems and show that both problems are
continuous under a Slater-type condition and, in fact, piecewise continuously differentiable under the constant rank constraint
qualification. Finally, we present some numerical results based on our unconstrained optimization reformulation. 相似文献
5.
Masao Fukushima 《Computational Management Science》2011,8(3):201-218
The generalized Nash equilibrium problem (GNEP) is a generalization of the standard Nash equilibrium problem, in which each
player’s strategy set may depend on the rival players’ strategies. The GNEP has recently drawn much attention because of its
capability of modeling a number of interesting conflict situations in, for example, an electricity market and an international
pollution control. However, a GNEP usually has multiple or even infinitely many solutions, and it is not a trivial matter
to choose a meaningful solution from those equilibria. The purpose of this paper is two-fold. First we present an incremental
penalty method for the broad class of GNEPs and show that it can find a GNE under suitable conditions. Next, we formally define
the restricted GNE for the GNEPs with shared constraints and propose a controlled penalty method, which includes the incremental
penalty method as a subprocedure, to compute a restricted GNE. Numerical examples are provided to illustrate the proposed
approach. 相似文献
6.
We consider an optimization reformulation approach for the generalized Nash equilibrium problem (GNEP) that uses the regularized
gap function of a quasi-variational inequality (QVI). The regularized gap function for QVI is in general not differentiable,
but only directionally differentiable. Moreover, a simple condition has yet to be established, under which any stationary
point of the regularized gap function solves the QVI. We tackle these issues for the GNEP in which the shared constraints
are given by linear equalities, while the individual constraints are given by convex inequalities. First, we formulate the
minimization problem involving the regularized gap function and show the equivalence to GNEP. Next, we establish the differentiability
of the regularized gap function and show that any stationary point of the minimization problem solves the original GNEP under
some suitable assumptions. Then, by using a barrier technique, we propose an algorithm that sequentially solves minimization
problems obtained from GNEPs with the shared equality constraints only. Further, we discuss the case of shared inequality
constraints and present an algorithm that utilizes the transformation of the inequality constraints to equality constraints
by means of slack variables. We present some results of numerical experiments to illustrate the proposed approach. 相似文献
7.
D. Dorsch H. T. Jongen V. Shikhman 《Journal of Optimization Theory and Applications》2013,159(3):606-634
In this article we study generalized Nash equilibrium problems (GNEP) and bilevel optimization side by side. This perspective comes from the crucial fact that both problems heavily depend on parametric issues. Observing the intrinsic complexity of GNEP and bilevel optimization, we emphasize that it originates from unavoidable degeneracies occurring in parametric optimization. Under intrinsic complexity, we understand the involved geometrical complexity of Nash equilibria and bilevel feasible sets, such as the appearance of kinks and boundary points, non-closedness, discontinuity and bifurcation effects. The main goal is to illustrate the complexity of those problems originating from parametric optimization and singularity theory. By taking the study of singularities in parametric optimization into account, the structural analysis of Nash equilibria and bilevel feasible sets is performed. For GNEPs, the number of players’ common constraints becomes crucial. In fact, for GNEPs without common constraints and for classical NEPs we show that—generically—all Nash equilibria are jointly nondegenerate Karush–Kuhn–Tucker points. Consequently, they are isolated. However, in presence of common constraints Nash equilibria will constitute a higher dimensional set. In bilevel optimization, we describe the global structure of the bilevel feasible set in case of a one-dimensional leader’s variable. We point out that the typical discontinuities of the leader’s objective function will be caused by follower’s singularities. The latter phenomenon occurs independently of the viewpoint of the optimistic or pessimistic approach. In case of higher dimensions, optimistic and pessimistic approaches are discussed with respect to possible bifurcation of the follower’s solutions. 相似文献
8.
The generalized Nash equilibrium problem (GNEP) is an extension of the standard Nash game where, in addition to the cost functions,
also the strategy spaces of each player depend on the strategies chosen by all other players. This problem is rather difficult
to solve and there are only a few methods available in the literature. One of the most popular ones is the so-called relaxation
method, which is known to be globally convergent under a set of assumptions. Some of these assumptions, however, are rather
strong or somewhat difficult to understand. Here, we present a modified relaxation method for the solution of a certain class
of GNEPs. The convergence analysis uses completely different arguments based on a certain descent property and avoids some
of the technical conditions for the original relaxation method. Moreover, numerical experiments indicate that the modified
relaxation method performs quite well on a number of different examples taken from the literature. 相似文献
9.
The generalized Nash equilibrium problem (GNEP) is a noncooperative game in which the strategy set of each player, as well as his payoff function, depend on the rival players strategies. As a generalization of the standard Nash equilibrium problem (NEP), the GNEP has recently drawn much attention due to its capability of modeling a number of interesting conflict situations in, for example, an electricity market and an international pollution control. In this paper, we propose an improved two-step (a prediction step and a correction step) method for solving the quasi-variational inequality (QVI) formulation of the GNEP. Per iteration, we first do a projection onto the feasible set defined by the current iterate (prediction) to get a trial point; then, we perform another projection step (correction) to obtain the new iterate. Under certain assumptions, we prove the global convergence of the new algorithm. We also present some numerical results to illustrate the ability of our method, which indicate that our method outperforms the most recent projection-like methods of Zhang et al. (2010). 相似文献
10.
The generalized Nash equilibrium problem (GNEP) is a generalization of the standard Nash equilibrium problem (NEP),in which both the utility function and the strategy space of each player depend on the strategies chosen by all other players.This problem has been used to model various problems in applications.However,the convergent solution algorithms are extremely scare in the literature.In this paper,we present an incremental penalty method for the GNEP,and show that a solution of the GNEP can be found by solving a sequence of smooth NEPs.We then apply the semismooth Newton method with Armijo line search to solve latter problems and provide some results of numerical experiments to illustrate the proposed approach. 相似文献
11.
12.
A globally convergent trust region algorithm for optimization with general constraints and simple bounds 总被引:3,自引:0,他引:3
1.IntroductionInthispaper,weconsiderthefollowingnonlinearprogr~ngproblemwherec(x)=(c,(x),c2(2),',We(.))',i(x)andci(x)(i=1,2,',m)arerealfunctions*ThisworkissupPOrtedbytheNationalNaturalScienceFOundationofChinaandtheManagement,DecisionandinformationSystemLab,theChineseAcademyofSciences.definedinD={xEReIISx5u}.Weassumethath相似文献
13.
Problem-dependent upper and lower bounds are given for the stepsize taken by long Taylor series methods for solving initial value problems in ordinary differential equations. Taylor series methods recursively generate the terms of the Taylor series and estimate the radius of convergence as well as the order and location of the primary singularities. A stepsize must then be chosen which is as large as possible to minimize the required number of steps, while remaining small enough to maintain the truncation error less than some tolerance.One could use any of four different measures of trunction error in an attempt to control the global error : i) absolute truncation error per step, ii) absolute trunction error per unit step, iii) relative truncation error per step, and iv) relative truncation error per unit step. For each of these measures, we give bounds for error and for the stepsize which yields a prescribed error. The bounds depend on the series length, radius of convergence, order, and location of the primary singularities. The bounds are shown to be optimal for functions with only one singularity of any order on the circle of convergence. 相似文献
14.
Error bounds, which refer to inequalities that bound the distance of vectors in a test set to a given set by a residual function, have proven to be extremely useful in analyzing the convergence rates of a host of iterative methods for solving optimization problems. In this paper, we present a new framework for establishing error bounds for a class of structured convex optimization problems, in which the objective function is the sum of a smooth convex function and a general closed proper convex function. Such a class encapsulates not only fairly general constrained minimization problems but also various regularized loss minimization formulations in machine learning, signal processing, and statistics. Using our framework, we show that a number of existing error bound results can be recovered in a unified and transparent manner. To further demonstrate the power of our framework, we apply it to a class of nuclear-norm regularized loss minimization problems and establish a new error bound for this class under a strict complementarity-type regularity condition. We then complement this result by constructing an example to show that the said error bound could fail to hold without the regularity condition. We believe that our approach will find further applications in the study of error bounds for structured convex optimization problems. 相似文献
15.
Minglu Ye 《Optimization》2017,66(7):1119-1134
The generalized Nash equilibrium problem (GNEP) is an n-person noncooperative game in which each player’s strategy set depends on the rivals’ strategy set. In this paper, we presented a half-space projection method for solving the quasi-variational inequality problem which is a formulation of the GNEP. The difference from the known projection methods is due to the next iterate point in this method is obtained by directly projecting a point onto a half-space. Thus, our next iterate point can be represented explicitly. The global convergence is proved under the minimal assumptions. Compared with the known methods, this method can reduce one projection of a vector onto the strategy set per iteration. Numerical results show that this method not only outperforms the known method but is also less dependent on the initial value than the known method. 相似文献
16.
We consider the generalized Nash equilibrium problem (GNEP), where not only the players’ cost functions but also their strategy spaces depend on the rivals’ decision variables. Existence results for GNEPs are typically shown by using a fixed point argument for a certain set-valued function. Here we use a regularization of this set-valued function in order to obtain a single-valued function that is easier to deal with from a numerical point of view. We show that the fixed points of the latter function constitute an important subclass of the generalized equilibria called normalized equilibria. This fixed point formulation is then used to develop a nonsmooth Newton method for computing a normalized equilibrium. The method uses a so-called computable generalized Jacobian that is much easier to compute than Clarke generalized Jacobian or B-subdifferential. We establish local superlinear/quadratic convergence of the method under the constant rank constraint qualification, which is weaker than the frequently used linear independence constraint qualification, and a suitable second-order condition. Some numerical results are presented to illustrate the performance of the method. 相似文献
17.
Robert Schaback 《Numerische Mathematik》2010,114(4):629-651
A general framework for proving error bounds and convergence of a large class of unsymmetric meshless numerical methods for
solving well-posed linear operator equations is presented. The results provide optimal convergence rates, if the test and
trial spaces satisfy a stability condition. Operators need not be elliptic, and the problems can be posed in weak or strong
form without changing the theory. Non-stationary kernel-based trial and test spaces are shown to fit into the framework, disregarding
the operator equation. As a special case, unsymmetric meshless kernel-based methods solving weakly posed problems with distributional
data are treated in some detail. This provides a foundation of certain variations of the “Meshless Local Petrov-Galerkin”
technique of S.N. Atluri and collaborators. 相似文献
18.
《Applied Mathematics Letters》2001,14(7):891-896
In the present paper, we study model singularly perturbed convection-diffusion problems with exponential boundary layers. It has been believed for some time that only a complete splitting of the exact solution into regular and layer parts provides the information necessary for the study of the uniform convergence properties of numerical methods for these problems on layer-adapted grids (such as Shishkin meshes). In the present paper, we give new proofs of uniform interpolation error estimates for linear and bilinear interpolation; these proofs are based on the older a priori bounds derived by Kellogg and Tsan [1]. 相似文献
19.
牛顿-正则化方法与一类差分方程反问题的求解 总被引:7,自引:0,他引:7
在用牛顿迭代法求解非线性算子方程时,总要求非线性算子的导算子是有界可逆的,即线性化方程是适定的.但在实际数值计算中.即使满足这个条件,也可能出现数值不稳定的现象.为了克服这个困难,[1]将牛顿法与求解线性不适定问题的BG方法(平均核方法)结合起来,在每一步迭代中利用BG方法稳定求解.考虑到Tikhonov的正则化方 相似文献
20.
Convergence rate analysis of iteractive algorithms for solving variational inequality problems 总被引:3,自引:0,他引:3
M.V. Solodov 《Mathematical Programming》2003,96(3):513-528
We present a unified convergence rate analysis of iterative methods for solving the variational inequality problem. Our results
are based on certain error bounds; they subsume and extend the linear and sublinear rates of convergence established in several
previous studies. We also derive a new error bound for $\gamma$-strictly monotone variational inequalities. The class of algorithms
covered by our analysis in fairly broad. It includes some classical methods for variational inequalities, e.g., the extragradient,
matrix splitting, and proximal point methods. For these methods, our analysis gives estimates not only for linear convergence
(which had been studied extensively), but also sublinear, depending on the properties of the solution. In addition, our framework
includes a number of algorithms to which previous studies are not applicable, such as the infeasible projection methods, a
separation-projection method, (inexact) hybrid proximal point methods, and some splitting techniques. Finally, our analysis
covers certain feasible descent methods of optimization, for which similar convergence rate estimates have been recently obtained
by Luo [14].
Received: April 17, 2001 / Accepted: December 10, 2002
Published online: April 10, 2003
RID="⋆"
ID="⋆" Research of the author is partially supported by CNPq Grant 200734/95–6, by PRONEX-Optimization, and by FAPERJ.
Key Words. Variational inequality – error bound – rate of convergence
Mathematics Subject Classification (2000): 90C30, 90C33, 65K05 相似文献