On Vector Variational Inequalities: Application to Vector Equilibria

We motivate the study of a vector variational inequality by a practical flow equilibrium problem on a network, namely a generalization of the well-known Wardrop equilibrium principle. Both weak and strong forms of the vector variational inequality are discussed and their relationships to a vector optimization problem are established under various convexity assumptions.  

Vector network equilibrium problems and nonlinear scalarization methods

The conventional equilibrium problem found in many economics and network models is based on a scalar cost, or a single objective. Recently, equilibrium problems based on a vector cost, or multicriteria, have received considerable attention. In this paper, we study a scalarization method for analyzing network equilibrium problems with vector-valued cost function. The method is based on a strictly monotone function originally proposed by Gerstewitz. Conditions that are both necessary and sufficient for weak vector equilibrium are derived, with the prominent feature that no convexity assumptions are needed, in contrast to other existing scalarization methods.  

Viscous flow through corrugated tube by boundary element method

Summary The creeping flow of a Newtonian fluid through a sinusoidally-corrugated tube is solved by the Boundary Element Method. Agreement with another numerical method is noted. In addition, it is shown that previous perturbation theory is valid only when the corrugation amplitude is small (<0.3a) and the wavelength of the corrugation is large (>3a), wherea is the mean radius of the tube.

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Zusammenfassung Das Problem der schleichenden Bewegung eines Newton'schen Fluids durch ein Rohr mit sinusförmig gewellter Wand wird mit Hilfe der Boundary Element-Methode gelöst. Übereinstimmung mit einer anderen numerischen Methode wird festestellt. Zudem wird gezeigt, daß eine früher gefundene Störungstheorie nur gültig ist wenn die Wellenamplitude klein (<0.3a) und die Wellenlänge groß (>3a) ist (a=mittlerer Rohrradius).

  

Nonlinear Lagrangian Theory for Nonconvex Optimization

The Lagrangian function in the conventional theory for solving constrained optimization problems is a linear combination of the cost and constraint functions. Typically, the optimality conditions based on linear Lagrangian theory are either necessary or sufficient, but not both unless the underlying cost and constraint functions are also convex.We propose a somewhat different approach for solving a nonconvex inequality constrained optimization problem based on a nonlinear Lagrangian function. This leads to optimality conditions which are both sufficient and necessary, without any convexity assumption. Subsequently, under appropriate assumptions, the optimality conditions derived from the new nonlinear Lagrangian approach are used to obtain an equivalent root-finding problem. By appropriately defining a dual optimization problem and an alternative dual problem, we show that zero duality gap will hold always regardless of convexity, contrary to the case of linear Lagrangian duality.  

Convexification of a Noninferior Frontier

In a recent paper by Li (Ref. 1), a scheme was proposed to convexify an efficient frontier for a vector optimization problem by rescaling each component of the vector objective functions by its p-power. For sufficiently large p, it was shown that the transformed efficient frontier is cone-convex; hence, the usual linear scalarization (or supporting hyperplane) method can be used to find the efficient solutions. An outstanding question remains: What is the minimum value of p such that the efficient frontier can be convexified? In this note, we answer the above question by deriving some theoretical lower bounds for p.  

On the intersection of two particular convex sets

In this note, necessary and sufficient conditions are given for the intersection of them−1 simplex co {ξ¹,...,ξ ^m } ofm affinely independent vectors ξ¹,...,ξ ^m of ℝ ⁿ and the negative orthant ℝ ₋ ⁿ to be empty, i.e., $$co\{ \xi ^1 ,...,\xi ^m \} \cap \mathbb{R}_ - ^n = \emptyset ,$$ wherem≤n. It is also shown that the special casem=2 can be checked easily. These results suggest that the above-mentioned emptiness can be checked recursively. Some numerical examples are given to illustrate the results. Potential applications of these results include the compatible multicommodity flow problems and satisficing solution problems.  

Direct training method for a continuous-time nonlinear optimal feedback controller

The solutions of most nonlinear optimal control problems are given in the form of open-loop optimal control which is computed from a given fixed initial condition. Optimal feedback control can in principle be obtained by solving the corresponding Hamilton-Jacobi-Bellman dynamic programming equation, though in general this is a difficult task. We propose a practical and effective alternative for constructing an approximate optimal feedback controller in the form of a feedforward neural network, and we justify this choice by several reasons. The controller is capable of approximately minimizing an arbitrary performance index for a nonlinear dynamical system for initial conditions arising from a nontrivial bounded subset of the state space. A direct training algorithm is proposed and several illustrative examples are given.This research was carried out with the support of a grant from the Australian Research Council.We thank the anonymous reviewers for their helpful comments.  

A computational method for a class of optimal relaxed control problems

In the present paper, we propose a computational scheme for solving a class of optimal relaxed control problems, using the concept of control parametrization. Furthermore, some important convergence properties of the proposed computational scheme are investigated. For illustration, a numerical example is also included.  

On minimax eigenvalue problems via constrained optimization

In this paper, we consider the problem of minimizing the maximum eigenvalues of a matrix. The aim is to show that this optimization problem can be transformed into a standard nonlinearly constrained optimization problem, and hence is solvable by existing software packages. For illustration, two examples are solved by using the proposed method.  

On constrained optimization problems with nonsmooth cost functionals

In this paper we present a computational procedure for minimizing a class ofL ₁-functionals subject to conventional as well as functional constraints. The computational procedure is based on the idea of enforced smoothing together with a method of converting the functional constraints into conventional equality constraints. For illustration, two examples are solved using the proposed procedure.