informs ® doi 10.1287/trsc.1030.0043 © 2004 INFORMS The contribution of the paper is a complete analysis of the sensitivity of elastic demand traffic (Wardrop) equilibria. The existence of a directional derivative of the equilibrium solution (link flow, least travel cost, demand) in any direction is given a characterization, and the same is done for its gradient. The gradient, if it exists, is further interpreted as a limiting case of the gradient of the logit-based SUE solution, as the dispersion parameter tends to infinity. In the absence of the gradient, we show how to compute a subgradient. All these computations (directional derivative, (sub)gradient) are performed by solving similar traffic equilibrium problems with affine link cost and demand functions, and they can be performed by the same tool as (or one similar to) the one used for the original traffic equilibrium model; this fact is of clear advantage when applying sensitivity analysis within a bilevel (or mathematical program with equilibrium constraints, MPEC) application, such as for congestion pricing, OD estimation, or network design. A small example illustrates the possible nonexistence of a gradient and the computation of a subgradient. Key words: traffic equilibrium; stochastic user equilibrium; sensitivity analysis; directional derivative; bilevel optimization

In the broad context of modeling for system design, it is normally assumed that all decision-makers cooperate fully and thus avoid conflict. However, this is not always possible, in which case the design process is best modelled and studied as a multi-player game. The objective in this paper is to present and illustrate such a general game-theoretic framework for design modeling. In particular, explicit solutions are computed and interpreted for conservative or minmax, Pareto, Nash and Stackelberg games in a simple pressure vessel design example in which two players interact strategically, one of whom wants to minimize the weight and other wishes to maximize the volume. In another example in which a rotating disk is to be optimized for two objectives related to its design and manufacture, the Pareto solutions show how a novel manufacturing cost function can improve, under assumptions of a cooperative game, the optimal shape that is obtained under a single objective formulation related ...

In [1], a modified relaxation method was proposed for mathematical programs with complementarity constraints and some new sufficient conditions for M- or B-stationarity were shown. However, due to an ignored sign in the Lagrangian function of the relaxed problem, the proofs of Theorems 3.4 and 3.5 in [1] are incorrect. In what follows, we give the corrected proofs. Throughout, we use the same notations as in [1]. Theorem 3.4. Let {ɛk} ⊆ (0, +∞) be convergent to 0 and z k ∈ Fɛk be a stationary point of problem (3) with ɛ = ɛk and multiplier vectors λ k, µ k, δ k, and γ k. Suppose that, for each k, ∇ 2 zLɛk (zk, λ k, µ k, δ k, γ k) is bounded below with constant αk on the corresponding tangent space Tɛk (zk). Let ¯z be an accumulation point of the sequence {z k}. If the sequence {αk} is bounded and the MPEC-LICQ holds at ¯z, then ¯z is an M-stationary point of problem (1). Proof. Assume that limk→ ∞ z k = ¯z without loss of generality. First of all, we note from Theorem 3.3 that ¯z is a C-stationary point of problem (1). To prove the theorem, we assume to the contrary that ¯z is not M-stationary to problem (1). Then, it follows from the definitions of C-stationarity and M-stationarity that there must exist an i0 ∈ IG(¯z) ∩ IH(¯z) such that

If a strong sufficient optimality condition of second order together with the Mangasarian-Fromowitz and the constant rank constraint qualifications are satisfied for a parametric optimization problem, then a local optimal solution is strongly stable in the sense of Kojima and the corresponding optimal solution function is locally Lipschitz continuous. In the article the possibilities for the computation of subgradients of this function are discussed. We will give formulae for the guaranteed computation of the entire subdifferential, provided that an additional assumption is satisfied. An example will show the necessity of this assumption. Moreover, this assumption is difficult to be verified. Without it, a subgradient can be computed with non-polynomial complexity in the worst case. A last approach yields a subgradient with probability one in polynomial time. 1

We consider a mathematical program with complementarity constraints (MPCC). Our purpose is to develop methods that enable us to compute a solution or a point with some kind of stationarity to MPCC by solving a finite number of nonlinear programs. To this end, we first introduce an active set identification technique. Then, by applying this technique to a smoothing continuation method presented by Fukushima and Pang (1999), we propose a hybrid method for solving MPCC. Under reasonable assumptions, the hybrid algorithm is shown to possess a finite termination property. Numerical experience shows that the proposed approach is quite e#ective.

The present paper serves to introduce a modelling technique and solution methodology for a robust and cost-optimizing approach to structural optimization. The structural design that responds the best, on average, to a given set of loads is obtained, where each load has its own probability of occurrence. This model is of special interest when a structural failure will lead to a reconstruction cost, rather than loss of life. The stochastic model is presented, along with its mathematical properties, generalizing some recently published results. A heuristic algorithm for solving the problem is presented, along with an e ective parallelization strategy. Numerical experiments are provided. 1

this paper we continue our line of investigation of augmented Lagrangian techniques as an efficient tool for both analysis and numerical treatment of optimization problems [IK1, IK2, IK3]. Here we focus an optimal control problems of the form (P)

The bilevel programming problem (BLPP) is equivalent to a two-person Stackelberg game in which the leader and follower pursue individual objectives. Play is sequential and the choices of one affect the choices and attainable payoffs of the other. The purpose of this paper is to investigate an extension of the linear BLPP where both players' objective functions are bilinear. To overcome certain discontinuities in the master problem, a regularized term is added to the follower's objective function. Using ideas from parametric programming, the directional derivatives of the regularized follower's solution function are computed along with its generalized Jacobian. This allows us to develop a bundle trust region algorithm. Theoretical results related to the existence of solutions are presented as well as a convergence analysis of the proposed methodology. Key words: bilevel programming, bundle algorithm, Lipschitz continuity, generalized gradients, nondifferentiable optimization. 1 Freiberg University of Mining and Technology, Germany, dempe@math.tu-freiberg.de 2 Graduate Program in Operations Research, University of Texas, Austin, U.S.A., jbard@mail.utexas.edu 1

A new smoothing approach based on entropic perturbation is proposed for solving mathematical programs with equilibrium constraints. Some of the desirable properties of the smoothing function are shown. The viability of the proposed approach is supported by a computational study on a set of well-known test problems.

Technical note: On the halfplane and cone algorithms for bilevel programming problems by Clegg and Smith