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105
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.
Optimization by direct search: New perspectives on some classical and modern methods
 SIAM Review
, 2003
"... Abstract. Direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives. Direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because t ..."
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Cited by 126 (14 self)
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Abstract. Direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives. Direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis. Nonetheless, users remained loyal to these methods, most of which were easy to program, some of which were reliable. In the past fifteen years, these methods have seen a revival due, in part, to the appearance of mathematical analysis, as well as to interest in parallel and distributed computing. This review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited. Our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results. The underlying principles allow generalization to handle bound constraints and linear constraints. We also discuss extensions to problems with nonlinear constraints.
Feature Selection via Mathematical Programming
, 1997
"... The problem of discriminating between two finite point sets in ndimensional feature space by a separating plane that utilizes as few of the features as possible, is formulated as a mathematical program with a parametric objective function and linear constraints. The step function that appears in th ..."
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Cited by 59 (22 self)
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The problem of discriminating between two finite point sets in ndimensional feature space by a separating plane that utilizes as few of the features as possible, is formulated as a mathematical program with a parametric objective function and linear constraints. The step function that appears in the objective function can be approximated by a sigmoid or by a concave exponential on the nonnegative real line, or it can be treated exactly by considering the equivalent linear program with equilibrium constraints (LPEC). Computational tests of these three approaches on publicly available realworld databases have been carried out and compared with an adaptation of the optimal brain damage (OBD) method for reducing neural network complexity. One feature selection algorithm via concave minimization (FSV) reduced crossvalidation error on a cancer prognosis database by 35.4% while reducing problem features from 32 to 4. Feature selection is an important problem in machine learning [18, 15, 1...
Algorithms For Complementarity Problems And Generalized Equations
, 1995
"... Recent improvements in the capabilities of complementarity solvers have led to an increased interest in using the complementarity problem framework to address practical problems arising in mathematical programming, economics, engineering, and the sciences. As a result, increasingly more difficult pr ..."
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Cited by 41 (5 self)
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Recent improvements in the capabilities of complementarity solvers have led to an increased interest in using the complementarity problem framework to address practical problems arising in mathematical programming, economics, engineering, and the sciences. As a result, increasingly more difficult problems are being proposed that exceed the capabilities of even the best algorithms currently available. There is, therefore, an immediate need to improve the capabilities of complementarity solvers. This thesis addresses this need in two significant ways. First, the thesis proposes and develops a proximal perturbation strategy that enhances the robustness of Newtonbased complementarity solvers. This strategy enables algorithms to reliably find solutions even for problems whose natural merit functions have strict local minima that are not solutions. Based upon this strategy, three new algorithms are proposed for solving nonlinear mixed complementarity problems that represent a significant improvement in robustness over previous algorithms. These algorithms have local Qquadratic convergence behavior, yet depend only on a pseudomonotonicity assumption to achieve global convergence from arbitrary starting points. Using the MCPLIB and GAMSLIB test libraries, we perform extensive computational tests that demonstrate the effectiveness of these algorithms on realistic problems. Second, the thesis extends some previously existing algorithms to solve more general problem classes. Specifically, the NE/SQP method of Pang & Gabriel (1993), the semismooth equations approach of De Luca, Facchinei & Kanz...
The essence of ESSENCE: A constraint language for specifying combinatorial problems
 In Proceedings of the 20th International Joint Conference on Artificial Intelligence
, 2005
"... Abstract. Essence is a new language for specifying combinatorial (decision or optimisation) problems at a high level of abstraction. The key feature enabling this abstraction is the provision of decision variables whose values can be combinatorial objects, such as tuples, sets, multisets, relations, ..."
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Cited by 40 (10 self)
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Abstract. Essence is a new language for specifying combinatorial (decision or optimisation) problems at a high level of abstraction. The key feature enabling this abstraction is the provision of decision variables whose values can be combinatorial objects, such as tuples, sets, multisets, relations, partitions and functions. Essence also allows these combinatorial objects to be nested to arbitrary depth, thus providing, for example, sets of partitions, sets of sets of partitions, and so forth. 1
QPCOMP: A Quadratic Programming Based Solver for Mixed Complementarity Problems
 Mathematical Programming
, 1997
"... QPCOMP is an extremely robust algorithm for solving mixed nonlinear complementarity problems that has fast local convergence behavior. Based in part on the NE/SQP method of Pang and Gabriel[14], this algorithm represents a significant advance in robustness at no cost in efficiency. In particular, th ..."
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Cited by 31 (15 self)
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QPCOMP is an extremely robust algorithm for solving mixed nonlinear complementarity problems that has fast local convergence behavior. Based in part on the NE/SQP method of Pang and Gabriel[14], this algorithm represents a significant advance in robustness at no cost in efficiency. In particular, the algorithm is shown to solve any solvable Lipschitz continuous, continuously differentiable, pseudomonotone mixed nonlinear complementarity problem. QPCOMP also extends the NE/SQP method for the nonlinear complementarity problem to the more general mixed nonlinear complementarity problem. Computational results are provided, which demonstrate the effectiveness of the algorithm. 1 Introduction This paper describes a new algorithm for solving the mixed nonlinear complementarity problem (MCP), which provides a significant improvement in robustness over previous superlinearly or quadratically convergent algorithms, while preserving these fast local convergence properties. The MCP is defined in...
The design of ESSENCE: a constraint language for specifying combinatorial problems
 In: Proceedings of IJCAI07
, 2007
"... ESSENCE is a new formal language for specifying combinatorial problems in a manner similar to natural rigorous specifications that use a mixture of natural language and discrete mathematics. ESSENCE provides a high level of abstraction, much of which is the consequence of the provision of decision v ..."
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Cited by 31 (2 self)
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ESSENCE is a new formal language for specifying combinatorial problems in a manner similar to natural rigorous specifications that use a mixture of natural language and discrete mathematics. ESSENCE provides a high level of abstraction, much of which is the consequence of the provision of decision variables whose values can be combinatorial objects, such as tuples, sets, multisets, relations, partitions and functions. ESSENCE also allows these combinatorial objects to be nested to arbitrary depth, thus providing, for example, sets of partitions, sets of sets of partitions, and so forth. Therefore, a problem that requires finding a complex combinatorial object can be directly specified by using a decision variable whose type is precisely that combinatorial object. 1
FATCOP: A fault tolerant CondorPVM mixed integer program solver. Mathematical Programming
, 1999
"... Abstract. We describe FATCOP, a new parallel mixed integer program solver written in PVM. The implementation uses the Condor resource management system to provide a virtual machine composed of otherwise idle computers. The solver differs from previous parallel branchandbound codes by implementing ..."
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Cited by 27 (4 self)
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Abstract. We describe FATCOP, a new parallel mixed integer program solver written in PVM. The implementation uses the Condor resource management system to provide a virtual machine composed of otherwise idle computers. The solver differs from previous parallel branchandbound codes by implementing a general purpose parallel mixed integer programming algorithm in an opportunistic multiple processor environment, as opposed to a conventional dedicated environment. It shows how to make effective use of resources as they become available while ensuring the program tolerates resource retreat. The solver performs well on test problems arising from real applications and is particularly useful for solving long running hard mixed integer programming problems.
Disciplined convex programming
 Global Optimization: From Theory to Implementation, Nonconvex Optimization and Its Application Series
, 2006
"... ..."
A globally convergent linearly constrained Lagrangian method for nonlinear optimization
 SIAM J. Optim
, 2002
"... Abstract. For optimization problems with nonlinear constraints, linearly constrained Lagrangian (LCL) methods solve a sequence of subproblems of the form “minimize an augmented Lagrangian function subject to linearized constraints. ” Such methods converge rapidly near a solution but may not be relia ..."
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Cited by 22 (5 self)
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Abstract. For optimization problems with nonlinear constraints, linearly constrained Lagrangian (LCL) methods solve a sequence of subproblems of the form “minimize an augmented Lagrangian function subject to linearized constraints. ” Such methods converge rapidly near a solution but may not be reliable from arbitrary starting points. Nevertheless, the wellknown software package MINOS has proved effective on many large problems. Its success motivates us to derive a related LCL algorithm that possesses three important properties: it is globally convergent, the subproblem constraints are always feasible, and the subproblems may be solved inexactly. The new algorithm has been implemented in Matlab, with an option to use either MINOS or SNOPT (Fortran codes) to solve the linearly constrained subproblems. Only first derivatives are required. We present numerical results on a subset of the COPS, HS, and CUTE test problems, which include many large examples. The results demonstrate the robustness and efficiency of the stabilized LCL procedure.