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31
An overview of bilevel optimization
, 2007
"... This paper is devoted to bilevel optimization, a branch of mathematical programming of both practical and theoretical interest. Starting with a simple example, we proceed towards a general formulation. We then present fields of application, focus on solution approaches, and make the connection wit ..."
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Cited by 68 (3 self)
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This paper is devoted to bilevel optimization, a branch of mathematical programming of both practical and theoretical interest. Starting with a simple example, we proceed towards a general formulation. We then present fields of application, focus on solution approaches, and make the connection with MPECs (Mathematical Programs with Equilibrium Constraints).
An implementable activeset algorithm for computing a Bstationary point of a mathematical program with linear complementarity constraints
 SIAM J. Optim
"... Abstract. In [3], an ɛactive set algorithm was proposed for solving a mathematical program with a smooth objective function and linear inequality/complementarity constraints. It is asserted therein that, under a uniform LICQ on the ɛfeasible set, this algorithm generates iterates whose cluster poi ..."
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Cited by 30 (4 self)
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Abstract. In [3], an ɛactive set algorithm was proposed for solving a mathematical program with a smooth objective function and linear inequality/complementarity constraints. It is asserted therein that, under a uniform LICQ on the ɛfeasible set, this algorithm generates iterates whose cluster points are Bstationary points of the problem. However, the proof has a gap and only shows that each cluster point is an Mstationary point. We discuss this gap and show that Bstationarity can be achieved if the algorithm is modified and an additional error bound condition holds. Key words. MPEC, Bstationary point, ɛactive set, error bound AMS subject classifications. 65K05, 90C30, 90C33
Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints
, 2005
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On the Global Solution of Linear Programs with Linear Complementarity Constraints
, 2007
"... This paper presents a parameterfree integerprogramming based algorithm for the global resolution of a linear program with linear complementarity constraints (LPEC). The cornerstone of the algorithm is a minimax integer program formulation that characterizes and provides certificates for the three ..."
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Cited by 20 (3 self)
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This paper presents a parameterfree integerprogramming based algorithm for the global resolution of a linear program with linear complementarity constraints (LPEC). The cornerstone of the algorithm is a minimax integer program formulation that characterizes and provides certificates for the three outcomes—infeasibility, unboundedness, or solvability—of an LPEC. An extreme point/ray generation scheme in the spirit of Benders decomposition is developed, from which valid inequalities in the form of satisfiability constraints are obtained. The feasibility problem of these inequalities and the carefully guided linear programming relaxations of the LPEC are the workhorse of the algorithm, which also employs a specialized procedure for the sparsification of the satifiability cuts. We establish the finite termination of the algorithm and report computational results using the algorithm for solving randomly generated LPECs of reasonable sizes. The results establish that the algorithm can handle infeasible, unbounded, and solvable LPECs effectively.
THE THEORY OF 2REGULARITY FOR MAPPINGS WITH LIPSCHITZIAN DERIVATIVES AND ITS APPLICATIONS TO OPTIMALITY CONDITIONS
, 2002
"... We study local structure of a nonlinear mapping near points where standard regularity and/or smoothness assumptions need not be satisfied. We introduce a new concept of 2regularity (a certain kind of secondorder regularity) for a once differentiable mapping whose derivative is Lipschitz continuous ..."
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Cited by 17 (14 self)
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We study local structure of a nonlinear mapping near points where standard regularity and/or smoothness assumptions need not be satisfied. We introduce a new concept of 2regularity (a certain kind of secondorder regularity) for a once differentiable mapping whose derivative is Lipschitz continuous. Under this 2regularity condition, we obtain the representation theorem and the covering theorem (i.e., stability with respect to “righthand side ” perturbations) under assumptions that are weaker than those previously employed in the literature for results of this type. These results are further used to derive a constructive description of the tangent cone to a set defined by (2regular) equality constraints and optimality conditions for related optimization problems. The class of mappings introduced and studied in the paper appears to be a convenient tool for treating complementarity structures by means of an appropriate equationbased reformulation. Optimality conditions for mathematical programs with (equivalently reformulated) complementarity constraints are also discussed.
Generalized stationary points and an interiorpoint method for mathematical programs with equilibrium constraints
 Industrial Engineering & Management Sciences, Northwestern University
, 2005
"... Abstract. Generalized stationary points of the mathematical program with equilibrium constraints (MPEC) are studied to better describe the limit points produced by interior point methods for MPEC. A primaldual interiorpoint method is then proposed, which solves a sequence of relaxed barrier proble ..."
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Cited by 15 (0 self)
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Abstract. Generalized stationary points of the mathematical program with equilibrium constraints (MPEC) are studied to better describe the limit points produced by interior point methods for MPEC. A primaldual interiorpoint method is then proposed, which solves a sequence of relaxed barrier problems derived from MPEC. Global convergence results are deduced without assuming strict complementarity or the linear independence constraint qualification for MPEC (MPECLICQ). Under certain general assumptions, the algorithm can always find some point with strong or weak stationarity. In particular, it is shown that every limit point of the generated sequence is a strong stationary point of MPEC if the penalty parameter of the merit function is bounded. Otherwise, a certain point with weak stationarity can be obtained. Preliminary numerical results are reported, which include a case analyzed by Leyffer for which the penalty interiorpoint algorithm failed to find a stationary point. Key words: Global convergence, interiorpoint methods, mathematical programming with equilibrium constraints, stationary point
Smoothing Implicit Programming Approaches For Stochastic Mathematical Programs With Linear Complementarity Constraints
, 2003
"... In this paper, we consider the stochastic mathematical program with equilibrium constraints (SMPEC) , which can be thought as a generalization of the mathematical program with equilibrium constraints. Many decision problems can be formulated as SMPECs in practice. We discuss both lowerlevel waitan ..."
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Cited by 13 (6 self)
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In this paper, we consider the stochastic mathematical program with equilibrium constraints (SMPEC) , which can be thought as a generalization of the mathematical program with equilibrium constraints. Many decision problems can be formulated as SMPECs in practice. We discuss both lowerlevel waitandsee and hereand now decision problems. For the lowerlevel waitandsee model, we propose a smoothing implicit programming method and establish a comprehensive convergence theory. For the hereandnow decision problem, we apply a penalty technique and suggest a similar method. We show that the two methods possess similar convergence properties.
M.: A modified relaxation scheme for mathematical programs with complementarity constraints
, 2005
"... Abstract. In this PaPer, we consider amathematical program with complementarity constraints. We present amodified relaxed program for this problem, which involves less constraints than the relaxation scheme studied by Scholtes (2000). We show that the linear independence constraint qualification hol ..."
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Cited by 8 (5 self)
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Abstract. In this PaPer, we consider amathematical program with complementarity constraints. We present amodified relaxed program for this problem, which involves less constraints than the relaxation scheme studied by Scholtes (2000). We show that the linear independence constraint qualification holds for the new relaxed problem under some mild conditions. We also consider alimiting behavior of the relaxed problem. We prove that any accumulation point of stationary points of the relaxed problems is $\mathrm{C} $ stationary to the original problem under the MPEC linear independence constraint qualification and, if the Hessian matrices of the Lagrangian functions of the relaxed problems are uniformly bounded below on the corresponding tangent space, it is $\mathrm{M}$stationary. We also obtain some sufficient conditions of $\mathrm{B}$stationarity for afeasible point of the original problem. In particular, some conditions described by the eigenvalues of the Hessian matrices mentioned above are new and can be verified easily. Key Words, mathematical program with complementarity constraints, (MPEC) linear independence constraint qualification, nondegeneracy, (B, M, C) stationarity, second0rder necessary conditions, upper level strict complementarity.
A Sequential NCP Algorithm for Solving Equilibrium Problems with Equilibrium Constraints
"... Abstract. This paper studies algorithms for equilibrium problems with equilibrium constraints (EPECs). We present a generalization of Scholtes’s regularization scheme for MPECs and extend his convergence results to this new relaxation method. We propose a sequential nonlinear complementarity (SNCP) ..."
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Cited by 6 (1 self)
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Abstract. This paper studies algorithms for equilibrium problems with equilibrium constraints (EPECs). We present a generalization of Scholtes’s regularization scheme for MPECs and extend his convergence results to this new relaxation method. We propose a sequential nonlinear complementarity (SNCP) algorithm to solve EPECs and establish the convergence of this algorithm. We present numerical results comparing the SNCP algorithm and diagonalization (nonlinear GaussSeidel and nonlinear Jacobi) methods on randomly generated EPEC test problems. The computational experience to date shows that both the SNCP algorithm and the nonlinear GaussSeidel method outperform the nonlinear Jacobi method. 1
A Hybrid Algorithm with Active Set Identification for Mathematical Programs with Complementarity Constraints
, 2002
"... We consider a mathematical program with complementarity constraints (MPCC). Our purpose is to develop methods that enable us to compute a solution or a point with some kind of stationarity to MPCC by solving a finite number of nonlinear programs. To this end, we first introduce an active set identif ..."
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Cited by 6 (2 self)
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We consider a mathematical program with complementarity constraints (MPCC). Our purpose is to develop methods that enable us to compute a solution or a point with some kind of stationarity to MPCC by solving a finite number of nonlinear programs. To this end, we first introduce an active set identification technique. Then, by applying this technique to a smoothing continuation method presented by Fukushima and Pang (1999), we propose a hybrid method for solving MPCC. Under reasonable assumptions, the hybrid algorithm is shown to possess a finite termination property. Numerical experience shows that the proposed approach is quite e#ective.