Ebook: Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods

The concept of "reformulation" has long been playing an important role in mathematical programming. A classical example is the penalization technique in constrained optimization that transforms the constraints into the objective function via a penalty function thereby reformulating a constrained problem as an equivalent or approximately equivalent unconstrained problem. More recent trends consist of the reformulation of various mathematical programming prob­ lems, including variational inequalities and complementarity problems, into equivalent systems of possibly nonsmooth, piecewise smooth or semismooth nonlinear equations, or equivalent unconstrained optimization problems that are usually differentiable, but in general not twice differentiable. Because of the recent advent of various tools in nonsmooth analysis, the reformulation approach has become increasingly profound and diversified. In view of growing interests in this active field, we planned to organize a cluster of sessions entitled "Reformulation - Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods" in the 16th International Symposium on Mathematical Programming (ismp97) held at Lausanne EPFL, Switzerland on August 24-29, 1997. Responding to our invitation, thirty-eight people agreed to give a talk within the cluster, which enabled us to organize thirteen sessions in total. We think that it was one of the largest and most exciting clusters in the symposium. Thanks to the earnest support by the speakers and the chairpersons, the sessions attracted much attention of the participants and were filled with great enthusiasm of the audience.

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The concept of `reformulation' has long played an important role in mathematical programming. A classical example is the penalization technique in constrained optimization. More recent trends consist of reformulation of various mathematical programming problems, including variational inequalities and complementarity problems, into equivalent systems of possibly nonsmooth, piecewise smooth or semismooth nonlinear equations, or equivalent unconstrained optimization problems that are usually differentiable, but in general not twice differentiable.
The book is a collection of peer-reviewed papers that cover such diverse areas as linear and nonlinear complementarity problems, variational inequality problems, nonsmooth equations and nonsmooth optimization problems, economic and network equilibrium problems, semidefinite programming problems, maximal monotone operator problems, and mathematical programs with equilibrium constraints.
The reader will be convinced that the concept of `reformulation' provides extremely useful tools for advancing the study of mathematical programming from both theoretical and practical aspects.
Audience: This book is intended for students and researchers in optimization, mathematical programming, and operations research.

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The concept of `reformulation' has long played an important role in mathematical programming. A classical example is the penalization technique in constrained optimization. More recent trends consist of reformulation of various mathematical programming problems, including variational inequalities and complementarity problems, into equivalent systems of possibly nonsmooth, piecewise smooth or semismooth nonlinear equations, or equivalent unconstrained optimization problems that are usually differentiable, but in general not twice differentiable.
The book is a collection of peer-reviewed papers that cover such diverse areas as linear and nonlinear complementarity problems, variational inequality problems, nonsmooth equations and nonsmooth optimization problems, economic and network equilibrium problems, semidefinite programming problems, maximal monotone operator problems, and mathematical programs with equilibrium constraints.
The reader will be convinced that the concept of `reformulation' provides extremely useful tools for advancing the study of mathematical programming from both theoretical and practical aspects.
Audience: This book is intended for students and researchers in optimization, mathematical programming, and operations research.
Content:
Front Matter....Pages i-viii
Solving Complementarity Problems by Means of a New Smooth Constrained Nonlinear Solver....Pages 1-24
A Non-Interior Predictor-Corrector Path-Following Method for LCP....Pages 25-43
Smoothing Newton Methods for Nonsmooth Dirichlet Problems....Pages 45-63
Frictional Contact Algorithms Based on Semismooth Newton Methods....Pages 65-79
Well-Posed Problems and Error Bounds in Optimization....Pages 81-116
Modeling and Solution Environments for MPEC: GAMS & MATLAB....Pages 117-126
Merit Functions and Stability for Complementarity Problems....Pages 127-147
Minimax and Triality Theory in Nonsmooth Variational Problems....Pages 149-159
Global and Local Superlinear Convergence Analysis of Newton-Type Methods for Semismooth Equations with Smooth Least Squares....Pages 161-179
Inexact Trust-Region Methods for Nonlinear Complementarity Problems....Pages 181-209
Regularized Newton Methods for Minimization of Convex Quadratic Splines with Singular Hessians....Pages 211-233
Regularized Linear Programs with Equilibrium Constraints....Pages 235-257
Reformulations of a Bicriterion Equilibrium Model....Pages 259-268
A Smoothing Function and Its Applications....Pages 269-291
On the Local Super-Linear Convergence of a Matrix Secant Implementation of the Variable Metric Proximal Point Algorithm for Monotone Operators....Pages 293-316
Reformulation of a Problem of Economic Equilibrium....Pages 317-334
A Globally Convergent Inexact Newton Method for Systems of Monotone Equations....Pages 335-354
On the Limiting Behavior of the Trajectory of Regularized Solutions of a P0-Complementarity Problem....Pages 355-369
Analysis of a Non-Interior Continuation Method Based on Chen-Mangasarian Smoothing Functions for Complementarity Problems....Pages 371-379
A New Merit Function and a Descent Method for Semidefinite Complementarity Problems....Pages 381-404
Numerical Experiments for a Class of Squared Smoothing Newton Methods for Box Constrained Variational Inequality Problems....Pages 405-420
Back Matter....Pages 421-441
....Pages 443-444