Outrata J V, Kocvara M, and Zowe J. Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Dordrecht, The Netherlands, Kluwer, 1998  Home/Search   Document Not in Database   Summary   Related Articles   Check

....utilized to date in the calculation of an approximation to the gradient. A vector taking the role of the gradient, whether it exists or not, is necessary to obtain when applying gradient based algorithms for bilevel programs, or mathematical programs with equilibrium constraints (MPEC) cf. [LPR96, OKZ98]. These are models in which the vector x is optimized with respect to an (upper level) objective of the parameter : x) subject to 2 P; f( x) 2 NC (x) where : 7 is a smooth function, and P is a nonempty and closed, typically also convex, set. Heuristic ....

....more complicated than to generate just one subgradient, as an arbitrary subgradient is not guaranteed to provide a descent direction. Bundle algorithms work by generating a collection of subgradients at nearby points and generating a search direction from their convex hull. We refer to the book [OKZ98] for a general discussion of such algorithms in the context of bilevel programming. In this section, we will focus on the basic question of when and how a subgradient can be calculated, and how its calculation is related to the directional derivative and gradient calculus. Throughout this ....

[Article contains additional citation context not shown here]

J. Outrata, M. Ko cvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.

....of his generalized gradient in 1973 (summarized in his seminal book [7] pioneered a rapid development, recently presented in detail in [8] and [20] Computational methods for nonsmooth optimization have also developed rapidly, with manyinteresting applications. For a recent look, see [19], which focuses on mechanical applications, or [17] which concentrates on optimal control. Nonsmooth optimization algorithms such as the subgradient methods outlined in [21] or the bundle methods described in [2, 15] typically assume a locally Lipschitz function, and at each iterate x # compute ....

J. Outrata, M. Kocvara, and J. Zowe. Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, Dordrecht, 1998. 27

....investigated a topic which is not covered in this paper. The monograph [LPR96] includes several algorithms for this class of problems, such as the implicit programming algorithm for the nested problem, and the penalty interior point algorithm for the simultaneous version of the problem, cf. also [OKZ98, HKP99a]. The nested problem can in principle be treated by any method from nondifferentiable optimization, for example, bundle methods ( OKZ98] and subgradient methods ( CPW01] Another possibility is to use smoothing (cf. FJQ99, Hil00] that is, to replace the equilibrium problem by a sequence of ....

.... the implicit programming algorithm for the nested problem, and the penalty interior point algorithm for the simultaneous version of the problem, cf. also [OKZ98, HKP99a] The nested problem can in principle be treated by any method from nondifferentiable optimization, for example, bundle methods ([OKZ98]) and subgradient methods ( CPW01] Another possibility is to use smoothing (cf. FJQ99, Hil00] that is, to replace the equilibrium problem by a sequence of smooth approximations. Then any standard first order algorithm, such as sequential explicit approximation methods, can be used for the ....

J. Outrata, M. Ko cvara, and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic Publishers, Dordrecht, 1998.

....of the problem variables into controls z 0 2 IR and states (z 1 ; z 2 ) 2 IR 2p . The equality constraints c i (z) 0; i 2 E are abbreviated as c E (z) 0 and similarly, c I (z) 0 represents the inequality constraints. Problems of this type arise frequently in applications, see [8, 16, 17] for references. Problem (1.1) is also referred to as a Mathematical Program with Complementarity Constraints (MPCC) Clearly, an MPEC with a more general complementarity condition like 0 G(z) H(z) 0 (1.2) can be written in the form (1.1) by introducing slack variables. It is easy to show ....

Outrata, J., Kocvara, M. and Zowe, J. Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht, 1998.

....and phrases. smoothing methods, n dimensional max function, recursive approximation, vertical linear complementarity (VLCP) A#liations of the authors are at the end of the manuscript. NCP) problems, linear complementarity (LCP) problems, and vertical linear complementarity (VLCP) problems [4, 9, 5, 11, 10]. Consequently, di#cult optimization problems, like mathematical programs with equilibrium constraints (MPEC) 8] become more tractable. The main purpose of this paper is to provide a generic approximation for the high dimensional max function. In order to achieve this, we propose a new method, ....

Outrata, J.V., Kocvara, M. and J. Zowe. Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results. Kluwer Academic Publishers, Dordrecht, 1998.

....z = z 0 ; z 1 ; z 2 ) z 0 2 IR n are the control and (z 1 ; z 2 ) 2 IR 2p are the state variables. The equality constraints c i (z) 0; i 2 E are abbreviated as c E (z) 0 and similarly, c I (z) 0 are the inequality constraints. Problems of this type arise frequently in applications, see [15, 24, 26] for references. Problem (1.1) is also referred to as a Mathematical Program with Complementarity Constraints (MPCC) Clearly, any MPEC with a more general complementarity condition such as for instance 0 y F (x; y) 0, can be written in the form (1.1) by introducing slack variables. It is ....

....are investigate. The Penalty Interior Point Algorithm of [24] aims to maintain z 1 0 and z 2 0 whilst using an SQP approach to solving NLP (1.2) Unfortunately, PIPA can converge to non stationary points under certain circumstances [23] The implicit programming approach due to Zowe et. al [26] (see also Dirkse and Ferris [10] solves the lower level complementarity problem for xed controls and computes sensitivities of the controls. At the upper level, a nonsmooth optimization problem is then solved using bundle trust region techniques. Jiang and Ralph [22] propose two smooth SQP ....

Outrata, J., Kocvara, M. and Zowe, J. Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht, 1998.

....NP hard (see Hansen, Jaumard and Savard [6] and exact algorithms have to rely on enumeration procedures, such as branch and bound. In the nonlinear case, weak) optimality conditions have been derived and descent algorithms for nonsmooth formulations have been proposed (see Kocvara and Outrata [7] for instance) These however may fail to yield even local optimum for the bilevel program, without strong assumptions. An interesting class of bilevel problems is obtained by letting y denote the follower s vector corresponding to goods or activities; y 1 represents the subvector of goods (or ....

Kocvara, M. and Outrata, J.V., "A nonsmooth approach to optimization problems with equilibrium constraints", in Complementarity and variational problems. State of the art, M.C. Ferris and J.S. Pang eds., SIAM, Philadelphia (1997).

....method. Although this is a promising technique, multigrid methods are usually strongly coupled to the type of discretization used, and hence are complex to implement in general purpose software. There are a large number of general purpose methods for solving linear complementarity problems [22, 7, 25]. We can divide these methods up to into essentially two categories: direct methods, such as pivoting techniques [7] and iterative methods, such as Newton iteration [25] and interior point algorithms [22] Some of these methods which have been applied specifically to American option pricing ....

....general purpose software. There are a large number of general purpose methods for solving linear complementarity problems [22, 7, 25] We can divide these methods up to into essentially two categories: direct methods, such as pivoting techniques [7] and iterative methods, such as Newton iteration [25] and interior point algorithms [22] Some of these methods which have been applied specifically to American option pricing include linear programming [9] pivoting methods, 14] and interior point methods [15] As pointed out in [15] pivoting methods (such as Lemke s algorithm [7] and LP ....

[Article contains additional citation context not shown here]

J. Outrata, M. Kocvara, and J. Zowe. Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, 1998.

....generalized gradient in 1973, summarized in his seminal book [6] pioneered a rapid development, recently presented in detail in Rockafellar and Wets book [16] Computational methods for nonsmooth optimization have also developed rapidly, with many interesting applications. For a recent look, see [15], which focuses on mechanical applications, or [14] which concentrates on optimal control. Nonsmooth optimization algorithms such as the subgradient methods outlined in Shor s monograph [17] or the bundle methods of Lemar echal and Wolfe [2, 12] typically assume a locally Lipschitz function, ....

J. Outrata, M. Kocvara, and J. Zowe. Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, Dordrecht, 1998.

....In addition, there are classes of problems that are explicitly formulated as degenerate nonlinear programs and whose Lagrange multiplier set not only is not a singleton, but also is unbounded. One such type of nonlinear program is mathematical programs with equilibrium constraints, or MPECs [18, 19, 24]. The complementarity part of the equilibrium constraints generally violate the Mangasarian Fromovitz constraint qualification (MFCQ) 21] MFCQ, in turn, is equivalent to the boundedness of the constraints [11] which means that such MPECs will have an unbounded Lagrange multiplier set. One ....

....and superlinearly convergent under only the assumptions of quadratic growth and a nonempty but not necessarily bounded Lagrange multiplier set. An important class of problems that do not generally have bounded Lagrange multiplier sets are the mathematical programs with equilibrium constraints [18, 24]. The results are achieved by relaxing the constraints and adding a linear penalty term with a sufficiently large parameter c to the objective function. The effect of the penalty term is to retain from the Lagrange multipliers of the original problem only those whose 1 norm is less than or equal ....

J. Outrata, M. Kocvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic Publishers, Dordrecht, 1998. 19

....in turn makes it approachable by a sequential quadratic programming algorithm. The robustness of the elastic mode when applied to MPCCs is demonstrated by several numerical examples. 1. Introduction. Complementarity constraints can be used to model numerous economics or mechanics applications [18, 25]. Solving optimization problems with complementarity constraints may prove difficult for classical nonlinear optimization, however, given that, at a solution x , such a problem cannot satisfy a constraint qualification [18] As a result, algorithms based on the linearization of the feasible ....

....this may lead to a large number of subcases to account for the alternatives involving degenerate complementarity constraints. Still other approaches have been defined for problems whose complementarity constraints originate in equilibrium conditions [18] A nonsmooth approach has been proposed in [25] for MPCCs in which the underlying complementarity constraints are assumed to originate in a strongly regular variational inequality. A bundle trust region algorithm is used, in which each element of the bundle is generated from the generalized gradient of the reduced objective function. The key ....

[Article contains additional citation context not shown here]

J. Outrata, M. Kocvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic Publishers, Dordrecht, 1998.

.... some objective function : P Theta jAj Theta jCj 7 [ f 1g of the form (ae ; v ; d ) where (v ; d ) is a pair of equilibrium flows and demands given the parameter vector values ae , then we speak of a mathematical program with equilibrium constraints (MPEC) e.g. [LPR96, OKZ98]) For such problems, having sufficiently strong differentiability properties of the solution mapping S is essential for its efficient solution. Attempts to define descent algorithms for the minimization of in the context of transportation analysis have most often been based on either ignoring ....

J. Outrata, M. Ko cvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.

....investigated a topic which is not covered in this paper. The monograph [LPR96] includes several algorithms for this class of problems, such as the implicit programming algorithm for the nested problem, and the penalty interior point algorithm for the simultaneous version of the problem, cf. also [OKZ98, HKP99a]. The nested problem can in principle be treated by any method from nondifferentiable optimization, for example, bundle methods ( OKZ98] and subgradient methods ( CPW99] Another possibility is to use smoothing (cf. FJQ99, Hil99] that is, to replace the equilibrium problem by a sequence of ....

.... the implicit programming algorithm for the nested problem, and the penalty interior point algorithm for the simultaneous version of the problem, cf. also [OKZ98, HKP99a] The nested problem can in principle be treated by any method from nondifferentiable optimization, for example, bundle methods ([OKZ98]) and subgradient methods ( CPW99] Another possibility is to use smoothing (cf. FJQ99, Hil99] that is, to replace the equilibrium problem by a sequence of smooth approximations. Then any standard first order algorithm, such as sequential explicit approximation methods, can be used for the ....

J. Outrata, M. Ko cvara, and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic Publishers, Dordrecht, 1998.

....large part to the fact that the feasible region de ned by f(x; y) Z j y solves VI(C(x) F (x; g is not convex, and in some cases, is not even closed. Nevertheless, a number of reasonable algorithms exist for solving MPECs, and research in this area remains vigorous. The reader is referred to [74, 87] for detailed treatments on MPECs. ....

J. Outrata, M. Kocvara, and J. Zowe. Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.

....NP hard (see Hansen, Jaumard and Savard [6] and exact algorithms have to rely on enumeration procedures, such as branch and bound. In the nonlinear case, weak) optimality conditions have been derived and descent algorithms for nonsmooth formulations have been proposed (see Kocvara and Outrata [7] for instance) These however may fail to yield even local optimum for the bilevel program, without strong assumptions. An interesting class of bilevel problems is obtained by letting y denote the follower s vector corresponding to goods or activities; y 1 represents the subvector of goods (or ....

Kocvara, M. and Outrata, J.V., "A nonsmooth approach to optimization problems with equilibrium constraints", in Complementarity and variational problems. State of the art, M.C. Ferris and J.S. Pang eds., SIAM, Philadelphia (1997).

....In addition, there are classes of problems that are explicitly formulated as degenerate nonlinear programs and whose Lagrange multiplier set not only is not a singleton, but also is unbounded. One such type of nonlinear program is mathematical programs with equilibrium constraints, or MPECs [19, 20, 25]. The complementarity part of the equilibrium constraints generally violate the Mangasarian Fromovitz constraint qualification (MFCQ) 22] MFCQ, in turn, is equivalent to the boundedness of the constraints [12] which means that such MPECs will have an unbounded Lagrange multiplier set. One ....

....superlinearly con 18 vergent under only the assumptions of quadratic growth and a nonempty but not necessarily bounded Lagrange multiplier set. An important class of problems that do not generally have bounded Lagrange multiplier sets are the mathematical programs with equilibrium constraints [19, 25]. The results are achieved by relaxing the constraints and adding a linear penalty term with a su#ciently large parameter c to the objective function. The e#ect of the penalty term is to retain from the Lagrange multipliers of the original problem only those whose 1 norm is less than or equal to ....

J. Outrata, M. Kocvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic Publishers, Dordrecht, 1998.

..... p, h l (z) 0 l = 1, q, 1) where f is a real valued continuously di#erentiable function on # n and G i , H i , g j , h l are real valued a#ne functions on # n . This problem has been of much interest and many algorithms have been proposed for its solution, as is evidenced by [1, 2, 5, 7, 9] and the extensive references therein. However, these algorithms in general are only guaranteed to compute either a B stationary point under the nondegeneracy (strict complementarity) assumption that is somewhat restrictive in practice, or a C stationary point for the problem, rather than the ....

Outrata, J. V., Kocvara, M. and Zowe, J., Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic Publishers, Dordrecht, 1998.

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Outrata J V, Kocvara M, and Zowe J. Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Dordrecht, The Netherlands, Kluwer, 1998

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J. Outrata, M. Kocvara, and J. Zowe. Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.

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J. Outrata, M. Ko cvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998.

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J. V. Outrata, M. Ko cvara, and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results, Kluwer, Dordrecht, The Netherlands, 1998.

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J. Outrata, M. Kocvara, and J. Zowe. Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results. Kluwer Academic Publishers (Dordrecht 1998).

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J. Outrata, M. Kocvara, and J. Zowe. Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, Dordrecht, 1998.

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Outrata, J.V., Kocvara, M. and Zowe, J., Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht (1998).

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J. Outrata, M. Kocvara, and J. Zowe. Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht, 1998.

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