/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

This paper describes modern techniques used to solve contact problems within Computational Mechanics. On the basis of a continuum description of contact, the mathematical structure of the contact formulation for finite element methods is derived. Emphasis is also placed on the constitutive behavior at the contact interface for normal and tangential (frictional) contact. Furthermore, different discretization schemes currently applied to solve engineering problems are formulated for small and finite strain problems. These include isoparametric interpolations, node-to-segment discretizations and also mortar and Nitsche techniques. Furthermore, several algorithms related to spatial contact search and fulfillment of the inequality constraints at the contact interface are discussed. Here, especially the penalty and Lagrange multiplier schemes are considered and also SQP- and linear-programming methods are reviewed.


Keywords:

contact mechanics;
friction;
penalty method;
Lagrange multiplier method;
contact algorithms;
finite element method;
finite deformations;
discretization methods

Computational Contact Mechanics

This paper is devoted to bilevel optimization, a branch of mathematical programming of both practical and theoretical interest. Starting with a simple example, we proceed towards a general formulation. We then present fields of application, focus on solution approaches, and make the connection with MPECs (Mathematical Programs with Equilibrium Constraints).

/pdf/an-overview-of-bilevel-optimization-4x7xo28qzy.pdf

An overview of bilevel optimization

Nonlinear Programming: A Unified Approach

This paper gives an extensive documentation of applications of finite-dimensional 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.

/pdf/engineering-and-economic-applications-of-complementarity-1nz165yv0c.pdf

Engineering and Economic Applications of Complementarity Problems

The class of simplicial decomposition (SD) schemes has shown to provide efficient tools for nonlinear network flows. When applied to the traffic assignment problem, shortest route subproblems are solved in order to generate extreme points of the polyhedron of feasible flows, and, alternately, master problems are solved over the convex hull of the generated extreme points. We review the development of simplicial decomposition and the closely related column generation methods for the traffic assignment problem; we then present a modified, disaggregated, representation of feasible solutions in SD algorithms for convex problems over Cartesian product sets, with application to the symmetric traffic assignment problem. The new algorithm, which is referred to as disaggregate simplicial decomposition (DSD), is given along with a specialized solution method for the disaggregate master problem. Numerical results for several well known test problems and a new one are presented. These experimentations indicate that o...

Simplicial decomposition with disaggregated representation for the traffic assignment problem

As a means to obtain a more accurate description of traffic flows than that provided by the basic model of traffic assignment, there have been suggestions to impose upper bounds on the link flows. This can be done either by introducing explicit link capacities or by employing travel time functions with asymptotes at the upper bounds. Although the latter alternative has the disadvantage of inherent numerical ill-conditioning, the capacitated assignment model has been studied and applied to a limited extent, the main reason being that the solutions can not be characterized by the classical Wardrop equilibrium conditions; they may, however, be characterized as Wardrop equilibria in terms of a well-defined, natural generalized travel cost. The introduction of link capacity side constraints makes the problem computationally more demanding. The availability of efficient algorithms for the basic model of traffic assignment motivates the use of dualization approaches for handling the capacity constraints. We propose and evaluate an augmented Lagrangean dual method in which the uncapacitated traffic assignment subproblems are solved with the disaggregate simplicial decomposition algorithm. This algorithm fully exploits the subproblem's structure and has very favourable reoptimization capabilities; both these properties are necessary for achieving computational efficiency in iterative dualization schemes. The dual method exhibits a linear rate of convergence under a standard nondegeneracy assumption. The efficiency of the overall algorithm is demonstrated through experiments with capacitated versions of well-known test problems, with the conclusion that the introduction of link capacities increases the computing times with no more than a factor of four. The introduction of capacities and the algorithm suggested can be used to derive tolls for the reduction of flows on overloaded links. The solution strategy can be applied also to other types of traffic assignment models where side constraints have been added in order to refine a descriptive or prescriptive assignment model.

https://publications.lib.chalmers.se/records/fulltext/local_141642.pdf

An augmented lagrangean dual algorithm for link capacity side constrained traffic assignment problems

We consider the introduction of side constraints for refining a descriptive or prescriptive traffic equilibrium assignment model, and analyze a general such a model. Side constraints can be introduced for several diverse reasons; we consider three basic ones. First, they can be used to describe the effects of a traffic control policy. Second, they can be used to improve an existing traffic equilibrium model for a given application by introducing, through them, further information about the traffic flow situation at hand. As such, these two strategies complement the refinement strategy based on the use of non-separable, and typically asymmetric, travel cost functions. Third, they can be used to describe flow restrictions that a central authority wishes to impose upon the users of the network. We study a general convexly side constrained traffic equilibrium assignment model, and establish several results pertaining to the above described areas of application. First, for the case of prescriptive side constraints that are associated with queueing effects, for example those describing signal controls, we establish a characterization of the solutions to the model through a Wardrop user equilibrium principle in terms of generalized travel costs and an equilibrium queueing delay result; in traffic networks with queueing the solutions may therefore be characterized as Wardrop equilibria in terms of well-defined and natural travel costs. Second, we show that the side constrained problem is equivalent to an equilibrium model with travel cost functions properly adjusted to take into account the information introduced through the side constraints. Third, we show that the introduction of side constraints can be used as a means to derive the link tolls that should be levied in order to achieve a set of traffic management goals. The introduction of side constraints makes the problem computationally more demanding, but this drawback can to some extent be overcome through the use of dualization approaches, which we also briefly discuss.

http://publications.lib.chalmers.se/records/fulltext/local_141626.pdf

Side constrained traffic equilibrium models: analysis, computation and applications

Lagrangean dualization and subgradient optimization techniques are frequently used within the field of computational optimization for finding approximate solutions to large, structured optimization problems. The dual subgradient scheme does not automatically produce primal feasible solutions; there is an abundance of techniques for computing such solutions (via penalty functions, tangential approximation schemes, or the solution of auxiliary primal programs), all of which require a fair amount of computational effort. We consider a subgradient optimization scheme applied to a Lagrangean dual formulation of a convex program, and construct, at minor cost, an ergodic sequence of subproblem solutions which converges to the primal solution set. Numerical experiments performed on a traffic equilibrium assignment problem under road pricing show that the computation of the ergodic sequence results in a considerable improvement in the quality of the primal solutions obtained, compared to those generated in the basic subgradient scheme.

http://www.math.chalmers.se/Math/Grundutb/CTH/tma521/0607/erg.pdf

Ergodic, primal convergence in dual subgradient schemes for convex programming

Recently Auchmuty (1989) has introduced a new class of merit functions, or optimization formulations, for variational inequalities in finite-dimensional space. We develop and generalize Auchmuty's results, and relate his class of merit functions to other works done in this field. Especially, we investigate differentiability and convexity properties, and present characterizations of the set of solutions to variational inequalities. We then present new descent algorithms for variational inequalities within this framework, including approximate solutions of the direction finding and line search problems. The new class of merit functions include the primal and dual gap functions, introduced by Zuhovickii et al. (1969a, 1969b), and the differentiable merit function recently presented by Fukushima (1992); also, the descent algorithm proposed by Fukushima is a special case from the class of descent methods developed in this paper. Through a generalization of Auchmuty's class of merit functions we extend those inherent in the works of Dafermos (1983), Cohen (1988) and Wu et al. (1991); new algorithmic equivalence results, relating these algorithm classes to each other and to Auchmuty's framework, are also given.

https://publications.lib.chalmers.se/records/fulltext/local_141647.pdf

Torbjörn Larsson

Papers

Simplicial decomposition with disaggregated representation for the traffic assignment problem

An augmented lagrangean dual algorithm for link capacity side constrained traffic assignment problems

Side constrained traffic equilibrium models: analysis, computation and applications

Ergodic, primal convergence in dual subgradient schemes for convex programming

A class of gap functions for variational inequalities