Franco Giannessi

Variational inequalities and network equilibrium problems

On a Separation Approach to Variational Inequalities C. Antoni. Traffic Scheduling in Telecommunication Systems and Network Flow M. Bonuccelli. On the Duality Theory for Finite Dimensional Variational Inequalities M. Castellani, G. Mastroeni. Some Properties of Periodic Solutions of Linear Control Systems via Quasivariational Inequalities P. Cubiotti. Generalized Quasivariational Inequalities and Traffic Equilibrium Problem M. De Luca. Vector Variational Inequality and Geometric Vector Optimization K.H. Elster, R. Elster. Testing a New Class of Algorithms for Nonlinear Complementarity Problems F. Facchinei, J. Soares. Equilibrium in Transport Networks with Capacity Constraints P. Ferrari. Separation of Sets and Gap Functions for Quasivariational Inequalities F. Giannessi. Stability of Monotone Variational Inequalities with Various Applications J. Gwinner. A Primaldual Proximal Point Algorithm for Variational Inequality Problems K. Iwaoka, et al. Relations between t, s, z-domain Descriptions of Periodically Switched Networks M. Koksal. On Side Constrained Models of Traffic Equilibria T. Larsson, M. Patriksson. Advantages and Drawbacks of Variational Inequalities Formulations P. Marcotte. 8 additional articles. Index.

Theorems of the alternative and optimality conditions

Several corrections to Ref. 1 are pointed out.

Equilibrium problems and variational models

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 -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 differentiability of the projection on closed convex subsets in Hilbert space.- 2.3. Time-dependent variational inequalities with memory terms.- 3. Ordinary Differential Inclusions with Convex Constraints: Sweeping Processes.- 3.1. Moving convex sets and systems with hysteresis.- 3.2. Sweeping processes and generalizations.- 4. Projected dynamical systems.- 4.1. Differentiability of the projection onto closed convex subsets revisited.- 4.2. Projected dynamical systems and stationarity.- 4.3. Well-posedness for projected dynamical systems.- 5. Some Asymptotic Results.- 5.1. Some classical results.- 5.2. An asymptotic result for full discretization.- 5.3. Some convergence results for continuous-time subgradient procedures for convex optimization.- References.- On the Contractibility of the Efficient and Weakly Efficient Sets in R2.- 1. Introduction.- 2. Preliminaries.- 3. Topological structure of the efficient sets of compact convex sets.- 4. Example.- References.- Existence Theorems for a Class of Variational Inequalities and Applications to a Continuous Model of Transportation.- 1. Introduction.- 2. Continuous transportation model.- 3. Existence Theorem.- References.- On Auxiliary Principle for Equilibrium Problems.- 1. Introduction.- 2. The auxiliary equilibrium problem.- 3. The auxiliary problem principle.- 4. Applications to variational inequalities and optimization problems.- 5. Concluding remarks.- References.- Multicriteria Spatial Price Networks: Statics and Dynamics.- 1. Introduction.- 2. The Multicriteria Spatial Price Model.- 3. Qualitative Properties.- 4. The Dynamics.- 5. The Discrete-Time Algorithm.- 6. Numerical Examples.- 7. Summary and Conclusions.- References.- Non regular data in unilateral variational problems.- 1. Introduction.- 2. The approach by truncation and approximation.- 3. Renormalized formulation.- 4. Multivalued operators and more general measures.- 5. Uniqueness and convergence.- References.- Equilibrium Concepts in Transportation Networks: Generalized Wardrop Conditions and Variational Formulations.- 1. Introduction.- 2. Equilibrium model in a traffic network.- References.- Variational Geometry and Equilibrium.- 1. Introduction.- 2. Variational Inequalities and Normals to Convex Sets.- 3. Quasi-Variational Inequalities and Normals to General Sets.- 4. Calculus and Solution Perturbations.- 5. Application to an Equilibrium Model with Aggregation.- References.- On the Calculation of Equilibrium in Time Dependent Traffic Networks.- 1. Introduction.- 2. Calculation of Equilibria.- 3. The algorithm.- 4. Applications and Examples.- 5. Conclusions.- References.- Mechanical Equilibrium and Equilibrium Systems.- 1. Introduction.- 2. Physical motivation.- 3. Statement of the mechanical force equilibrium problem.- 4. The principle of virtual work.- 5. Characterization of the constraints.- 6. Quasi-variational inequalities (QVI).- 7. Principle of virtual work in force fields under scleronomic and holonomic constraints.- 8. Dual form of the principle of virtual work in force field under scleronomic and holonomic constraints.- 9. Procedure for solving mechanical equilibrium problems.- 10. Existence of solutions.- References.- False Numerical Convergence in Some Generalized Newton Methods.- 1. Introduction.- 2. Some generalized Newton methods.- 3. False numerical convergence.- 4. An example.- 5. Avoiding false numerical convergence.- References.- Distance to the Solution Set of an Inequality with an Increasing Function.- 1. Introduction.- 2. Preliminaries.- 3. Distance to the solution set of the inequality with an arbitrary increasing function.- 4. Distance to the solution set of the inequality with an ICAR function.- 5. Inequalities with an increasing function defined on the entire space.- 6. Inequalities with a topical function.- References.- Transportation Networks with Capacity Constraints.- 1. Introduction.- 2. Wardrops generalized equilibrium condition.- 3. A triangular network.- 4. More about generalized equilibrium principle.- 5. Capacity constraints and paradox.- References.

On Minty Variational Principle

The well known Minty Variational Inequality is extended to the vector case. Such an inequality is shown to recover Vector Optimization problems in the convex case.

On the Theory of Vector Optimization and Variational Inequalities. Image Space Analysis and Separation

By exploiting recent results, it is shown that the theories of Vector Optimization and of Vector Variational Inequalities can be based on the image space analysis and theorems of the alternative or separation theorems. It is shown that, starting from such a general scheme, several theoretical aspects can be developed - like optimality conditions, duality, penalization - as well as methods of solution - like scalarization.

Semidifferentiable functions and necessary optimality conditions

In the last two decades, there has been an increasing interest in nonsmooth optimization, both from a theoretical viewpoint and because of several applications. Necessary optimality conditions, as well as other important topics, have received new attention (see, for instance, Refs. 1–11 and references therein).In recent papers (see, for instance, Refs. 12–18 and references therein), theorems of the alternative for generalized systems have been studied and their use in optimization has been exploited. As a consequence of this analysis, the concept of image of a constrained extremum problem has been developed; such a concept, whose introduction goes back to the work of Carathéodory, has only recently been recognized to be a powerful tool (Refs. 8, 10, 13, 14, 18–21). On the basis of these ideas, in the present paper we deal with a necessary condition for constrained extremum problems having a finite-dimensional image, while those having an infinite-dimensional one will be treated in a subsequent paper. The necessary condition is established within a class of semidifferentiable functions, which is introduced here and which embraces several classic types of functions (e.g., convex functions, differentiable functions, and even some discontinuous functions). The condition embodies the classic theorems of Lagrange, John, Karush, Kuhn-Tucker, and Euler.

Nonlinear Optimization and Related Topics

Preface. Generalized Lagrange multipliers: regularity and boundedness G. Bigi, M. Pappalardo. A primal-dual algorithm for minimizing a non-convex function subject to bound and linear equality constraints A.R. Conn, et al. Minimal convexificators of a positively homogeneous function and a characterization of its convexity and concavity V.F. Demyanov. Optimal control problems and penalization V.F. Demyanov, et al. A truncated Newton method for constrained optimization G. Di Pillo, et al. Fixed and virtual stability center methods for convex nonsmooth minimization A. Fuduli, M. Gaudioso. Iterative methods for ill-conditioned linear systems from optimization N.I.M. Gould. An algorithm for solving nonlinear programs with noisy inequality constraints M. Hintermuller. Generic existence uniqueness and stability in optimization problems A. Ioffe, R. Lucchetti. On a class of bilevel programs M. Labbe, et al. Separation methods for vector variational inequalities. Saddle point and gap function G. Mastroeni. Functions with primal-dual gradient structure and U-Hessians R. Mifflin, C. Sagastizabal. Quadratic and multidimensional assignment problems P.M. Pardalos, L.S. Pitsoulis. A new merit function and an SQP method for non-strictly monotone variational inequalities M. Patriksson. A logarithmic barrier approach to Fischer function J. Peng, et al. On an approach to optimization problems with a probabilistic cost and or constraints E. Polak, et al. Semiderivative functions and reformulation methods for solving complementarity and variational inequality problems L. Qi, et al. Global Lagrange multiplier rule and smooth exact penalty functions for equality constraints T. Rapcsak. Structural methods in thesolution of variational inequalities S.M. Robinson. Extended nonlinear programming R.T. Rockafellar. On the efficiency of splitting and projection methods for large strictly convex quadratic programs V. Ruggiero, L. Zanni. A comparison of rates of convergence of two inexact proximal point algorithms M.V. Solodov, et al. One way to construct a global search algorithm for d. c. minimization problems A.S. Strekalovsky. Error bounds and superlinear convergence analysis of some Newton-type methods in optimization P. Tseng. A new derivative-free descent method for the nonlinear complementarity problem K. Yamada, et al.

Theorems of the alternative for multifunctions with applications to optimization: general results

Theorems of the alternative and separation theorems have been shown to be very useful concepts in constrained extremum problems (see, for instance, Refs. 1–12). Their use has stressed the concept of image of a constrained extremum problem, which has turned out to be a powerful and promising tool for investigating the main aspects of optimization (see Refs. 13 and 19). It should be pointed out that, in this approach, a finite-dimensional image problem can be associated to the given extremum problem, even if this is infinite-dimensional and provided that its constraints are expressed by functionals. Such a development can be carried on by means of theorems of the alternative for systems of single-valued functions.In this paper, theorems of the alternative for systems of multifunctions are studied, some general properties are stated, and connections with known results investigated. It is shown how the present approach can be used to analyze extremum problems, where the image of the domain of the constraining functions belongs to a functional space. Such a development will be carried on in a subsequent paper.

Separation of sets and Wolfe duality

Lagrangian duality can be derived from separation in the Image Space, namely the space where the images of the objective and constraining functions of the given extremum problem run. By exploiting such a result, we analyse the relationships between Wolfe and Mond-Weir duality and prove their equivalence in the Image Space under suitable generalized convexity assumptions.

On the theory of Lagrangian duality

It is shown that a general Lagrangian duality theory for constrained extremum problems can be drawn from a separation scheme in the Image Space, namely in the space where the functions of the given problem run.