This paper proposes a new decomposition method for globally solving mathematical programming prob... more This paper proposes a new decomposition method for globally solving mathematical programming problems with affine equilibrium constraints (AMPEC). First, we view AMPEC as a bilevel programming problem where the lower level problem is a parametric affine variational inequality. Then, we use a regularization technique to formulate the resulting problem as a mathematical program with an additional constraint defined by the difference of two convex functions (DC function). A main feature of this DC decomposition is that the second component depends upon only the parameter in the lower level problem. This property allows us to develop branch-and-bound algorithms for globally solving AMPEC where the adaptive rectangular bisection takes place only in the space of the parameters. As an example, we use the proposed algorithm to solve a bilevel Nash-Cournot equilibrium market model. Computational results show the efficiency of the proposed algorithm.
In this paper we propose two iterative schemes for solving equilibrium problems which are called ... more In this paper we propose two iterative schemes for solving equilibrium problems which are called dual extragradient algorithms. In contrast with the primal extragradient methods in Quoc et al. (Optimization 57(6):749-776, 2008) which require to solve two general strongly convex programs at each iteration, the dual extragradient algorithms proposed in this paper only need to solve, at each iteration, one general strongly convex program, one projection problem and one subgradient calculation. Moreover, we provide the worst case complexity bounds of these algorithms, which have not been done in the primal extragradient methods yet. An application to Nash-Cournot equilibrium models of electricity markets is presented and implemented to examine the performance of the proposed algorithms.
In this paper we propose a new branch-and-bound algorithm by using an ellipsoidal partition for m... more In this paper we propose a new branch-and-bound algorithm by using an ellipsoidal partition for minimizing an indefinite quadratic function over a bounded polyhedral convex set which is not necessarily given explicitly by a system of linear inequalities and/or equalities. It is required that for this set there exists an efficient algorithm to verify whether a point is feasible, and
We generalize the projection method for strongly monotone multivalued variational inequalities wh... more We generalize the projection method for strongly monotone multivalued variational inequalities where the cost operator is not necessarily Lipschitz. At each iteration at most one projection onto the constrained set is needed. When the convex constrained set is not ...
We investigate the inexact Tikhonov and proximal point regularization methods for pseudomonotone ... more We investigate the inexact Tikhonov and proximal point regularization methods for pseudomonotone equilibrium problems. In this case, the regularized subproblems might not be strongly monotone, even not pseudomonotone. However, any iterative sequence of the regularized subproblems tends to the same solution, which, for the Tikhonov method, is the projection of the starting point onto the solution set of the original problem. This convergence result suggests algorithms for finding the limit point of the Tikhonov regularization method. Application to multivalued pseudomonotone variational inequalities is discussed.
We investigate the inexact Tikhonov and proximal point regularization methods for pseudomonotone ... more We investigate the inexact Tikhonov and proximal point regularization methods for pseudomonotone equilibrium problems. In this case, the regularized subproblems might not be strongly monotone, even not pseudomonotone. However, any iterative sequence of the regularized subproblems tends to the same solution, which, for the Tikhonov method, is the projection of the starting point onto the solution set of the original problem. This convergence result suggests algorithms for finding the limit point of the Tikhonov regularization method. Application to multivalued pseudomonotone variational inequalities is discussed.
We investigate the inexact Tikhonov and proximal point regularization methods for pseudomonotone ... more We investigate the inexact Tikhonov and proximal point regularization methods for pseudomonotone equilibrium problems. In this case, the regularized subproblems might not be strongly monotone, even not pseudomonotone. However, any iterative sequence of the regularized subproblems tends to the same solution, which, for the Tikhonov method, is the projection of the starting point onto the solution set of the original problem. This convergence result suggests algorithms for finding the limit point of the Tikhonov regularization method. Application to multivalued pseudomonotone variational inequalities is discussed.
This paper proposes a new decomposition method for globally solving mathematical programming prob... more This paper proposes a new decomposition method for globally solving mathematical programming problems with affine equilibrium constraints (AMPEC). First, we view AMPEC as a bilevel programming problem where the lower level problem is a parametric affine variational inequality. Then, we use a regularization technique to formulate the resulting problem as a mathematical program with an additional constraint defined by the difference of two convex functions (DC function). A main feature of this DC decomposition is that the second component depends upon only the parameter in the lower level problem. This property allows us to develop branch-and-bound algorithms for globally solving AMPEC where the adaptive rectangular bisection takes place only in the space of the parameters. As an example, we use the proposed algorithm to solve a bilevel Nash-Cournot equilibrium market model. Computational results show the efficiency of the proposed algorithm.
In this paper we propose two iterative schemes for solving equilibrium problems which are called ... more In this paper we propose two iterative schemes for solving equilibrium problems which are called dual extragradient algorithms. In contrast with the primal extragradient methods in Quoc et al. (Optimization 57(6):749-776, 2008) which require to solve two general strongly convex programs at each iteration, the dual extragradient algorithms proposed in this paper only need to solve, at each iteration, one general strongly convex program, one projection problem and one subgradient calculation. Moreover, we provide the worst case complexity bounds of these algorithms, which have not been done in the primal extragradient methods yet. An application to Nash-Cournot equilibrium models of electricity markets is presented and implemented to examine the performance of the proposed algorithms.
In this paper we propose a new branch-and-bound algorithm by using an ellipsoidal partition for m... more In this paper we propose a new branch-and-bound algorithm by using an ellipsoidal partition for minimizing an indefinite quadratic function over a bounded polyhedral convex set which is not necessarily given explicitly by a system of linear inequalities and/or equalities. It is required that for this set there exists an efficient algorithm to verify whether a point is feasible, and
We generalize the projection method for strongly monotone multivalued variational inequalities wh... more We generalize the projection method for strongly monotone multivalued variational inequalities where the cost operator is not necessarily Lipschitz. At each iteration at most one projection onto the constrained set is needed. When the convex constrained set is not ...
We investigate the inexact Tikhonov and proximal point regularization methods for pseudomonotone ... more We investigate the inexact Tikhonov and proximal point regularization methods for pseudomonotone equilibrium problems. In this case, the regularized subproblems might not be strongly monotone, even not pseudomonotone. However, any iterative sequence of the regularized subproblems tends to the same solution, which, for the Tikhonov method, is the projection of the starting point onto the solution set of the original problem. This convergence result suggests algorithms for finding the limit point of the Tikhonov regularization method. Application to multivalued pseudomonotone variational inequalities is discussed.
We investigate the inexact Tikhonov and proximal point regularization methods for pseudomonotone ... more We investigate the inexact Tikhonov and proximal point regularization methods for pseudomonotone equilibrium problems. In this case, the regularized subproblems might not be strongly monotone, even not pseudomonotone. However, any iterative sequence of the regularized subproblems tends to the same solution, which, for the Tikhonov method, is the projection of the starting point onto the solution set of the original problem. This convergence result suggests algorithms for finding the limit point of the Tikhonov regularization method. Application to multivalued pseudomonotone variational inequalities is discussed.
We investigate the inexact Tikhonov and proximal point regularization methods for pseudomonotone ... more We investigate the inexact Tikhonov and proximal point regularization methods for pseudomonotone equilibrium problems. In this case, the regularized subproblems might not be strongly monotone, even not pseudomonotone. However, any iterative sequence of the regularized subproblems tends to the same solution, which, for the Tikhonov method, is the projection of the starting point onto the solution set of the original problem. This convergence result suggests algorithms for finding the limit point of the Tikhonov regularization method. Application to multivalued pseudomonotone variational inequalities is discussed.
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