Werner Oettli

Time-dependent traffic equilibria

We consider the existence, characterization, and calculation of equilibria in transportation networks, when the route capacities and demand requirements depend on time. The problem is situated in a Banach space setting and formulated in terms of a variational inequality.

A Generalization of Vectorial Equilibria

A generalized form of vectorial equilibria is proposed, and, using an abstract monotonicity condition, an existence result is demonstrated.

Equivalents of Ekeland's principle

In this note we present a new result which is equivalent to the celebrated Ekelands variational principle, and a set of implications which includes a new non-convex minimisation principle due to Takahashi.

Existence of equilibria for monotone multivalued mappings

Abstract. Using a particular kind of pseudomonotonicity for multivalued mappings, an existence result is proved for equilibria, variational inequalities, and a combination of both.

Mathematical programs with a two-dimensional reverse convex constraint

We consider the problem min {f(x): x ∈ G, T(x) ∉ int D}, where f is a lower semicontinuous function, G a compact, nonempty set in ℝn, D a closed convex set in ℝ2 with nonempty interior and T a continuous mapping from ℝn to ℝ2. The constraint T(x) ∉ int D is a reverse convex constraint, so the feasible domain may be disconnected even when f, T are affine and G is a polytope. We show that this problem can be reduced to a quasiconcave minimization problem over a compact convex set in ℝ2 and hence can be solved effectively provided f, T are convex and G is convex or discrete. In particular we discuss a reverse convex constraint of the form 〈c, x〉 · 〈d, x〉≤1. We also compare the approach in this paper with the parametric approach.

Method for minimizing a convex-concave function over a convex set

A branch-and-bound method is proposed for minimizing a convex-concave function over a convex set. The minimization of a DC-function is a special case, where the subproblems connected with the bounding operation can be solved effectively.

The theorem of the alternative, the key-theorem, and the vector-maximum problem

Consequences of a general formulation of the theorem of the alternative are exploited.

Theorems of the alternative and duality for inf-sup problems

The duality theory of convex mathematical programming is extended to inf-sup problems. To this end a new, general inf-sup theorem for two different convexlike, respectively concavelike payoff functions is established under an abstract closedness assumption, thus avoiding the usual compactness requirement. This closedness assumption is then made more concrete; in particular for the partially homogeneous programs under study, regularity conditions of Karlin and Slater type are discussed. Finally, related theorems of the alternative as well as a result of Hahn-Banach type are derived, where the usual bilinear form is replaced by a more general coupling function.

Solvability of generalized nonlinear symmetric variational inequalities

This paper deals with the study of a general class of nonlinear variational inequalities. An existence result is given, and a perturbed iterative scheme is analyzed for solving such problems.

A Farkas lemma for difference sublinear systems and quasidifferentiable programming

A new generalized Farkas theorem of the alternative is presented for systems involving functions which can be expressed as the difference of sublinear functions. Various other forms of theorems of the alternative are also given using quasidifferential calculus. Comprehensive optimality conditions are then developed for broad classes of infinite dimensional quasidifferentiable programming problems. Applications to difference convex programming and infinitely constrained concave minimization problems are also discussed.