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25
Engineering and economic applications of complementarity problems
 SIAM Review
, 1997
"... Abstract. This paper gives an extensive documentation of applications of finitedimensional nonlinear complementarity problems in engineering and equilibrium modeling. For most applications, we describe the problem briefly, state the defining equations of the model, and give functional expressions f ..."
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Cited by 127 (24 self)
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Abstract. This paper gives an extensive documentation of applications of finitedimensional nonlinear complementarity problems in engineering and equilibrium modeling. For most applications, we describe the problem briefly, state the defining equations of the model, and give functional expressions for the complementarity formulations. The goal of this documentation is threefold: (i) to summarize the essential applications of the nonlinear complementarity problem known to date, (ii) to provide a basis for the continued research on the nonlinear complementarity problem, and (iii) to supply a broad collection of realistic complementarity problems for use in algorithmic experimentation and other studies.
LAGRANGE MULTIPLIERS AND OPTIMALITY
, 1993
"... Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions ..."
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Cited by 89 (7 self)
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Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger theoretical picture. A major line of research has been the nonsmooth geometry of onesided tangent and normal vectors to the set of points satisfying the given constraints. Another has been the gametheoretic role of multiplier vectors as solutions to a dual problem. Interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored. Lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows blackandwhite constraints to be replaced by penalty expressions. This paper traces such themes in the current theory of Lagrange multipliers, providing along the way a freestanding exposition of basic nonsmooth analysis as motivated by and applied to this subject.
MCPLIB: A Collection of Nonlinear Mixed Complementarity Problems
 Optimization Methods and Software
, 1994
"... The origins and some motivational details of a collection of nonlinear mixed complementarity problems are given. This collection serves two purposes. Firstly, it gives a uniform basis for testing currently available and new algorithms for mixed complementarity problems. Function and Jacobian evaluat ..."
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Cited by 65 (27 self)
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The origins and some motivational details of a collection of nonlinear mixed complementarity problems are given. This collection serves two purposes. Firstly, it gives a uniform basis for testing currently available and new algorithms for mixed complementarity problems. Function and Jacobian evaluations for the resulting problems are provided via a GAMS interface, making thorough testing of algorithms on practical complementarity problems possible. Secondly, it gives examples of how to formulate many popular problem formats as mixed complementarity problems and how to describe the resulting problems in GAMS format. We demonstrate the ease and power of formulating practical models in the MCP format. Given these examples, it is hoped that this collection will grow to include many problems that test complementarity algorithms more fully. The collection is available by anonymous ftp. Computational results using the PATH solver covering all of these problems are described. 1 Introduction R...
Generalized linearquadratic problems of deterministic and stochastic optimal control in discrete time
 SIAM J. Control Opt
, 1990
"... Abstract. Two fundamental classes of problems in largescale linear and quadratic programming are described. Multistage problems covering a wide variety of models in dynamic programming and stochastic programming are represented in a new way. Strong properties of duality are revealed which support t ..."
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Cited by 32 (7 self)
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Abstract. Two fundamental classes of problems in largescale linear and quadratic programming are described. Multistage problems covering a wide variety of models in dynamic programming and stochastic programming are represented in a new way. Strong properties of duality are revealed which support the development of iterative approximate techniques of solution in terms of saddlepoints. Optimality conditions are derived in a form that emphasizes the possibilities of decomposition.
Convergence rates in forwardbackward splitting
, 1989
"... Forwardbackward splitting methods provide a range of approaches to solving largescale optimization problems and variational inequalities in which structure conducive to decomposition can be utilized. Apart from special cases where the forward step is absent and a version of the proximal point alg ..."
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Cited by 21 (3 self)
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Forwardbackward splitting methods provide a range of approaches to solving largescale optimization problems and variational inequalities in which structure conducive to decomposition can be utilized. Apart from special cases where the forward step is absent and a version of the proximal point algorithm comes out, efforts at evaluating the convergence potential of such methods have so far relied on Lipschitz properties and strong monotonicity, or inverse strong monotonicity, of the mapping involved in the forward step, the perspective mainly being that of projection algorithms. Here convergence is analyzed by a technique that allows properties of the mapping in the backward step to be brought in as well. For the first time in such a general setting, global and local contraction rates are derived, moreover in a form making it possible to determine the optimal step size relative to certain constants associated with the given problem. Insights are thereby gained into the effects of shifting strong monotonicity between the forward and backward mappings when a splitting is selected.
Primaldual projected gradient algorithms for extended linearquadratic programming
 SIAM J. Optimization
"... Abstract. Many largescale problems in dynamic and stochastic optimization can be modeled with extended linearquadratic programming, which admits penalty terms and treats them through duality. In general the objective functions in such problems are only piecewise smooth and must be minimized or max ..."
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Cited by 16 (2 self)
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Abstract. Many largescale problems in dynamic and stochastic optimization can be modeled with extended linearquadratic programming, which admits penalty terms and treats them through duality. In general the objective functions in such problems are only piecewise smooth and must be minimized or maximized relative to polyhedral sets of high dimensionality. This paper proposes a new class of numerical methods for “fully quadratic ” problems within this framework, which exhibit secondorder nonsmoothness. These methods, combining the idea of finiteenvelope representation with that of modified gradient projection, work with local structure in the primal and dual problems simultaneously, feeding information back and forth to trigger advantageous restarts. Versions resembling steepest descent methods and conjugate gradient methods are presented. When a positive threshold of εoptimality is specified, both methods converge in a finite number of iterations. With threshold 0, it is shown under mild assumptions that the steepest descent version converges linearly, while the conjugate gradient version still has a finite termination property. The algorithms are designed to exploit features of primal and dual decomposability of the Lagrangian, which are typically available in a largescale setting, and they are open to considerable parallelization. Key words. Extended linearquadratic programming, largescale numerical optimization, finiteenvelope representation, gradient projection, primaldual methods, steepest descent methods, conjugate gradient methods. AMS(MOS) subject classifications. 65K05, 65K10, 90C20 1. Introduction. A
Robust Solution Of Mixed Complementarity Problems
, 1994
"... This thesis is concerned with algorithms and software for the solution of the Mixed Complementarity Problem, or MCP. The MCP formulation is useful for expressing systems of nonlinear inequalities and equations; the complementarity allows boundary conditions be to specified in a succinct manner. Prob ..."
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Cited by 9 (0 self)
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This thesis is concerned with algorithms and software for the solution of the Mixed Complementarity Problem, or MCP. The MCP formulation is useful for expressing systems of nonlinear inequalities and equations; the complementarity allows boundary conditions be to specified in a succinct manner. Problems of this type occur in many branches of the sciences, including mathematics, engineering, economics, operations research, and computer science. The algorithm we propose for the solution of MCP is a Newton based method containing a novel application of a nonmonotone stabilization technique previously applied to methods for solving smooth systems of equalities and for unconstrained minimization. In order to apply this technique, we have adapted and extended the path construction technique of Ralph (1994), resulting in the PATH algorithm. We present a global convergence result for the PATH algorithm that generalizes similar results obtained in the smooth case. The PATH solver is a sophisticated implementation of this algorithm that makes use of the sparse basis updating package of MINOS 5.4. Due to the widespread use of algebraic modeling languages in the practice of operations research, economics, and other fields from which complementarity problems are drawn, we have developed a complementarity facility for both the GAMS and AMPL modeling languages, as well as software interface libraries to be used in hooking up a complementarity solver as a solution subsystem. These interface libraries provide the algorithm developer with a convenient and efficient means of developing and testing an algorithm, ...
Operator Splitting Methods for Monotone Affine Variational Inequalities, with a Parallel Application to Optimal Control
 INFORMS J. Comput
, 1994
"... This paper applies splitting techniques developed for setvalued maximal monotone operators to monotone affine variational inequalities, including as a special case the classical linear complementarity problem. We give a unified presentation of several splitting algorithms for monotone operators, an ..."
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Cited by 8 (1 self)
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This paper applies splitting techniques developed for setvalued maximal monotone operators to monotone affine variational inequalities, including as a special case the classical linear complementarity problem. We give a unified presentation of several splitting algorithms for monotone operators, and then apply these results to obtain two classes of algorithms for affine variational inequalities. The second class resembles classical matrix splitting, but has a novel "underrelaxation " step, and converges under more general conditions. In particular, the convergence proofs do not require the affine operator to be symmetric. We specialize our matrixsplittinglike method to discretetime optimal control problems formulated as extended linearquadratic programs in the manner advocated by Rockafellar and Wets. The result is a highly parallel algorithm, which we implement and test on the Connection Machine CM5 computer family. The affine variational inequality problem is to find a vector x...
Computational schemes for largescale problems in extended linearquadratic programming
 Mathematical Programming
, 1990
"... Abstract. Numerical approaches are developed for solving largescale problems of extended linearquadratic programming that exhibit Lagrangian separability in both primal and dual variables simultaneously. Such problems are kin to largescale linear complementarity models as derived from application ..."
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Cited by 8 (1 self)
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Abstract. Numerical approaches are developed for solving largescale problems of extended linearquadratic programming that exhibit Lagrangian separability in both primal and dual variables simultaneously. Such problems are kin to largescale linear complementarity models as derived from applications of variational inequalities, and they arise from general models in multistage stochastic programming and discretetime optimal control. Because their objective functions are merely piecewise linearquadratic, due to the presence of penalty terms, they do not fit a conventional quadratic programming framework. They have potentially advantageous features, however, which so far have not been exploited in solution procedures. These features are laid out and analyzed for their computational potential. In particular, a new class of algorithms, called finiteenvelope methods, is described that does take advantage of the structure. Such methods reduce the solution of a highdimensional extended linearquadratic program to that of a sequence of lowdimensional ordinary quadratic programs.
A PredictorCorrector Algorithm For A Class Of Nonlinear Saddle Point Problems
 SIAM Journal on Control and Optimization
, 1994
"... . An interior pathfollowing algorithm is proposed for solving the nonlinear saddle point problem minimax c T x + OE(x) + b T y \Gamma /(y) \Gamma y T Ax subject to (x; y) 2 X \Theta Y ae R n \Theta R m ; where OE(x) and /(y) are smooth convex functions and X and Y are boxes (hyperrecta ..."
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Cited by 5 (2 self)
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. An interior pathfollowing algorithm is proposed for solving the nonlinear saddle point problem minimax c T x + OE(x) + b T y \Gamma /(y) \Gamma y T Ax subject to (x; y) 2 X \Theta Y ae R n \Theta R m ; where OE(x) and /(y) are smooth convex functions and X and Y are boxes (hyperrectangles). This problem is closely related to models in stochastic programming and optimal control studied by Rockafellar and Wets. Existence conditions on a central path are established. Starting from an initial solution near the central path with duality gap O(¯), the algorithm finds an ffloptimal solution of the problem in O( p m+ nj log ¯=fflj) iterations if both OE(x) and /(y) satisfy a scaled Lipschitz condition. Keywords. Interior point methods, optimal control, saddle point problem, stochastic programming. Abbreviated title. IP method for saddle point problems AMS subject classifications. 49J35, 65K10, 90C06, 90C15, 90C33 October, 1994 This research is partially supported by grant...