On Vector Quasi-Equilibrium Problems.- 1. Introduction.- 2. Preliminaries.- 3. Existence Results.- 4. Some Applications.- References.- The Log-Quadratic Proximal Methodology in Convex Optimization Algorithms and Variational Inequalities.- 1. Introduction.- 2. Lagrangians and Proximal Methods.- 2.1. The quadratic augmented Lagrangian.- 2.2. Proximal Minimization Algorithms.- 2.3. Entropic Proximal Methods and Modified Lagrangians.- 2.4. Difficulties with Entropic Proximal Methods.- 2.5. Toward Solutions to Difficulties.- 3. The Logarithmic-Quadratic Proximal Framework.- 3.1. The LQ-Function and its Conjugate: Basic Properties.- 3.2. The Logarithmic-Quadratic Proximal Minimization.- 4. The LQP in Action.- 4.1. Primal LQP for Variational Inequalities over Polyhedra.- 4.2. Lagrangian Methods for convex optimization and variational inequalities.- 4.3. Dual and Primal-Dual Decomposition schemes.- 4.4. Primal Decomposition: Block Gauss-Seidel Schemes for Linearly constrained Problems.- 4.5. Convex Feasibility Problems.- 4.6. Bundle Methods in Nonsmooth Optimization.- References.- The Continuum Model of Transportation Problem.- 1. Introduction.- 2. Calculus of the solution.- References.- The Economic Model for Demand-Supply Problems.- 1. Introduction.- 2. The first phase: formalization of the equilibrium.- 3. The second phase: formalization of the equilibrium.- 4. The dependence of the second phase on the first one.- 5. General model.- 6. Example.- References.- Constrained Problems of Calculus of Variations Via Penalization Technique.- 1. Introduction.- 2. Statement of the problem.- 3. An equivalent statement of the problem.- 4. Local minima.- 5. Penalty functions.- 6. Exact penalty functions.- 6.1. Properties of the function ?.- 6.2. Properties of the function G.- 6.3. The rate of descent of the function ?.- 6.4. An Exact Penalty function.- 7. Necessary conditions for an Extremum.- 7.1. Necessary conditions generated by classical variations.- 7.2. Discussion and Remarks.- References.- Variational Problems with Constraints Involving Higher-Order Derivatives.- 1. Introduction.- 2. Statement of the problem.- 3. An equivalent statement of the problem.- 4. Local minima.- 5. Properties of the function ?.- 5.1. A classical variation of z.- 5.2. The case z ? Z.- 5.3. The case z ? Z.- 6. Exact penalty functions.- 6.1. Properties of the function G.- 6.2. An Exact Penalty function.- 7. Necessary conditions for an Extremum.- References.- On the strong solvability of a unilateral boundary value problem for Nonlinear Parabolic Operators in the Plane.- 1. Introduction.- 2. Hypotheses and results.- 3. Preliminary results.- 4. Proof of the theorems.- References.- Solving a Special Class of Discrete Optimal Control Problems Via a Parallel Interior-Point Method.- 1. Introduction.- 2. Framework of the Method.- 3. Global convergence.- 4. A special class of discrete optimal control problems.- 5. Numerical experiments.- 6. Conclusions.- References.- Solving Large Scale Fixed Charge Network Flow Problems.- 1. Introduction.- 2. Problem Definition and Formulation.- 3. Solution Procedure.- 3.1. The DSSP.- 3.2. Local Search.- 4. Computational Results.- 5. Concluding Remarks.- References.- Variable Projection Methods for Large-Scale Quadratic Optimization in data Analysis Applications.- 1. Introduction.- 2. Large QP Problems in Training Support Vector Machines.- 3. Numerical Solution of Image Restoration Problem.- 4. A Bivariate Interpolation Problem.- 5. Conclusions.- References.- Strong solvability of boundary value problems in elasticity with Unilateral Constraints.- 1. Introduction.- 2. Basic assumptions and main results.- 3. Preliminary results.- 4. Proof of the theorems.- References.- Time Dependent Variational Inequalities - Some Recent Trends.- 1. Introduction.- 2. Time - an additional parameter in variational inequalities.- 2.1. Time-dependent variational inequalities and quasi-variational inequalities.- 2.2. Some classic results on the differentiab
The volume, devoted to variational analysis and its applications, collects selected and refereed contributions, which provide an outline of the field. The meeting of the title "Equilibrium Problems and Variational Models", which was held in Erice (Sicily) in the period June 23 - July 2 2000, was the occasion of the presentation of some of these papers; other results are a consequence of a fruitful and constructive atmosphere created during the meeting. New results, which enlarge the field of application of variational analysis, are presented in the book; they deal with the vectorial analysis, time dependent variational analysis, exact penalization, high order deriva tives, geometric aspects, distance functions and log-quadratic proximal methodology. The new theoretical results allow one to improve in a remarkable way the study of significant problems arising from the applied sciences, as continuum model of transportation, unilateral problems, multicriteria spatial price models, network equilibrium problems and many others. As noted in the previous book "Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models", edited by F. Giannessi, A. Maugeri and P.M. Pardalos, Kluwer Academic Publishers, Vol. 58 (2001), the progress obtained by variational analysis has permitted to han dle problems whose equilibrium conditions are not obtained by the mini mization of a functional. These problems obey a more realistic equilibrium condition expressed by a generalized orthogonality (complementarity) con dition, which enriches our knowledge of the equilibrium behaviour. Also this volume presents important examples of this formulation.
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